This document provides definitions and examples of different types of matrices including: real matrix, square matrix, row matrix, column matrix, null matrix, sub-matrix, diagonal matrix, scalar matrix, unit matrix, upper triangular matrix, lower triangular matrix, triangular matrix, single element matrix, equal matrices, singular and non-singular matrices. It also discusses elementary row and column transformations, rank of a matrix, solutions to homogeneous and non-homogeneous systems of linear equations, characteristic equations, eigenvectors and eigenvalues.
It contains the basics of matrix which includes matrix definition,types of matrices,operations on matrices,transpose of matrix,symmetric and skew symmetric matrix,invertible matrix,
application of matrix.
Matrices can be added, subtracted, and multiplied according to certain rules.
- Matrices can only be added or subtracted if they are the same size. The sum or difference of matrices A and B yields a matrix C of the same size.
- Matrices can be multiplied by a scalar. Multiplying a matrix A by a scalar k results in a new matrix kA where each element is multiplied by k.
- Matrix multiplication allows combining information from two matrices but has specific rules regarding the dimensions of the matrices.
The document defines and provides examples of different types of matrices, including:
- Square matrices, where the number of rows equals the number of columns.
- Rectangular matrices, where the number of rows does not equal the number of columns.
- Row matrices, with only one row.
- Column matrices, with only one column.
- Null or zero matrices, with all elements equal to zero.
- Diagonal matrices, with all elements equal to zero except those on the main diagonal.
The document also discusses transpose, adjoint, and addition of matrices.
Matrices can be added, subtracted, and multiplied under certain conditions.
Addition and subtraction require matrices to be the same size.
Matrix multiplication requires the number of columns of the first matrix to equal the number of rows of the second matrix.
Matrices can also be multiplied by scalars.
Here are the key steps to find the eigenvalues of the given matrix:
1) Write the characteristic equation: det(A - λI) = 0
2) Expand the determinant: (1-λ)(-2-λ) - 4 = 0
3) Simplify and factor: λ(λ + 1)(λ + 2) = 0
4) Find the roots: λ1 = 0, λ2 = -1, λ3 = -2
Therefore, the eigenvalues of the given matrix are -1 and -2.
The document defines determinants as values that can be computed from the elements of a square matrix. Determinants are used throughout mathematics, including in solving systems of linear equations, change of variables rules for integrals, eigenvalue problems, and expressing volumes of parallelepipeds. The determinant of a matrix product equals the product of the determinants, showing that the determinant is a multiplicative map. A matrix is invertible if and only if its determinant is non-zero.
The document defines the limit of a function and how to determine if the limit exists at a given point. It provides an intuitive definition, then a more precise epsilon-delta definition. Examples are worked through to show how to use the definition to prove limits, including finding appropriate delta values given an epsilon and showing a function satisfies the definition.
It contains the basics of matrix which includes matrix definition,types of matrices,operations on matrices,transpose of matrix,symmetric and skew symmetric matrix,invertible matrix,
application of matrix.
Matrices can be added, subtracted, and multiplied according to certain rules.
- Matrices can only be added or subtracted if they are the same size. The sum or difference of matrices A and B yields a matrix C of the same size.
- Matrices can be multiplied by a scalar. Multiplying a matrix A by a scalar k results in a new matrix kA where each element is multiplied by k.
- Matrix multiplication allows combining information from two matrices but has specific rules regarding the dimensions of the matrices.
The document defines and provides examples of different types of matrices, including:
- Square matrices, where the number of rows equals the number of columns.
- Rectangular matrices, where the number of rows does not equal the number of columns.
- Row matrices, with only one row.
- Column matrices, with only one column.
- Null or zero matrices, with all elements equal to zero.
- Diagonal matrices, with all elements equal to zero except those on the main diagonal.
The document also discusses transpose, adjoint, and addition of matrices.
Matrices can be added, subtracted, and multiplied under certain conditions.
Addition and subtraction require matrices to be the same size.
Matrix multiplication requires the number of columns of the first matrix to equal the number of rows of the second matrix.
Matrices can also be multiplied by scalars.
Here are the key steps to find the eigenvalues of the given matrix:
1) Write the characteristic equation: det(A - λI) = 0
2) Expand the determinant: (1-λ)(-2-λ) - 4 = 0
3) Simplify and factor: λ(λ + 1)(λ + 2) = 0
4) Find the roots: λ1 = 0, λ2 = -1, λ3 = -2
Therefore, the eigenvalues of the given matrix are -1 and -2.
The document defines determinants as values that can be computed from the elements of a square matrix. Determinants are used throughout mathematics, including in solving systems of linear equations, change of variables rules for integrals, eigenvalue problems, and expressing volumes of parallelepipeds. The determinant of a matrix product equals the product of the determinants, showing that the determinant is a multiplicative map. A matrix is invertible if and only if its determinant is non-zero.
The document defines the limit of a function and how to determine if the limit exists at a given point. It provides an intuitive definition, then a more precise epsilon-delta definition. Examples are worked through to show how to use the definition to prove limits, including finding appropriate delta values given an epsilon and showing a function satisfies the definition.
This presentation describes Matrices and Determinants in detail including all the relevant definitions with examples, various concepts and the practice problems.
This document discusses linear independence, basis, and dimension in linear algebra. It defines linear independence as vectors being linearly independent if the only solution that produces the zero vector is the trivial solution with all coefficients equal to zero. A basis is defined as a set of linearly independent vectors that span the vector space. The dimension of a vector space is the number of vectors in any basis of that space. The dimensions of the four fundamental subspaces (row space, column space, nullspace, and left nullspace) of a matrix are defined in terms of the rank of the matrix.
This document provides an overview of matrices including:
- How to describe matrices using m rows and n columns
- Common types of matrices such as row, column, zero, square, diagonal, and unit matrices
- Basic matrix operations including addition, subtraction, scalar multiplication
- Rules for matrix multiplication including that matrices must be conformable
- The transpose of a matrix which is obtained by interchanging rows and columns
- Properties of transposed matrices including (A+B)T = AT + BT and (AB)T = BTAT
This document provides an overview of matrix algebra concepts for business students. It defines key terms like matrix, order, types of matrices including identity, diagonal and triangular matrices, and matrix operations such as addition, subtraction and multiplication. It also explains determinants, which evaluate whether a system of linear equations has a unique solution. Determinants are calculated by taking the difference of products of diagonal elements of a square matrix. This document serves as a basic introduction and recap of matrix algebra.
This document contains a presentation on vector analysis and matrices submitted by mechanical engineering students at Sonargaon University. It includes definitions of vectors, types of vectors, vector operations of addition, subtraction, dot product and cross product. It also defines different types of matrices, matrix operations of addition and subtraction, and scalar multiplication. Applications of vectors and matrices are discussed for calculating forces, velocities, and in cryptography to encrypt data for privacy.
This document provides information about determinants of square matrices:
- It defines the determinant of a matrix as a scalar value associated with the matrix. Determinants are computed using minors and cofactors.
- Properties of determinants are described, such as how determinants change with row/column operations or identical rows/columns.
- Examples are provided to demonstrate computing determinants by expanding along rows or columns and using cofactors and minors.
- Applications of determinants include finding the area of triangles and solving systems of linear equations.
The document discusses matrices and their operations. It defines what a matrix is, provides examples of different types of matrices, and covers key matrix operations like addition, subtraction, scalar multiplication, and matrix multiplication. It also defines important matrix concepts such as the transpose of a matrix, inverse of a matrix, and properties related to these operations and concepts.
The document defines complex numbers and their properties. It states that a complex number has the form x + iy, where x and y are real numbers and i = √-1. Complex numbers can be represented in rectangular form as x + iy or in polar form as r(cosθ + i sinθ), where r is the modulus or absolute value and θ is the argument. The document also defines conjugate complex numbers and describes how to calculate the sum, difference, and product of two complex numbers.
The document defines and provides examples of different types of sets:
1. Empty/null sets contain no elements. Singleton sets contain only one element. Finite sets contain a finite number of elements, while infinite sets are not finite.
2. Subsets contain elements of another set. Proper subsets are subsets that are not equal to the original set. Power sets are the set of all subsets of a given set.
3. Examples are given of empty, singleton, finite, infinite, equivalent, equal, subset, and proper subset sets. Cardinal numbers represent the number of elements in a set.
This document discusses basic matrix operations including:
- Defining a matrix as a rectangular arrangement of numbers in rows and columns with an order specified by the number of rows and columns.
- Adding and subtracting matrices requires they have the same order and involves adding or subtracting corresponding entries.
- Multiplying a matrix by a scalar involves multiplying each entry in the matrix by the scalar value.
- Matrix multiplication is not commutative and can only be done if the number of columns in the first matrix equals the number of rows in the second matrix. It involves multiplying entries and summing the products based on their positions.
This document provides information about sequences and series in mathematics. It defines sequences, limits of sequences, convergence and divergence of sequences, infinite series, tests to determine convergence of series like the divergence test, limit comparison test, ratio test, root test, and power series. Examples of applying these concepts to specific series are also included.
The document defines row echelon form and reduced row echelon form for matrices. Row echelon form requires that leading 1's occur farther to the right in lower rows. Reduced row echelon form further requires that all entries above leading 1's are zero. The document also discusses Gauss elimination method and elementary row operations for transforming a matrix into row echelon or reduced row echelon form.
The document defines basic concepts about sets including:
- A set is a collection of distinct objects called elements. Sets can be represented using curly brackets or the set builder method.
- Common set symbols are defined such as belongs to (∈), is a subset of (⊆), and is not a subset of (⊄).
- Types of sets like empty sets, singleton sets, finite sets, and infinite sets are described.
- Operations between sets such as union, intersection, difference, and complement are explained using Venn diagrams.
- Laws for sets like commutative, associative, distributive, double complement, and De Morgan's laws are listed.
- An example problem calculates
This document discusses operational research and assignment techniques. It begins by defining operational research as a scientific approach to problem solving aimed at reducing costs. It then provides examples of how operational research is used in various sectors like transportation, healthcare, and banking. The document outlines the characteristics and limitations of operational research. It proceeds to define assignment techniques as a way to allocate jobs and resources in the lowest cost manner. The remainder of the document details the specific steps involved in solving an assignment problem using a matrix-based approach to find an optimal solution.
Lesson 7-8: Derivatives and Rates of Change, The Derivative as a functionMatthew Leingang
The derivative is one of the fundamental quantities in calculus, partly because it is ubiquitous in nature. We give examples of it coming about, a few calculations, and ways information about the function an imply information about the derivative
The document contains a list of 6 group members with their names and student identification numbers. The group members are:
1. Ridwan bin shamsudin, student ID: D20101037472
2. Mohd. Hafiz bin Salleh, student ID: D20101037433
3. Muhammad Shamim Bin Zulkefli, student ID: D20101037460
4. Jasman bin Ronie, student ID: D20101037474
5. Hairieyl Azieyman Bin Azmi, student ID: D20101037426
6. Mustaqim Bin Musa, student ID:
This document discusses two methods for solving systems of linear equations:
1) The inverse matrix method, which involves writing the system in matrix form AX = B and solving for X by computing A-1B, where A-1 is the inverse of the coefficient matrix A.
2) Cramer's rule, which uses determinants to find the values of the variables by taking ratios of determinants formed from the coefficients and constants. The method works by solving one variable at a time.
This document provides a lesson on matrix inverses and solving systems of equations using inverse matrices. It begins with examples of determining whether two matrices are inverses of each other and finding the inverse of a given matrix. It then explains how to use inverse matrices to solve systems of equations by writing the system as a matrix equation and multiplying both sides by the inverse. Examples are provided to demonstrate solving systems using inverse matrices and decoding encoded messages using a given encoding matrix.
This document discusses matrices and their properties. It begins by defining a matrix as a rectangular array of numbers or functions. It then describes 14 different types of matrices including real, square, row, column, null, sub, diagonal, scalar, unit, upper/lower triangular, and singular/non-singular matrices. It also covers elementary row and column transformations, the rank of a matrix, the consistency of linear systems of equations, and the characteristic equation.
This document discusses matrices and their properties. It begins by defining a matrix as a rectangular array of numbers or functions. It then describes 14 different types of matrices including real, square, row, column, null, sub, diagonal, scalar, unit, upper/lower triangular, and singular/non-singular matrices. It also covers elementary row and column transformations, the rank of a matrix, the consistency of linear systems of equations, and the characteristic equation.
This presentation describes Matrices and Determinants in detail including all the relevant definitions with examples, various concepts and the practice problems.
This document discusses linear independence, basis, and dimension in linear algebra. It defines linear independence as vectors being linearly independent if the only solution that produces the zero vector is the trivial solution with all coefficients equal to zero. A basis is defined as a set of linearly independent vectors that span the vector space. The dimension of a vector space is the number of vectors in any basis of that space. The dimensions of the four fundamental subspaces (row space, column space, nullspace, and left nullspace) of a matrix are defined in terms of the rank of the matrix.
This document provides an overview of matrices including:
- How to describe matrices using m rows and n columns
- Common types of matrices such as row, column, zero, square, diagonal, and unit matrices
- Basic matrix operations including addition, subtraction, scalar multiplication
- Rules for matrix multiplication including that matrices must be conformable
- The transpose of a matrix which is obtained by interchanging rows and columns
- Properties of transposed matrices including (A+B)T = AT + BT and (AB)T = BTAT
This document provides an overview of matrix algebra concepts for business students. It defines key terms like matrix, order, types of matrices including identity, diagonal and triangular matrices, and matrix operations such as addition, subtraction and multiplication. It also explains determinants, which evaluate whether a system of linear equations has a unique solution. Determinants are calculated by taking the difference of products of diagonal elements of a square matrix. This document serves as a basic introduction and recap of matrix algebra.
This document contains a presentation on vector analysis and matrices submitted by mechanical engineering students at Sonargaon University. It includes definitions of vectors, types of vectors, vector operations of addition, subtraction, dot product and cross product. It also defines different types of matrices, matrix operations of addition and subtraction, and scalar multiplication. Applications of vectors and matrices are discussed for calculating forces, velocities, and in cryptography to encrypt data for privacy.
This document provides information about determinants of square matrices:
- It defines the determinant of a matrix as a scalar value associated with the matrix. Determinants are computed using minors and cofactors.
- Properties of determinants are described, such as how determinants change with row/column operations or identical rows/columns.
- Examples are provided to demonstrate computing determinants by expanding along rows or columns and using cofactors and minors.
- Applications of determinants include finding the area of triangles and solving systems of linear equations.
The document discusses matrices and their operations. It defines what a matrix is, provides examples of different types of matrices, and covers key matrix operations like addition, subtraction, scalar multiplication, and matrix multiplication. It also defines important matrix concepts such as the transpose of a matrix, inverse of a matrix, and properties related to these operations and concepts.
The document defines complex numbers and their properties. It states that a complex number has the form x + iy, where x and y are real numbers and i = √-1. Complex numbers can be represented in rectangular form as x + iy or in polar form as r(cosθ + i sinθ), where r is the modulus or absolute value and θ is the argument. The document also defines conjugate complex numbers and describes how to calculate the sum, difference, and product of two complex numbers.
The document defines and provides examples of different types of sets:
1. Empty/null sets contain no elements. Singleton sets contain only one element. Finite sets contain a finite number of elements, while infinite sets are not finite.
2. Subsets contain elements of another set. Proper subsets are subsets that are not equal to the original set. Power sets are the set of all subsets of a given set.
3. Examples are given of empty, singleton, finite, infinite, equivalent, equal, subset, and proper subset sets. Cardinal numbers represent the number of elements in a set.
This document discusses basic matrix operations including:
- Defining a matrix as a rectangular arrangement of numbers in rows and columns with an order specified by the number of rows and columns.
- Adding and subtracting matrices requires they have the same order and involves adding or subtracting corresponding entries.
- Multiplying a matrix by a scalar involves multiplying each entry in the matrix by the scalar value.
- Matrix multiplication is not commutative and can only be done if the number of columns in the first matrix equals the number of rows in the second matrix. It involves multiplying entries and summing the products based on their positions.
This document provides information about sequences and series in mathematics. It defines sequences, limits of sequences, convergence and divergence of sequences, infinite series, tests to determine convergence of series like the divergence test, limit comparison test, ratio test, root test, and power series. Examples of applying these concepts to specific series are also included.
The document defines row echelon form and reduced row echelon form for matrices. Row echelon form requires that leading 1's occur farther to the right in lower rows. Reduced row echelon form further requires that all entries above leading 1's are zero. The document also discusses Gauss elimination method and elementary row operations for transforming a matrix into row echelon or reduced row echelon form.
The document defines basic concepts about sets including:
- A set is a collection of distinct objects called elements. Sets can be represented using curly brackets or the set builder method.
- Common set symbols are defined such as belongs to (∈), is a subset of (⊆), and is not a subset of (⊄).
- Types of sets like empty sets, singleton sets, finite sets, and infinite sets are described.
- Operations between sets such as union, intersection, difference, and complement are explained using Venn diagrams.
- Laws for sets like commutative, associative, distributive, double complement, and De Morgan's laws are listed.
- An example problem calculates
This document discusses operational research and assignment techniques. It begins by defining operational research as a scientific approach to problem solving aimed at reducing costs. It then provides examples of how operational research is used in various sectors like transportation, healthcare, and banking. The document outlines the characteristics and limitations of operational research. It proceeds to define assignment techniques as a way to allocate jobs and resources in the lowest cost manner. The remainder of the document details the specific steps involved in solving an assignment problem using a matrix-based approach to find an optimal solution.
Lesson 7-8: Derivatives and Rates of Change, The Derivative as a functionMatthew Leingang
The derivative is one of the fundamental quantities in calculus, partly because it is ubiquitous in nature. We give examples of it coming about, a few calculations, and ways information about the function an imply information about the derivative
The document contains a list of 6 group members with their names and student identification numbers. The group members are:
1. Ridwan bin shamsudin, student ID: D20101037472
2. Mohd. Hafiz bin Salleh, student ID: D20101037433
3. Muhammad Shamim Bin Zulkefli, student ID: D20101037460
4. Jasman bin Ronie, student ID: D20101037474
5. Hairieyl Azieyman Bin Azmi, student ID: D20101037426
6. Mustaqim Bin Musa, student ID:
This document discusses two methods for solving systems of linear equations:
1) The inverse matrix method, which involves writing the system in matrix form AX = B and solving for X by computing A-1B, where A-1 is the inverse of the coefficient matrix A.
2) Cramer's rule, which uses determinants to find the values of the variables by taking ratios of determinants formed from the coefficients and constants. The method works by solving one variable at a time.
This document provides a lesson on matrix inverses and solving systems of equations using inverse matrices. It begins with examples of determining whether two matrices are inverses of each other and finding the inverse of a given matrix. It then explains how to use inverse matrices to solve systems of equations by writing the system as a matrix equation and multiplying both sides by the inverse. Examples are provided to demonstrate solving systems using inverse matrices and decoding encoded messages using a given encoding matrix.
This document discusses matrices and their properties. It begins by defining a matrix as a rectangular array of numbers or functions. It then describes 14 different types of matrices including real, square, row, column, null, sub, diagonal, scalar, unit, upper/lower triangular, and singular/non-singular matrices. It also covers elementary row and column transformations, the rank of a matrix, the consistency of linear systems of equations, and the characteristic equation.
This document discusses matrices and their properties. It begins by defining a matrix as a rectangular array of numbers or functions. It then describes 14 different types of matrices including real, square, row, column, null, sub, diagonal, scalar, unit, upper/lower triangular, and singular/non-singular matrices. It also covers elementary row and column transformations, the rank of a matrix, the consistency of linear systems of equations, and the characteristic equation.
This document outlines topics related to matrices, including:
- Types of matrices such as real, square, row, column, null, sub, diagonal, scalar, unit, upper triangular, lower triangular, and singular matrices
- Characteristic equations, eigenvectors, and eigenvalues of matrices
- Properties of eigenvalues including that the sum of eigenvalues is the trace and the product is the determinant
- Examples of finding the sum and product of eigenvalues without directly calculating them
The document provides definitions and examples of key matrix concepts.
This document provides an overview of key topics in mathematical methods including:
- Matrices and linear systems of equations
- Eigenvalues and eigenvectors, real and complex matrices, and quadratic forms
- Algebraic and transcendental equations and interpolation methods
- Curve fitting, numerical differentiation and integration, and numerical solutions to ODEs
- Fourier series, Fourier transforms, and partial differential equations
It also lists several textbooks and references on mathematical methods.
Some types of matrices, Eigen value , Eigen vector, Cayley- Hamilton Theorem & applications, Properties of Eigen values, Orthogonal matrix , Pairwise orthogonal, orthogonal transformation of symmetric matrix, denationalization of a matrix by orthogonal transformation (or) orthogonal deduction, Quadratic form and Canonical form , conversion from Quadratic to Canonical form, Order, Index Signature, Nature of canonical form.
Linear Algebra Presentation including basic of linear AlgebraMUHAMMADUSMAN93058
This document discusses linear algebra concepts including systems of linear equations, matrices, and matrix operations. It covers topics such as matrix addition, subtraction, multiplication, and transposition. Matrix-vector products and partitioned matrices are also explained. Elementary row operations are defined as interchanging rows, multiplying a row by a non-zero number, and adding a multiple of one row to another. The document concludes by defining row reduced echelon form (RREF) and row echelon form (REF) of a matrix.
1) A matrix is a rectangular array of numbers arranged in rows and columns. The dimensions of a matrix are specified by the number of rows and columns.
2) The inverse of a square matrix A exists if and only if the determinant of A is not equal to 0. The inverse of A, denoted A^-1, is the matrix that satisfies AA^-1 = A^-1A = I, where I is the identity matrix.
3) For two matrices A and B to be inverses, their product must result in the identity matrix regardless of order, i.e. AB = BA = I. This shows that one matrix undoes the effect of the other.
Beginning direct3d gameprogrammingmath05_matrices_20160515_jintaeksJinTaek Seo
This document provides an overview of linear systems and matrices. It defines key linear algebra concepts such as linear functions, linear maps, homogeneous and non-homogeneous linear systems, and plane equations. It also explains how to represent linear systems using matrices and describes common matrix operations including addition, scalar multiplication, transposition, and matrix multiplication. Finally, it discusses inverses, determinants, and using matrices to represent transformations such as rotations in 2D space.
This document provides an overview of matrices and determinants. It begins by defining a matrix and listing its key properties. It then describes 9 different types of matrices including square, diagonal, identity, and triangular matrices. The document outlines how to perform addition, subtraction, and multiplication of matrices. It also covers transposing matrices and calculating determinants. Finally, it discusses minors, cofactors, adjoints, inverses, and using Cramer's rule to solve systems of linear equations. Worked examples and practice problems are provided throughout.
The document defines various types of matrices and matrix operations. It discusses the definitions of a matrix, types of matrices including rectangular, column, row, square, diagonal and unit matrices. It also defines transpose, symmetric, skew-symmetric and cofactor matrices. The document provides examples of calculating the minor and cofactor of matrix elements, as well as the adjoint and inverse of matrices using the formula for the inverse of a square matrix in terms of its determinant and adjoint.
This document defines and describes different types of matrices. It begins by defining a matrix as an arrangement of numbers, symbols or expressions in rows and columns. It then discusses the order of a matrix, elements within a matrix, and examples of 3x3 matrices. Several basic types of matrices are defined, including row matrices, column matrices, square matrices, rectangular matrices, diagonal matrices, null matrices, symmetric matrices, and skew-symmetric matrices. Related matrices such as the transpose, adjoint, and inverse of a matrix are also explained. The document concludes by defining the rank of a matrix and describing the properties of an echelon matrix.
This document discusses matrices and determinants. It begins by defining a matrix as a rectangular array of numbers or other objects arranged in rows and columns. It then discusses types of matrices, equality of matrices, and algebraic operations on matrices. The document also covers symmetric and skew-symmetric matrices, determinants of different matrix orders, properties of determinants such as how they change when rows/columns are swapped, and the product of determinants.
This document provides an overview of matrices and matrix operations. It defines what a matrix is and discusses matrix order and elements. It then covers basic matrix operations like scalar multiplication, addition, and multiplication. It introduces the concepts of transpose, special matrices like diagonal and triangular matrices, and the null and identity matrices. The document aims to define fundamental matrix concepts and arithmetic operations.
The document defines matrices and their properties. It discusses the key concepts of matrices including:
- Types of matrices such as row, column, rectangular, square, null, diagonal, and identity matrices.
- Properties of matrices including order, rows, columns, equality, transpose, addition, multiplication, and inverse.
- Special matrices like symmetric, skew-symmetric, and scalar matrices.
- How to evaluate determinants of square matrices and solve systems of linear equations using matrices.
For a system involving two variables (x and y), each linear equation determines a line on the xy-plane. Because a solution to a linear system must satisfy all of the equations, the solution set is the intersection of these lines, and is hence either a line, a single point, or the empty set
A matrix is a rectangular array of numbers arranged in rows and columns. There are several types of matrices including square, rectangular, diagonal, identity, and triangular matrices. Operations that can be performed on matrices include addition, subtraction, multiplication by a scalar, and determining the transpose, determinant, and inverse of a matrix. A C program is shown that uses nested for loops to input and output the elements of a matrix.
This document discusses methods for finding the rank of a matrix. It begins by introducing the concept of linear independence and dependence of vectors. It then explains that the rank of a matrix is the maximum number of linearly independent columns. Two methods are described for determining the rank: using the determinant, and reducing the matrix to row echelon form. An example applying each method is provided. The document concludes by thanking the audience.
Vector integrals generalize integration to vector-valued functions. Line integrals measure the accumulated effect of a vector field along a curve, surface integrals measure flux through a surface, and volume integrals measure the total effect within a region. Vector integrals are used widely in physics and engineering to calculate quantities such as work, magnetic flux, and fluid flow.
Formal letters have a specific structure and format. They are used for professional communication regarding work concerns, orders, applications and addressing problems. A formal letter includes the sender's address, date, recipient's address, salutation, subject, body organized in paragraphs, complimentary close and signature. The body should be brief, clear, precise and polite. Common complimentary closes are "Yours sincerely" or "Yours faithfully". Formal letters follow a block, modified block or semi-block format to organize the elements in a clear, readable way.
Linear probing is a technique for resolving collisions in hash tables that uses open addressing. When a collision occurs inserting a key-value pair, linear probing searches through the hash table sequentially from the hashed index to find the next empty slot to insert the pair. This provides high performance due to locality of reference. The linear probing hash table stores data directly in the array, handling collisions by probing through subsequent indices in the table until an empty slot is found to insert the new pair. Insertion, search, and deletion operations take constant expected time on average when using a random hash function.
This document discusses solving systems of linear equations using matrix methods. It begins by defining a matrix and explaining how to write the augmented matrix of a system of linear equations. It then describes three row operations that can be performed on matrices: exchanging rows, multiplying a row by a non-zero number, and adding a multiple of one row to another. Examples are provided of using these row operations to solve systems with two and three variables and put the augmented matrix in row echelon form. The document concludes by discussing how to recognize inconsistent or dependent systems.
Doubly linked lists are lists where each node contains a pointer to the next node and previous node. Each node stores data and pointers to the next and previous nodes. Finding a node requires searching the list using the next pointers until the desired node is found or the end is reached. Inserting a node involves finding the correct location and adjusting the next and previous pointers of the neighboring nodes. Headers and trailers can simplify insertion and deletion by avoiding special cases for the first and last nodes. Large integers can be represented using a special linked list implementation that treats each digit as a node.
1. The document discusses three external water softening processes: lime-soda process, zeolite process, and ion exchange process.
2. The lime-soda process uses lime and soda ash to precipitate calcium and magnesium ions from water. The zeolite process uses naturally occurring minerals to exchange sodium ions for calcium and magnesium ions. The ion exchange process uses cation and anion exchange resins to remove ions from water.
3. Each process has advantages like reducing hardness and corrosion but also limitations such as producing waste sludge or requiring pre-treatment of the raw water.
This document describes a chemistry lab experiment to determine the concentration of HCl using conductometric titration with NaOH. The key steps are:
1) 40ml of HCl is titrated with 0.1M NaOH solution while measuring the conductance.
2) The conductance is plotted against the volume of NaOH added.
3) The neutralization point is determined from the graph by extrapolating the straight lines.
4) Using the titration data and volume of HCl, the molarity of HCl is calculated to be 0.015M.
This document provides an introduction and overview of biodiesel. It discusses what biodiesel is, how it is made from vegetable and animal fats, its fuel properties, issues related to vehicle operation and engine impacts, current fuel costs, distribution challenges, existing policies and programs supporting biodiesel, and areas for future attention and research. The purpose is to inform a technical subcommittee about biodiesel and address increasing interest in its use.
- Engineering curves include conic sections such as ellipses, parabolas, and hyperbolas which are obtained by cutting a right circular cone in different ways.
- An ellipse is obtained when the cutting plane is inclined to the axis and cuts all generators, producing a closed curve. A parabola results when the cutting plane is parallel to one generator. A hyperbola occurs when the cutting plane is inclined to one side of the axis.
- Conic sections can be defined using a focus and directrix, with eccentricity determining whether the curve is elliptical (e < 1), parabolic (e = 1), or hyperbolic (e > 1).
1. Fuels are combustible substances containing mainly carbon that produce heat energy when burned. Coal, petroleum, and natural gas are important primary fossil fuels found naturally.
2. Fuels can be classified based on their physical state as solid, liquid, or gaseous. They can also be classified as primary fuels found in nature or secondary fuels derived from primary fuels.
3. The calorific value of a fuel is the amount of heat released during complete combustion. It is usually measured in kilocalories per kilogram. Higher calorific value includes the heat of condensation of water vapor produced, while lower calorific value does not.
This document summarizes Kirchoff's laws, which are two circuit analysis laws developed by Gustav Kirchoff. Kirchoff's Current Law (KCL) states that the algebraic sum of currents entering a node is zero. Kirchoff's Voltage Law (KVL) states that the algebraic sum of the voltages around any closed loop in a circuit must be zero. The document provides examples of applying KCL and KVL to solve for currents in circuits containing meshes and nodes. It determines the current through an 8 ohm resistor by setting up KCL and KVL equations and solving the system of equations.
KuberTENes Birthday Bash Guadalajara - K8sGPT first impressionsVictor Morales
K8sGPT is a tool that analyzes and diagnoses Kubernetes clusters. This presentation was used to share the requirements and dependencies to deploy K8sGPT in a local environment.
Low power architecture of logic gates using adiabatic techniquesnooriasukmaningtyas
The growing significance of portable systems to limit power consumption in ultra-large-scale-integration chips of very high density, has recently led to rapid and inventive progresses in low-power design. The most effective technique is adiabatic logic circuit design in energy-efficient hardware. This paper presents two adiabatic approaches for the design of low power circuits, modified positive feedback adiabatic logic (modified PFAL) and the other is direct current diode based positive feedback adiabatic logic (DC-DB PFAL). Logic gates are the preliminary components in any digital circuit design. By improving the performance of basic gates, one can improvise the whole system performance. In this paper proposed circuit design of the low power architecture of OR/NOR, AND/NAND, and XOR/XNOR gates are presented using the said approaches and their results are analyzed for powerdissipation, delay, power-delay-product and rise time and compared with the other adiabatic techniques along with the conventional complementary metal oxide semiconductor (CMOS) designs reported in the literature. It has been found that the designs with DC-DB PFAL technique outperform with the percentage improvement of 65% for NOR gate and 7% for NAND gate and 34% for XNOR gate over the modified PFAL techniques at 10 MHz respectively.
CHINA’S GEO-ECONOMIC OUTREACH IN CENTRAL ASIAN COUNTRIES AND FUTURE PROSPECTjpsjournal1
The rivalry between prominent international actors for dominance over Central Asia's hydrocarbon
reserves and the ancient silk trade route, along with China's diplomatic endeavours in the area, has been
referred to as the "New Great Game." This research centres on the power struggle, considering
geopolitical, geostrategic, and geoeconomic variables. Topics including trade, political hegemony, oil
politics, and conventional and nontraditional security are all explored and explained by the researcher.
Using Mackinder's Heartland, Spykman Rimland, and Hegemonic Stability theories, examines China's role
in Central Asia. This study adheres to the empirical epistemological method and has taken care of
objectivity. This study analyze primary and secondary research documents critically to elaborate role of
china’s geo economic outreach in central Asian countries and its future prospect. China is thriving in trade,
pipeline politics, and winning states, according to this study, thanks to important instruments like the
Shanghai Cooperation Organisation and the Belt and Road Economic Initiative. According to this study,
China is seeing significant success in commerce, pipeline politics, and gaining influence on other
governments. This success may be attributed to the effective utilisation of key tools such as the Shanghai
Cooperation Organisation and the Belt and Road Economic Initiative.
DEEP LEARNING FOR SMART GRID INTRUSION DETECTION: A HYBRID CNN-LSTM-BASED MODELgerogepatton
As digital technology becomes more deeply embedded in power systems, protecting the communication
networks of Smart Grids (SG) has emerged as a critical concern. Distributed Network Protocol 3 (DNP3)
represents a multi-tiered application layer protocol extensively utilized in Supervisory Control and Data
Acquisition (SCADA)-based smart grids to facilitate real-time data gathering and control functionalities.
Robust Intrusion Detection Systems (IDS) are necessary for early threat detection and mitigation because
of the interconnection of these networks, which makes them vulnerable to a variety of cyberattacks. To
solve this issue, this paper develops a hybrid Deep Learning (DL) model specifically designed for intrusion
detection in smart grids. The proposed approach is a combination of the Convolutional Neural Network
(CNN) and the Long-Short-Term Memory algorithms (LSTM). We employed a recent intrusion detection
dataset (DNP3), which focuses on unauthorized commands and Denial of Service (DoS) cyberattacks, to
train and test our model. The results of our experiments show that our CNN-LSTM method is much better
at finding smart grid intrusions than other deep learning algorithms used for classification. In addition,
our proposed approach improves accuracy, precision, recall, and F1 score, achieving a high detection
accuracy rate of 99.50%.
A review on techniques and modelling methodologies used for checking electrom...nooriasukmaningtyas
The proper function of the integrated circuit (IC) in an inhibiting electromagnetic environment has always been a serious concern throughout the decades of revolution in the world of electronics, from disjunct devices to today’s integrated circuit technology, where billions of transistors are combined on a single chip. The automotive industry and smart vehicles in particular, are confronting design issues such as being prone to electromagnetic interference (EMI). Electronic control devices calculate incorrect outputs because of EMI and sensors give misleading values which can prove fatal in case of automotives. In this paper, the authors have non exhaustively tried to review research work concerned with the investigation of EMI in ICs and prediction of this EMI using various modelling methodologies and measurement setups.
Introduction- e - waste – definition - sources of e-waste– hazardous substances in e-waste - effects of e-waste on environment and human health- need for e-waste management– e-waste handling rules - waste minimization techniques for managing e-waste – recycling of e-waste - disposal treatment methods of e- waste – mechanism of extraction of precious metal from leaching solution-global Scenario of E-waste – E-waste in India- case studies.
5. An m n matrix is usually written as
A matrix is usually denoted by a single capital letter A, B or C
etc.
11 12 13 1
21 22 23 2
31 32 33 3
1 2 3
....
....
....
.... .... .... .... ....
....
n
n
n
m m m mn
a a a a
a a a a
a a a a
a a a a
6. Thus, an m n matrix A may be written as
A = , where i = 1, 2, 3, … , m ;
j = 1, 2, 3, … , n
In Algebra, the matrices have their largest
application in the theory of simultaneous
equations and linear transformations.
6
ij
a
8. (2) Square Matrix:- A matrix in which the number
of rows is equal to the number of columns is called
a square matrix, otherwise, it is said to be
rectangular matrix.
i.e., a matrix A = is a square matrix if m = n
a rectangular matrix if m n
ij m n
a
9. A square matrix having n rows and n columns is
called “ n – rowed square matrix”,
is a 3 – rowed square matrix
The elements of a square matrix are
called its diagonal elements and the diagonal
along with these elements are called principal or
leading diagonal.
11 12 13
21 22 23
31 32 33
a a a
a a a
a a a
33
,
22
11 a
a
,
a
10. The sum of the diagonal elements of a
square matrix is called its trace or spur.
Thus, trace of the n rowed square matrix
A= is
ij
a
n
1
i
ij
nn
33
22
11 a
a
....
a
a
a
11. (3) Row Matrix :-
A matrix having only one row and any
number of columns,
i.e., a matrix of order 1 n is called a row
matrix.
[2 5 -3 0] is a row matrix.
Example:-
12. (4) Column Matrix:-
A matrix having only one column
and any number of rows,
i.e., a matrix of order m 1 is called a column
matrix.
is a column matrix.
1
0
2
Example:-
13. (5) Null Matrix:-
A matrix in which each element is zero
is called a null matrix or void matrix or zero matrix.
A null matrix of order m n is denoted by
=
13
n
m
O
Example :-
0
0
0
0
0
0
0
0
4
2
O
14. (6) Sub - matrix :-
A matrix obtained from a given matrix A by
deleting some of its rows or columns or both is
called a sub – matrix of A.
is a sub – matrix of
4
1
0
3
2
4
6
1
7
0
5
3
5
2
1
0
Example:-
15. (7) Diagonal Matrix :-
A square matrix in which all non – diagonal
elements are zero is called a diagonal matrix.
i.e., A = [a ] is a diagonal matrix if a = 0 for i j.
is a diagonal matrix.
ij n
n ij
Example:-
0
0
0
0
1
0
0
0
2
16. (8) Scalar Matrix:-
A diagonal matrix in which all the diagonal
elements are equal to a scalar, say k, is called a
scalar matrix.
i.e., A = [a ] is a scalar matrix if
is a scalar matrix.
2
0
0
0
2
0
0
0
2
ij n
n
j
i
when
k
j
i
when
0
aij
Example :-
17. (9) Unit Matrix or Identity Matrix:-
A scalar matrix in which each diagonal
element is 1 is called a unit or identity matrix. It is
denoted by .
i.e., A = [a ] is a unit matrix if
is a unit matrix.
n
I
ij n
n
j
i
when
1
j
i
when
0
aij
Example
1
0
0
1
18. (10) Upper Triangular Matrix.
A square matrix in which all the elements
below the principal diagonal are zero is called an
upper triangular matrix.
i.e., A = [a ] is an upper triangular matrix if a = 0
for i > j
is an upper triangular
matrix
3
0
0
5
1
0
4
3
2
ij n
n ij
Example:-
19. (11) Lower Triangular Matrix.
A square matrix in which all the elements
above the principal diagonal are zero is called a
lower triangular matrix.
i.e., A = [a ] is a lower triangular matrix if a = 0
for i < j
is a lower triangular
matrix.
ij n
n ij
Example:-
0
2
3
0
6
5
0
0
1
20. (12) Triangular Matrix:-
A triangular matrix is either upper
triangular or lower triangular.
(13) Single Element Matrix:-
A matrix having only one element is
called a single element matrix.
i.e., any matrix [3] is a single element matrix.
21. (14) Equal Matrices:-
Two matrices A and B are said to be equal iff
they have the same order and their corresponding
elements are equal.
i.e., if A = and B = , then A = B
iff a) m = p and n = q
b) a = b for all i and j.
q
p
ij]
b
[
n
m
ij]
a
[
ij ij
22. (15) Singular and Non – Singular Matrices:-
A square matrix A is said to be singular if |A|
= 0 and non – singular if |A| 0.
A = is a singular
matrix since |A| = 0.
1
1
0
4
3
2
4
3
2
Example :-
23. 1.3 ELEMENTARY ROW AND COLUMN
TRANSFORMATION
Some operations on matrices called as elementary
transformations. There are six types of elementary
transformations, three of then are row transformations and
other three of them are column transformations. There are
as follows
1) Interchange of any two rows or columns.
2) Multiplication of the elements of any row (or column)
by a non zero number k.
3) Multiplication to elements of any row or column by a
scalar k and addition of it to the corresponding elements
of any other row or column.
23
24. 1.4 RANK OF MATRIX
A positive integer ‘r’ is said to be the rank of a non- zero
matrix A if
1) There exists at least one non-zero minor of order r of A
and whose determinate is not equal to zero.
2) Every minor of order greater than r of A is zero.
The rank of a matrix A is denoted by ρ(A) .
24
26. 1.5 Consistency of Linear System of
Equation and their solution
Solution of the linear system AX= B
26
n
n
nn
n
n
n
n
n
n
b
x
a
x
a
x
a
b
x
a
x
a
x
a
b
x
a
x
a
x
a
2
2
1
1
2
2
2
22
1
21
1
1
2
12
1
11
n
n
nn
n
n
n
n
b
b
b
x
x
x
a
a
a
a
a
a
a
a
a
2
1
2
1
2
1
2
22
21
1
12
11
If the system has at least one solution then the equations
are said to be consistent otherwise they are said to be
inconsistent.
27. Solve the System
x - 2y + 3z = 2
2x - 3z = 3
x + y +z = 0
Ans x = 21/19
y = -16/19
z = -5/19
In Gauss Method, we reduce the Coefficient matrix to
triangular form by E-row transformations.
27
28. s
28
A system of Non-
Homogeneous Linear Equation
SOLUTION EXIST
SYSTEM IS
CONSISTENT
Rank A=Rank C
NO SOLUTION
SYSTEM IS
INCONSISTENT
Rank A ≠ Rank C
UNIQUE SOLUTION
Rank A=Rank C = n
(n=Number of
unknown variables)
Infinite Number of solution
Rank A= Rank C< n
(n = Number of unknown
variables)
29. Solution of the linear system AX= 0
This system of equation is called homogeneous
equations
AX=0 all ways consistent
2x + y + 3z = 0
x + y + 3z = 0
4x + 3y + 2z = 0
29
30. s
30
A system of Homogeneous
Linear Equation
AX=0
Always has a solution
Unique Or Trivial Solution
Rank A = n (n=Number of
unknown variables)
(Each Unknown equal to
zero)
Infinite Or Non-Trivial
solution
Rank A< n
(n = Number of unknown
variables)
Find R(A)
31. Vectors :
Any ordered n-tuple of numbers is called n-vectors . By an
ordered n-tuple we means a set consisting of n-numbers in
which the place of each number is fixed . If x1, x2, x3, ….. xn
is called n-vector.
Eg. (1 ,2 ,3 ,4) is a 4 vectors
Linearly Dependent :
Vectors (Matrices ) x1, x2, x3, ….. xn are said to be dependent
.if there exist r scalars λ1, λ2, λ3, ….. λn . Not all zero . If the
relation of the type
31
32. λ1 x1+ λ2 x2+ λ3 x3+….. λn xn =0
λ1 ≠ λ2 ≠ λ3 =0
λ4 =λ5=….. λn =0
Linearly Independent :
Vectors (Matrices ) x1, x2, x3, ….. xn are said to be
independent .if there exist r scalars λ1, λ2, λ3, ….. λn . All
are zero . If the relation of the type
λ1 x1+ λ2 x2+ λ3 x3+….. λn xn =0
λ1 = λ2 = λ3 =….. λn = 0
32
33. Linearly dependence and independence of
vectors by Rank Method
1) Rank of MATRIX = Number of vector
(then vector are Linearly Independent)
2) Rank of MATRIX < Number of vector
(then vector are Linearly dependent)
t
33
34. 1.6 CHARACTERISTIC EQUATION
If A is any square matrix of order n, we can
form the matrix , where is the nth order
unit matrix.
The determinant of this matrix equated to zero,
i.e.,
I
λ
A I
0
λ
a
...
a
a
...
...
...
...
a
...
λ
a
a
a
...
a
λ
a
λ
A
nn
n2
n1
2n
22
21
1n
12
11
I
36. Consider the linear transformation Y = AX ...(1)
which transforms the column vector X into the column
vector Y. We often required to find those vectors X
which transform into scalar multiples of themselves.
Let X be such a vector which transforms into
λX by the transformation (1).
1.7 EIGEN VALUES AND EIGEN
VECTORS
37. Then Y = X ... (2)
From (1) and (2), AX = X AX- X = 0
(A - )X = 0 ...(3)
This matrix equation gives n homogeneous linear
equations
... (4)
I
0
λ)x
(a
...
x
a
x
a
....
....
....
....
0
x
a
...
λ)x
(a
x
a
0
x
a
...
x
a
λ)x
(a
n
nn
2
n2
1
n1
n
2n
2
22
1
21
n
1n
2
12
1
11
I
38. These equations will have a non-trivial solution only
if the co-efficient matrix A - is singular
i.e., if |A - | = 0 ... (5)
Corresponding to each root of (5), the
homogeneous system (3) has a non-zero solution
X = is called an eigen vector or latent
vector
I
I
4
2
1
x
...
x
x
39. Properties of Eigen Values:-
1.The sum of the eigen values of a matrix is the sum
of the elements of the principal diagonal.
2.The product of the eigen values of a matrix A is
equal to its determinant.
3.If is an eigen value of a matrix A, then 1/ is
the eigen value of A-1 .
4.If is an eigen value of an orthogonal matrix, then
1/ is also its eigen value.
39
40. PROPERTY 1:- If λ1, λ2,…, λn are the eigen values of
A, then
i. k λ1, k λ2,…,k λn are the eigen values of the matrix
kA, where k is a non – zero scalar.
ii. are the eigen values of the inverse
matrix A-1.
iii. are the eigen values of Ap, where p
is any positive integer.
n
2
1 λ
1
,...,
λ
1
,
λ
1
p
n
p
2
p
1 λ
...,
,
λ
,
λ
40
41. 7. Find the eigen values and eigen vectors of the
matrix A =
Solution:- The characteristic equation of the given
matrix is
0
2
1
6
1
2
3
2
2
0
|
I
A
|
0
2
1
6
1
2
3
2
2
43. Corresponding to = - 3, the eigen vectors are
given by
0
1
2
1
0
3
2
3
0
3
2
0
3
2
1
6
4
2
3
2
1
0
2
1
1
2
2
1
3
2
3
2
1
1
k
k
k
k
k
k
x
x
x
x
x
or
X
I
1
2
1
1
2
2
1
3
1
X
by
given
are
vectors
eigen
The
2k
3k
x
then
k
x
,
k
x
Let
x
equation
t
independen
one
only
get
We
3
A
44. Corresponding to λ = 5, the eigen vectors are
given by (A – 5 I )X2 = 0.
0
5
2
0
3
2
0
3
2
7
0
0
0
5
2
1
6
4
2
3
2
7
3
2
1
3
2
1
3
2
1
3
2
1
x
x
x
x
x
x
x
x
x
x
x
x
46. 1.8 CAYLEY HAMILTON THEOREM
Every square matrix satisfies its own
characteristic equation.
Let A = [aij]n×n be a square matrix
then,
n
n
nn
2
n
1
n
n
2
22
21
n
1
12
11
a
...
a
a
....
....
....
....
a
...
a
a
a
...
a
a
A
47. Let the characteristic polynomial of A be (λ)
Then,
The characteristic equation is
11 12 1n
21 22 2n
n1 n2 nn
φ(λ) = A - λI
a - λ a ... a
a a - λ ... a
=
... ... ... ...
a a ... a - λ
| A - λI|= 0
48. Note 1:- Premultiplying equation (1) by A-1 , we
have
n n-1 n-2
0 1 2 n
n n-1 n-2
0 1 2 n
We are to prove that
p λ +p λ +p λ +...+p = 0
p A +p A +p A +...+p I= 0 ...(1)
I
n-1 n-2 n-3 -1
0 1 2 n-1 n
-1 n-1 n-2 n-3
0 1 2 n-1
n
0 =p A +p A +p A +...+p +p A
1
A =- [p A +p A +p A +...+p I]
p
49. This result gives the inverse of A in terms of
(n-1) powers of A and is considered as a practical
method for the computation of the inverse of the
large matrices.
Note 2:- If m is a positive integer such that m > n
then any positive integral power Am of A is linearly
expressible in terms of those of lower degree.
50. Verify Cayley – Hamilton theorem for the matrix
A = . Hence compute A-1 .
Solution:- The characteristic equation of A is
2
1
1
1
2
1
1
1
2
tion)
simplifica
(on
0
4
9λ
6λ
λ
or
0
λ
2
1
1
1
λ
2
1
1
1
λ
2
i.e.,
0
λI
A
2
3
Example 1:-
54. 1.9 DIAGONALISATION OF A
MATRIX
Diagonalisation of a matrix A is the
process of reduction A to a diagonal form.
If A is related to D by a similarity transformation,
such that D = M-1AM then A is reduced to the
diagonal matrix D through modal matrix M. D is
also called spectral matrix of A.
54
55. REDUCTION OF A MATRIX TO
DIAGONAL FORM
If a square matrix A of order n has n linearly
independent eigen vectors then a matrix B can
be found such that B-1AB is a diagonal matrix.
Note:- The matrix B which diagonalises A is called
the modal matrix of A and is obtained by
grouping the eigen vectors of A into a square
matrix.
55
56. Similarity of matrices:-
A square matrix B of order n is said to be a
similar to a square matrix A of order n if
B = M-1AM for some non singular
matrix M.
This transformation of a matrix A by a non –
singular matrix M to B is called a similarity
transformation.
Note:- If the matrix B is similar to matrix A, then B
has the same eigen values as A.
56
57. Reduce the matrix A = to diagonal form by
similarity transformation. Hence find A3.
Solution:- Characteristic equation is
=> λ = 1, 2, 3
Hence eigen values of A are 1, 2, 3.
3
0
0
1
2
0
2
1
1
57
0
λ
-
3
0
0
1
λ
-
2
0
2
1
λ
1-
Example:-
58. Corresponding to λ = 1, let X1 = be the eigen
vector then
3
2
1
x
x
x
0
0
1
k
X
x
0
x
,
k
x
0
2x
0
x
x
0
2x
x
0
0
0
x
x
x
2
0
0
1
1
0
2
1
0
0
X
)
I
(A
1
1
3
2
1
1
3
3
2
3
2
3
2
1
1
58
59. Corresponding to λ = 2, let X2 = be the eigen
vector then,
3
2
1
x
x
x
0
1
-
1
k
X
x
-k
x
,
k
x
0
x
0
x
0
2x
x
x
0
0
0
x
x
x
1
0
0
1
0
0
2
1
1
-
0
X
)
(A
2
2
3
2
2
2
1
3
3
3
2
1
3
2
1
2
0
,
I
2
59
60. Corresponding to λ = 3, let X3 = be the eigen
vector then,
3
2
1
x
x
x
2
2
-
3
k
X
x
k
-
x
,
k
x
0
x
0
2x
x
x
0
0
0
x
x
x
0
0
0
1
1
-
0
2
1
2
-
0
X
)
(A
3
3
1
3
3
3
2
3
3
2
1
3
2
1
3
3
2
2
3
,
2
I
3
k
x
60
64. Orthogonal Transformation
Let A be a Symmetric Matrix, then
AA ʹ = I
We know that ,diagonalisation transformation of a
Symmetric Matrix is D = M-1AM
If we Normalize each Eigen vector and use then
to form the Normalized Model Matrix N
Then N is an Orthogonal Matrix
D = NʹAN
Transforming A into D by means of transformation
D = NʹAN is called Orthogonal transformation. 64
I
I
I
I