This document defines and explains key concepts related to eigenvalues and eigenvectors of matrices. It states that an eigenvector is a vector that remains pointing in the same direction when multiplied by a matrix, represented by the equation AX = λX. It then provides properties of eigenvalues for different types of matrices, such as real eigenvalues for symmetric matrices. It also describes how eigenvalues change based on operations to the original matrix, such as scaling. The document concludes by explaining how to calculate eigenvalues from the characteristic equation and derive eigenvectors.
Numerical solution of eigenvalues and applications 2SamsonAjibola
This project aims at studying the methods of numerical solution of eigenvalue problems and their applications. An accurate mathematical method is needed to solve direct and inverse eigenvalue problems related to different applications such as engineering analysis and design, statistics, biology e.t.c. Eigenvalue problems are of immense interest and play a pivotal role not only in many fields of engineering but also in pure and applied mathematics, thus numerical methods are developed to solve eigenvalue problems. The primary objective of this work is to showcase these various eigenvalue algorithms such as QR algorithm, power method, Krylov subspace iteration (Lanczos and Arnoldi) and explain their effects and procedures in solving eigenvalue problems.
Numerical solution of eigenvalues and applications 2SamsonAjibola
This project aims at studying the methods of numerical solution of eigenvalue problems and their applications. An accurate mathematical method is needed to solve direct and inverse eigenvalue problems related to different applications such as engineering analysis and design, statistics, biology e.t.c. Eigenvalue problems are of immense interest and play a pivotal role not only in many fields of engineering but also in pure and applied mathematics, thus numerical methods are developed to solve eigenvalue problems. The primary objective of this work is to showcase these various eigenvalue algorithms such as QR algorithm, power method, Krylov subspace iteration (Lanczos and Arnoldi) and explain their effects and procedures in solving eigenvalue problems.
this the slide for learning derogatory and non derogatory matrices. You can eaily refer this. we have also stated some basic learnt topics in matrices so that you won't confusing over the steps of example. this is the part of linear algebra. the slide template is taken from slidesgo.com but it is designed by me. we have also mentioned the polynomials that we have to find and the necessary steps that we need to follow to show or prove whether the matrix is derogatory or not. i hope you like this slide.
Eigen values and eigen vectors engineeringshubham211
mathematics...for engineering mathematics.....learn maths...............................The individual items in a matrix are called its elements or entries.[4] Provided that they are the same size (have the same number of rows and the same number of columns), two matrices can be added or subtracted element by element. The rule for matrix multiplication, however, is that two matrices can be multiplied only when the number of columns in the first equals the number of rows in the second. Any matrix can be multiplied element-wise by a scalar from its associated field. A major application of matrices is to represent linear transformations, that is, generalizations of linear functions such as f(x) = 4x. For example, the rotation of vectors in three dimensional space is a linear transformation which can be represented by a rotation matrix R: if v is a column vector (a matrix with only one column) describing the position of a point in space, the product Rv is a column vector describing the position of that point after a rotation. The product of two transformation matrices is a matrix that represents the composition of two linear transformations. Another application of matrices is in the solution of systems of linear equations. If the matrix is square, it is possible to deduce some of its properties by computing its determinant. For example, a square matrix has an inverse if and only if its determinant is not zero. Insight into the geometry of a linear transformation is obtainable (along with other information) from the matrix's eigenvalues and eigenvectors.
Applications of matrices are found in most scientific fields. In every branch of physics, including classical mechanics, optics, electromagnetism, quantum mechanics, and quantum electrodynamics, they are used to study physical phenomena, such as the motion of rigid bodies. In computer graphics, they are used to project a 3-dimensional image onto a 2-dimensional screen. In probability theory and statistics, stochastic matrices are used to describe sets of probabilities; for instance, they are used within the PageRank algorithm that ranks the pages in a Google search.[5] Matrix calculus generalizes classical analytical notions such as derivatives and exponentials to higher dimensions.
A major branch of numerical analysis is devoted to the development of efficient algorithms for matrix computations, a subject that is centuries old and is today an expanding area of research. Matrix decomposition methods simplify computations, both theoretically and practically. Algorithms that are tailored to particular matrix structures, such as sparse matrices and near-diagonal matrices, expedite computations in finite element method and other computations. Infinite matrices occur in planetary theory and in atomic theory. A simple example of an infinite matrix is the matrix representing the derivative operator, which acts on the Taylor series of a function
...
this the slide for learning derogatory and non derogatory matrices. You can eaily refer this. we have also stated some basic learnt topics in matrices so that you won't confusing over the steps of example. this is the part of linear algebra. the slide template is taken from slidesgo.com but it is designed by me. we have also mentioned the polynomials that we have to find and the necessary steps that we need to follow to show or prove whether the matrix is derogatory or not. i hope you like this slide.
Eigen values and eigen vectors engineeringshubham211
mathematics...for engineering mathematics.....learn maths...............................The individual items in a matrix are called its elements or entries.[4] Provided that they are the same size (have the same number of rows and the same number of columns), two matrices can be added or subtracted element by element. The rule for matrix multiplication, however, is that two matrices can be multiplied only when the number of columns in the first equals the number of rows in the second. Any matrix can be multiplied element-wise by a scalar from its associated field. A major application of matrices is to represent linear transformations, that is, generalizations of linear functions such as f(x) = 4x. For example, the rotation of vectors in three dimensional space is a linear transformation which can be represented by a rotation matrix R: if v is a column vector (a matrix with only one column) describing the position of a point in space, the product Rv is a column vector describing the position of that point after a rotation. The product of two transformation matrices is a matrix that represents the composition of two linear transformations. Another application of matrices is in the solution of systems of linear equations. If the matrix is square, it is possible to deduce some of its properties by computing its determinant. For example, a square matrix has an inverse if and only if its determinant is not zero. Insight into the geometry of a linear transformation is obtainable (along with other information) from the matrix's eigenvalues and eigenvectors.
Applications of matrices are found in most scientific fields. In every branch of physics, including classical mechanics, optics, electromagnetism, quantum mechanics, and quantum electrodynamics, they are used to study physical phenomena, such as the motion of rigid bodies. In computer graphics, they are used to project a 3-dimensional image onto a 2-dimensional screen. In probability theory and statistics, stochastic matrices are used to describe sets of probabilities; for instance, they are used within the PageRank algorithm that ranks the pages in a Google search.[5] Matrix calculus generalizes classical analytical notions such as derivatives and exponentials to higher dimensions.
A major branch of numerical analysis is devoted to the development of efficient algorithms for matrix computations, a subject that is centuries old and is today an expanding area of research. Matrix decomposition methods simplify computations, both theoretically and practically. Algorithms that are tailored to particular matrix structures, such as sparse matrices and near-diagonal matrices, expedite computations in finite element method and other computations. Infinite matrices occur in planetary theory and in atomic theory. A simple example of an infinite matrix is the matrix representing the derivative operator, which acts on the Taylor series of a function
...
Maximum likelihood estimation (MLE) is a popular method for parameter estimation in both applied probability and statistics but MLE cannot solve the problem of incomplete data or hidden data because it is impossible to maximize likelihood function from hidden data. Expectation maximum (EM) algorithm is a powerful mathematical tool for solving this problem if there is a relationship between hidden data and observed data. Such hinting relationship is specified by a mapping from hidden data to observed data or by a joint probability between hidden data and observed data. In other words, the relationship helps us know hidden data by surveying observed data. The essential ideology of EM is to maximize the expectation of likelihood function over observed data based on the hinting relationship instead of maximizing directly the likelihood function of hidden data. Pioneers in EM algorithm proved its convergence. As a result, EM algorithm produces parameter estimators as well as MLE does. This tutorial aims to provide explanations of EM algorithm in order to help researchers comprehend it. Moreover some improvements of EM algorithm are also proposed in the tutorial such as combination of EM and third-order convergence Newton-Raphson process, combination of EM and gradient descent method, and combination of EM and particle swarm optimization (PSO) algorithm.
4. Linear Algebra for Machine Learning: Eigenvalues, Eigenvectors and Diagona...Ceni Babaoglu, PhD
The seminar series will focus on the mathematical background needed for machine learning. The first set of the seminars will be on "Linear Algebra for Machine Learning". Here are the slides of the fourth part which is discussing eigenvalues, eigenvectors and diagonalization.
Here is the link of the first part which was discussing linear systems: https://www.slideshare.net/CeniBabaogluPhDinMat/linear-algebra-for-machine-learning-linear-systems/1
Here are the slides of the second part which was discussing basis and dimension:
https://www.slideshare.net/CeniBabaogluPhDinMat/2-linear-algebra-for-machine-learning-basis-and-dimension
Here are the slides of the third part which is discussing factorization and linear transformations.
https://www.slideshare.net/CeniBabaogluPhDinMat/3-linear-algebra-for-machine-learning-factorization-and-linear-transformations-130813437
Students, digital devices and success - Andreas Schleicher - 27 May 2024..pptxEduSkills OECD
Andreas Schleicher presents at the OECD webinar ‘Digital devices in schools: detrimental distraction or secret to success?’ on 27 May 2024. The presentation was based on findings from PISA 2022 results and the webinar helped launch the PISA in Focus ‘Managing screen time: How to protect and equip students against distraction’ https://www.oecd-ilibrary.org/education/managing-screen-time_7c225af4-en and the OECD Education Policy Perspective ‘Students, digital devices and success’ can be found here - https://oe.cd/il/5yV
How to Make a Field invisible in Odoo 17Celine George
It is possible to hide or invisible some fields in odoo. Commonly using “invisible” attribute in the field definition to invisible the fields. This slide will show how to make a field invisible in odoo 17.
How to Split Bills in the Odoo 17 POS ModuleCeline George
Bills have a main role in point of sale procedure. It will help to track sales, handling payments and giving receipts to customers. Bill splitting also has an important role in POS. For example, If some friends come together for dinner and if they want to divide the bill then it is possible by POS bill splitting. This slide will show how to split bills in odoo 17 POS.
We all have good and bad thoughts from time to time and situation to situation. We are bombarded daily with spiraling thoughts(both negative and positive) creating all-consuming feel , making us difficult to manage with associated suffering. Good thoughts are like our Mob Signal (Positive thought) amidst noise(negative thought) in the atmosphere. Negative thoughts like noise outweigh positive thoughts. These thoughts often create unwanted confusion, trouble, stress and frustration in our mind as well as chaos in our physical world. Negative thoughts are also known as “distorted thinking”.
The Art Pastor's Guide to Sabbath | Steve ThomasonSteve Thomason
What is the purpose of the Sabbath Law in the Torah. It is interesting to compare how the context of the law shifts from Exodus to Deuteronomy. Who gets to rest, and why?
Instructions for Submissions thorugh G- Classroom.pptxJheel Barad
This presentation provides a briefing on how to upload submissions and documents in Google Classroom. It was prepared as part of an orientation for new Sainik School in-service teacher trainees. As a training officer, my goal is to ensure that you are comfortable and proficient with this essential tool for managing assignments and fostering student engagement.
How to Create Map Views in the Odoo 17 ERPCeline George
The map views are useful for providing a geographical representation of data. They allow users to visualize and analyze the data in a more intuitive manner.
Read| The latest issue of The Challenger is here! We are thrilled to announce that our school paper has qualified for the NATIONAL SCHOOLS PRESS CONFERENCE (NSPC) 2024. Thank you for your unwavering support and trust. Dive into the stories that made us stand out!
Unit 8 - Information and Communication Technology (Paper I).pdfThiyagu K
This slides describes the basic concepts of ICT, basics of Email, Emerging Technology and Digital Initiatives in Education. This presentations aligns with the UGC Paper I syllabus.
Synthetic Fiber Construction in lab .pptxPavel ( NSTU)
Synthetic fiber production is a fascinating and complex field that blends chemistry, engineering, and environmental science. By understanding these aspects, students can gain a comprehensive view of synthetic fiber production, its impact on society and the environment, and the potential for future innovations. Synthetic fibers play a crucial role in modern society, impacting various aspects of daily life, industry, and the environment. ynthetic fibers are integral to modern life, offering a range of benefits from cost-effectiveness and versatility to innovative applications and performance characteristics. While they pose environmental challenges, ongoing research and development aim to create more sustainable and eco-friendly alternatives. Understanding the importance of synthetic fibers helps in appreciating their role in the economy, industry, and daily life, while also emphasizing the need for sustainable practices and innovation.
2. Eigen vector of a matrix A is a vector
represented by a matrix X such that when X
is multiplied with matrix A, then the
direction of the resultant matrix remains
same as vector X.
Mathematically, above statement can be
represented as:
AX = λX
3. Eigen values of real symmetric and
hermitian matrices are real.
Eigen values of real skew symmetric and
skew hermitian matrices are either pure
imaginary or zero.
Eigen values of unitary and orthogonal
matrices are of unit modulus |λ| = 1.
4. If λ1, λ2…….λn are the eigen values of A, then
kλ1, kλ2…….kλn are eigen values of kA
If λ1, λ2…….λn are the eigen values of A, then
1/λ1, 1/λ2…….1/λn are eigen values of A-1
If λ1, λ2…….λn are the eigen values of A, then
λ1
k, λ2
k…….λn
k are eigen values of Ak
Eigen values of A = Eigen Values of AT
(Transpose)
5. Sum of Eigen Values = Trace of A (Sum of
diagonal elements of A)
Product of Eigen Values = |A|
Maximum number of distinct eigen values of
A = Size of A
If A and B are two matrices of same order
then, Eigen values of AB = Eigen values of BA
6. AX = λX
=> AX – λX = 0
=> (A – λI) X = 0 …..(1)
Above condition will be true only if (A – λI) is
singular. That means,
|A – λI| = 0 …..(2)
(2) is known as characteristic equation of the
matrix.
7. The roots of the characteristic equation are
the eigen values of the matrix A.
Now, to find the eigen vectors, we simply put
each eigen value into (1) and solve it by
Gaussian elimination, that is, convert the
augmented matrix (A – λI) = 0 to row echelon
form and solve the linear system of
equations thus obtained.
8. In linear algebra, an eigenvector or
characteristic vector of a linear
transformation is a nonzero vector that
changes at most by a scalar factor when that
linear transformation is applied to it. The
corresponding eigenvalue is the factor by
which the eigenvector is scaled.