Eigen values for 1st year btech

1. EIGEN VALUES & EIGEN VECTOR
• Name: UtsavKishoreOjha
• RollNo. :232179342
• UniversityRollNo. : 12500123180
• Stream: Cse
• Section: C

2. CONTENTS OF THE SLIDE
• Introduction
• Definition of Matrix
• Key Charecteristics of a Matrix
• Example of Matrix
• Applications of Matrix
• Operations of a Matrix
• Properties of Eigen Values
• Properties of Eigen Vectors
• Finding Eigen Values
• Finding Eigen Vectors
• Conclusion

3. INTRODUCTION
In the world of linear algebra,eigenvaluesand eigenvectorsare a power couple!They help us
understand how a matrix, which represents a linear transformation, affects certain special vectors.
Eigenvectors:Imagine a magic mirror that stretches or shrinks an object but keepsits direction. The
object's shape changes, but it still points in the same way. In mathematical terms, these objects are
eigenvectors,and the amount by which they're stretched (or shrunk) is the eigenvalue.
Eigenvalues:Think of the eigenvalue as a scaling factor. When you apply the matrix (the fancy
mirror) to an eigenvector,you get the eigenvector back, just slightly modified.If the eigenvalue is
positive, the vector gets stretched; if it's negative, it gets flippedand shrunk; if it's zero,the vector
disappears!
Here's the key equation:
A * x = λ * x
where:A is the matrix (the magic mirror)
x is the eigenvector (the object)
λ is the eigenvalue(the scaling factor)

4. DEFINITION OF MATRIX
A matrix is a rectangular array of numbers, symbols, or
expressions,arranged in rows and columns. It's used to
representvarious mathematical objects and
relationships, and it plays a central role in linear algebra,
statistics, engineering,and numerous other fields.

5. KEY CHARACTERISTICS OF A MATRIX
Dimensions:- A matrix is described by its dimensions,
typically written as "m x n," where m is the number of rows
and n is the number of columns. For example, a 2 x 3 matrix
has 2 rows and 3 columns.
Elements:- The individual items within the matrix are called
elements or entries. Each element is identified by its row and
column position, using subscript notation. For example, a2,4
refers to the element in the 2nd row and 4th column.
Notation:- Matrices are commonly denoted by capital letters,
like A, B, C, etc. ..

6. EXAMPLES OF MATRIX
Square matrix:- A matrix with the same number of rows and
columns (e.g., a 3 x 3 matrix).
Row matrix:-A matrix with only one row (e.g., a 1 x 4 matrix).
Column matrix:-A matrix with only one column (e.g., a 5 x 1
matrix).
Zero matrix: A matrix where all elements are zero.
Identity matrix: A square matrix with 1s on the diagonal
and 0s elsewhere.

7. APPLICATIONS OF MATRIX
Linear transformations:- Matrices can represent linear
transformations, such as rotations, reflections, and scaling in
geometric spaces.
Systems of linear equations:- Matrices are used to express and
solve systems of linear equations.
Graph theory:- Matrices can represent graphs and networks.
Probability and statistics:- Matrices are used in probability theory
and statistics to represent data and relationships.
Image processing:- Matrices are used to store and manipulate
digital images.
Machine learning:- Matrices are fundamental for various machine
learning algorithms.

8. COMMON OPERATIONS
ON MATRIX
Addition:- Matrices with the same dimensions can be added
element-wise.
Subtraction:- Matrices with the same dimensions can be
subtracted element-wise.
Multiplication:- Matrices can be multiplied, but specific rules
for compatibility and order apply.
Transpose:- The transpose of a matrix is formed by swapping
its rows and columns

9. PROPERTIES OF EIGEN
VALUES
Special scalar values:-When applied to a matrix equation,they
cause the resulting vector to be a scaled versionof the original
vector.
Not unique: A single matrix can have multiple eigenvalues,each
associated with a specific eigenvector.
Can be real or complex:- Depending on the matrix, eigenvalues can
be real numbers, imaginarynumbers, or complex numbers with
both real and imaginary parts.
Influence transformations:- They determine how a linear
transformationstretches, shrinks, or reflects a vector.
Sum and product have meaning:-The sum of a matrix's eigenvalues
equals the trace of the matrix, and their product equals the
determinantof the matrix.

10. PROPERTIES OF EIGEN
VECTOR
Non-zero vectors:- When multiplied by their correspondingmatrix,
they remain in the same direction but are scaled by theeigenvalue.
Not unique (except for multiplicity 1):- A single eigenvalue can have
multiple eigenvectors, forming aneigen space. However, if an
eigenvalue has a multiplicity of 1, there is only one corresponding
eigenvector.
Orthogonal(for distinct eigenvalues):- If multiple eigenvaluesare
distinct, the correspondingeigenvectors are orthogonal
(perpendicular)to each other.
Span the vector space (sometimes):- In some cases, all
eigenvectors of a matrix can be combined to form a basis for the
entire vector space the matrix operates on.
Useful for diagonalization:- Findingeigenvaluesand eigenvectors
allows you to diagonalize a matrix, simplifyingits analysisand
understanding its behavior.

11. FINDING EIGEN VALUES
(STEP - BY - STEP)
 Verify Square Matrix:-Eigenvalues exist only for square matrices (matrices with the same
number of rows and columns).If your matrix is not square, you cannot proceed with this
process.
 Construct the Characteristic Equation:-Subtract λ (lambda, a scalar variable) times the
identity matrix (I) from the original matrix (A). This creates the matrix (A- λI).Calculate the
determinant of this matrix, represented asdet(A - λI).Set the determinant equal to zero,
forming the characteristic equation:det(A - λI) = 0.
 Solve the Characteristic Equation:-Solve the characteristic equation for λ. The solutions
for λ will be the eigenvalues of the matrixA.For 2x2 matrices, the quadratic formula can be
used. For larger matrices, computational tools or techniques like factoring or Gaussian
elimination might be needed.
 Example (2x2 Matrix):-Consider the matrix A = [3 2; 4 1].Following steps 1-2, we get the
characteristic equation: (3-λ)(1-λ) - 8 = 0.Solving for λ using the quadratic formula yields λ
= 5 or λ = -1.Therefore, the eigenvalues of matrix A are 5 and -1.
 Finding Eigenvectors (Optional):-If you also need the eigenvectors, you would substitute
each eigenvalue back into the equation (A - λI)v = 0 and solve for the corresponding vector
v.
• Remember:->Eigenvalues may be real or complex numbers.>For larger matrices,
computational tools can assist in solving the characteristic equation and finding
eigenvalues.

12. FINDING EIGEN VECTOR
(STEP - BY- STEP)
 Choose an Eigenvalue:-Selectone of the eigenvalues(λ) that you found for the matrix.
 Set Up the Equation:-Create the matrix (A - λI), where A is the original matrix, λ is the chosen
eigenvalue,and I is the identity matrix of the same size as A. Write the equation (A - λI)v = 0, where
v represents the eigenvector you'reseeking.
 Solvethe System of Equations:-Rewrite the equation (A - λI)v = 0 as a systemof linear equations,
one for each row of the matrix. Solve this system of equations using techniques like Gaussian
elimination or row reduction. The solutions will be the components of the eigenvector v that
correspond to the chosen eigenvalueλ.
 Normalize(Optional):-If desired,normalize the eigenvector by dividing it by its magnitude to make
it a unit vector (having a length of 1). This is often done for consistency and easier interpretation.
 Repeat for Other Eigenvalues:-Ifthe matrix has multiple eigenvalues,repeat steps 1-4 for each
eigenvalueto find its correspondingeigenvectors.
Example (Using the matrix from the previousexample):-
Let's find the eigenvector corresponding to λ = 5 for the matrix A = [3 2; 4 1].
Setting up the equation (A - λI)v = 0, we get [-2 2; 4 -4]v = 0.
Simplifying this system of equations, we get -x + y = 0, which means x = y.
Choosing a value for x (e.g.,x = 1), we get the eigenvector v = [1; 1].
Normalizing v, we get the unit eigenvector v = [1/√2; 1/√2]..

13. CONCLUSION
Eigenvalues and eigenvectors are fundamental concepts in linear
algebra, with far-reaching applications in numerous fields like
engineering, physics, computer science, and data analysis.
Understanding eigenvalues and eigenvectors empowers us to analyze
complex systems, solve differential equations, compress data, and unlock
a plethora of technological advances. From designing earthquake-
resistant structures to powering facial recognition algorithms,
eigenvalues and eigenvectors shape the world around us in profound
ways.
We saw that eigenvalues represent scaling factors and eigenvectors
define directions preserved by a linear transformation.
Finding eigenvalues involves solving the characteristic equation, while
eigenvectors are determined by solving systems of equations associated
with each eigen values. The world of eigenvalues and eigenvectors
extends beyond this presentation, with topics like diagonalization,
spectral analysis, and applications in various domains waiting to be
explored.

14. REFERENCE

www.scibd.com
www.mathway.com

15. Thank You