1. Linear algebra: matrix Eigen-value Problems
Part 3
Eng. Hassan S. Migdadi

2. Eigenvalues and eigenvectors Vector spaces Linear transformations
Eigenvalue Problems
Eigenvalues and eigenvectors
Vector spaces
Linear transformations
Matrix diagonalization

3. The Eigenvalue Problem
Consider a nxn matrix A
Vector equation: Ax = lx
Seek solutions for x and l
l satisfying the equation are the eigenvalues
Eigenvalues can be real and/or imaginary; distinct and/or repeated
x satisfying the equation are the eigenvectors
Nomenclature
The set of all eigenvalues is called the spectrum
Absolute value of an eigenvalue:
The largest of the absolute values of the eigenvalues is called the spectral radius

4. Determining Eigenvalues
Vector equation
Ax = lx  (A-lI)x = 0
A-lI is called the characteristic matrix
Non-trivial solutions exist if and only if:
This is called the characteristic equation
Characteristic polynomial
nth-order polynomial in l
Roots are the eigenvalues {l1, l2, …, ln}

5. Eigenvalue Example Characteristic matrix Characteristic equation
Eigenvalues: l1 = -5, l2 = 2

6. Eigenvalue Properties
Eigenvalues of A and AT are equal
Singular matrix has at least one zero eigenvalue
Eigenvalues of A-1: 1/l1, 1/l2, …, 1/ln
Eigenvalues of diagonal and triangular matrices are equal to the diagonal elements
Trace
Determinant

7. Determining Eigenvectors
First determine eigenvalues: {l1, l2, …, ln}
Then determine eigenvector corresponding to each eigenvalue:
Eigenvectors determined up to scalar multiple
Distinct eigenvalues
Produce linearly independent eigenvectors
Repeated eigenvalues
Produce linearly dependent eigenvectors
Procedure to determine eigenvectors more complex (see text)
Will demonstrate in Matlab

8. Eigenvector Example Eigenvalues Determine eigenvectors: Ax = lx
Eigenvector for l1 = -5
Eigenvector for l1 = 2

9. Matlab Examples
>> A=[ 1 2; 3 -4]; >> e=eig(A) e = 2 -5 >> [X,e] = eig(A) X =
>> A=[2 5; 0 2]; >> e=eig(A) e = 2 >> [X,e]=eig(A) X =

10. Vector Spaces Real vector space V Properties of a real vector space
Set of all n-dimensional vectors with real elements
Often denoted Rn
Element of real vector space denoted
Properties of a real vector space
Vector addition
Scalar multiplication

11. Vector Spaces cont. Linearly independent vectors Basis
Elements:
Linear combination:
Equation satisfied only for cj = 0
Basis
n-dimensional vector space V contains exactly n linearly independent vectors
Any n linearly independent vectors form a basis for V
Any element of V can be expressed as a linear combination of the basis vectors
Example: unit basis vectors in R3

12. Inner Product Spaces Inner product
Properties of an inner product space
Two vectors with zero inner product are called orthogonal
Relationship to vector norm
Euclidean norm
General norm
Unit vector: ||a|| = 1

13. Linear Transformation
Properties of a linear operator F
Linear operator example: multiplication by a matrix
Nonlinear operator example: Euclidean norm
Linear transformation
Invertible transformation
Often called a coordinate transformation

14. Orthogonal Transformations
Orthogonal matrix
A square matrix satisfying: AT = A-1
Determinant has value +1 or -1
Eigenvalues are real or complex conjugate pairs with absolute value of unity
A square matrix is orthonormal if:
Orthogonal transformation
y = Ax where A is an orthogonal matrix
Preserves the inner product between any two vectors
The norm is also invariant to orthogonal transformation

15. Similarity Transformations
Eigenbasis
If a nxn matrix has n distinct eigenvalues, the eigenvectors form a basis for Rn
The eigenvectors of a symmetric matrix form an orthonormal basis for Rn
If a nxn matrix has repeated eigenvalues, the eigenvectors may not form a basis for Rn (see text)
Similar matrices
Two nxn matrices are similar if there exists a nonsingular nxn matrix P such that:
Similar matrices have the same eigenvalues
If x is an eigenvector of A, then y = P-1x is an eigenvector of the similar matrix

16. Matrix Diagonalization
Assume the nxn matrix A has an eigenbasis
Form the nxn modal matrix X with the eigenvectors of A as column vectors: X = [x1, x2, …, xn]
Then the similar matrix D = X-1AX is diagonal with the eigenvalues of A as the diagonal elements
Companion relation: XDX-1 = A

17. Matrix Diagonalization Example

18. Matlab Example
>> A=[-1 2 3; ; ]; >> [X,e]=eig(A) X = e = >> D=inv(X)*A*X D =