Review of Basic Matrix Operations

Reading:

Appendix C of the text.
The matrix math supplement to Appendix C ¨C a handout available at 123 Latrobe Hall.
Review your linear - algebra textbook as required

Show your work. Don't just write down the final answer - be methodical and clear about the entire
sequence of steps you used to arrive at your answers.

Given the following matrices

1. Compute by hand (show your work) the determinant, eigenvalues, and eigenvectors of A. Check your
result ¨C verify that

2. Compute by hand (show your work) the determinant, eigenvalues, and eigenvectors of B. Check your
result ¨C verify that
3. Compute by hand (show your work) the determinant, eigenvalues, and eigenvectors of C. Check your
result ¨C verify that

4. Compute by hand (show your work) the determinent, eigenvalues, and eigenvectors of D. Check your
result ¨C verify that

5. Use Matlab to compute the eigenvalues of A. Print your work.

6. Use Matlab to compute the eigenvalues of B. Print your work.

7. Use Matlab to compute the eigenvalues of C. Print your work.

8. Use Matlab to compute the eigenvalues of D. Print your work.

MATLAB Function Reference eig Eigenvalues and eigenvectors

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Syntax

d = eig(A)
[V,D] = eig(A)
[V,D] = eig(A,¡¯nobalance¡¯)
d = eig(A,B)
[V,D] = eig(A,B)

Description

d = eig(A) returns a vector of the eigenvalues of matrix A.

[V,D] = eig(A) produces matrices of eigenvalues (D) and eigenvectors (V) of matrix A, so that A*V
= V*D. Matrix D is the canonical form of A--a diagonal matrix with A's eigenvalues on the main
diagonal. Matrix V is the modal matrix--its columns are the eigenvectors of A.

The eigenvectors are scaled so that the norm of each is 1.0. Use [W,D] = eig(A¡¯); W = W¡¯ to
compute the left eigenvectors, which satisfy
W*A = D*W.

[V,D] = eig(A,¡¯nobalance¡¯) finds eigenvalues and eigenvectors without a preliminary balancing
step. Ordinarily, balancing improves the conditioning of the input matrix, enabling more accurate
computation of the eigenvectors and eigenvalues. However, if a matrix contains small elements that
are really due to roundoff error, balancing may scale them up to make them as significant as the other
elements of the original matrix, leading to incorrect eigenvectors. Use the nobalance option in this
event. See the balance function for more details.

d = eig(A,B) returns a vector containing the generalized eigenvalues, if A and B are square
matrices.

[V,D] = eig(A,B) produces a diagonal matrix D of generalized eigenvalues and a full matrix V
whose columns are the corresponding eigenvectors so that A*V = B*V*D. The eigenvectors are scaled
so that the norm of each is 1.0.

Remarks

The eigenvalue problem is to determine the nontrivial solutions of the equation :

where A is an n-by-n matrix, x is a length n column vector, and λ is a scalar. The n values of λ that
satisfy the equation are the eigenvalues , and the corresponding values of x are the right eigenvectors.
In MATLAB, the function eig solves for the eigenvalues λ, and optionally the eigenvectors x.

The generalized eigenvalue problem is to determine the nontrivial solutions of the equation

where both A and B are n-by-n matrices and λ is a scalar. The values of λ that satisfy the equation
are the generalized eigenvalues and the corresponding values of x are the generalized right
eigenvectors.

If B is nonsingular, the problem could be solved by reducing it to a standard eigenvalue problem

Because B can be singular, an alternative algorithm, called the QZ method, is necessary.

When a matrix has no repeated eigenvalues, the eigenvectors are always independent and the
eigenvector matrix V diagonalizes the original matrix A if applied as a similarity transformation.
However, if a matrix has repeated eigenvalues, it is not similar to a diagonal matrix unless it has a
full (independent) set of eigenvectors. If the eigenvectors are not independent then the original matrix
is said to be defective. Even if a matrix is defective, the solution from eig satisfies A*X = X*D.

Examples

The matrix

B = [3 -2 -.9 2*eps;-2 4 -1 -eps;-eps/4 eps/2 -1 0;-.5 -.5 .1 1];

has elements on the order of roundoff error. It is an example for which the nobalance option is
necessary to compute the eigenvectors correctly. Try the statements

[VB,DB] = eig(B)
B*VB - VB*DB
[VN,DN] = eig(B,¡¯nobalance¡¯)
B*VN - VN*DN

Algorithm

For real matrices, eig(X) uses the EISPACK routines BALANC, BALBAK, ORTHES, ORTRAN, and HQR2.
BALANC and BALBAK balance the input matrix. ORTHES converts a real general matrix to Hessenberg
form using orthogonal similarity transformations. ORTRAN accumulates the transformations used by
ORTHES. HQR2 finds the eigenvalues and eigenvectors of a real upper Hessenberg matrix by the QR
method. The EISPACK subroutine HQR2 is modified to make computation of eigenvectors optional.

When eig is used with two input arguments, the EISPACK routines QZHES, QZIT, QZVAL, and QZVEC
solve for the generalized eigenvalues via the QZ algorithm. Modifications handle the complex case .

When eig is used with one complex argument, the solution is computed using the QZ algorithm as
eig(X,eye(X)). Modifications to the QZ routines handle the special case B = I.

For detailed descriptions of these algorithms, see the EISPACK Guide.

Diagnostics

If the limit of 30n iterations is exhausted while seeking an eigenvalue:
Solution will not converge.

See Also