MA 2071

Using Matlab for Linear Algebra

������������������ note:� Matlab commands entered by user shown in��� >>blue font

1. Creating Matrices

rule:� name = [� entries separated by commas with ; at end of each row]

example:����� >> A = [1,2,3; 4,5,6; 7,8,9]�� creates a 3x3 matrix

A =

���� 1���� 2���� 3

���� 4���� 5���� 6

���� 7���� 8���� 9

note no semicolon at the end of the command I typed. This means the output

will be displayed (here a good idea). If you put a semicolon then the output is

suppressed� (sometimes a good idea where there is a ton of output).

example:� >> B = [3, -1,2]��� creates a 1x3� row matrix (not a column)

example:� >> C = [1 , 0 , 0]�� creates a 3x1 column matrix.� The apostrophe at the end is the Matlab equivalent of matrix transpose.� (if you don�t remember that bit of trivia, a transpose operation on a matrix makes rows into columns and vice versa. In the textbook it is indicated by a superscript of ^T).

If you need to use the identity matrix, Matlab has a function to build it for you called� eye.� You tell it the size and you have it.

>>I3 = eye(3)����������� ( I3 was my choice for a name � use anything you want except lowercase i)

I3 =

1� 0� 0

0� 1� 0

0� 0� 1

2. Performing the Gauss-Jordan algorithm to get the Final Form (RREF) of a matrix is

��������� very easy!� If A is your matrix which you have already defined and entered then simply use rref:

������������������ >> A = [1,2,3;4,5,6;7,8,9]��

������������������ >> rref(A)

������������������ ans =��� 1� 0� -1

����������������������������������� �0� 1�� 2

����������������������������������� �0� 0�� 0

which is the RREF or Final Form of this matrix. The matrix does not have to be square.

����������������������������

3.� Matrix Arithmetic

is demonstrated fairly easily by the following examples, some of which refer to the matrices created earlier.

multiplication:

����������� >>� H = A * C

����������� H =

����������������������� 1

����������������������� 4

����������������������� 7

inversion:

����������� >>� J = [2, 3; 4, 1 ]

����������� J =

����������������������� 2� 3

����������������������� 4� 1

����������� >> K = inv(A)

����������� K =

�� �������������������� -0.1000��� 0.3000������������ (note that Matlab uses floating point displays)

� ��������������������� ��0.4000�� -0.2000

(check) >> J*K

ans =

�� �������� �1.0000�� -0.0000

����� ����� ���0��� �� ���1.0000

rank:����� (number of nonzero rows in RREF)

����������� >>rank(J)

����������� ans =

����������������������� 2

determinants:

����������� >>det(J)

����������� ans =

����������������������� -10

3. Eigenvalue and Eigenvector Calculations

>>� eig(A)���� computes the eigenvalues of A puts them in a column matrix

ans =�������������������� ��������������(for the matrix defined as A at the top of this page)

�� 16.1168

�� -1.1168

�� -0.0000

>> [P,D] = eig(A)������������ creates a square matrix P with eigenvectors as columns and

another matrix D with eigenvalues on the diagonal

P =

��� 0.2320��� 0.7858��� 0.4082

��� 0.5253��� 0.0868�� -0.8165

��� 0.8187�� -0.6123��� 0.4082

D =

�� 16.1168�������� 0�������� 0

�������� 0�� -1.1168�������� 0

�������� 0�������� 0�� -0.0000

this turns out to be absolutely perfect for the approach that Kolman takes in the textbook!!� The Principal Axis Theorem of Chapter 8 states that� these new matrices are related to the original matrix A by the key formula

����������������������������������� A = P D P^-1

and we can easily demonstrate this in Matlab with minimal effort by now entering the command

>> P*D*inv(P)

and getting the output

ans =

��� 1.0000��� 2.0000��� 3.0000

��� 4.0000��� 5.0000��� 6.0000

��� 7.0000��� 8.0000��� 9.0000

which is the original matrix A !!!