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 1x3row 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 calledeye.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 =

 

100

010

001

 

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 =��� 10-1

����������������������������������� 01�� 2

����������������������������������� 00�� 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 =

����������������������� 23

����������������������� 41

����������� >> 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 thatthese 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 !!!