Download Computational Linear Algebra

Math 234
Computational Linear Algebra
Linear Algebra on the Calculator
To enter a matrix into a TI-83 calculator, access the MATRIX menu by
pressing the buttons 2nd x-1. First create a matrix by using the right arrow
to go to the EDIT menu. Select the name of the matrix you want to edit
and then press ENTER. Note that you first set the matrix dimensions.
Press QUIT (2nd MODE) to go back to the standard calculator screen. If we want to enter a
matrix name on the screen, go back to the MATRIX screen (2nd x-1) and select the name you
want. Note this screen tells you which matrices have values and their sizes. Press ENTER and
the name will appear on the screen.
For example, typing [A]^2 will compute the square. We can add, subtract, multiply, and invert
(A-1) matrices easily. Other matrix operations such as the transpose are available under the
MATH screen of the MATRIX menu.
The calculator will give you an error message if the quantity is undefined. For example, the
matrix A we entered is not invertible. If we tried to compute the inverse of A by using the x-1
button we get:
Linear Algebra on the Computer
The software Freemat is a free version of Matlab, a powerful linear
algebra solver. You can download Freemat at:
freemat.sourceforge.net
Freemat processes commands through the command line. For example,
if you type
--> x = 2 + 3
and press ENTER, Freemat will respond:
x=5
To type in a matrix variable, we surround the values with square brackets. The semicolon ";"
appears at the end of each row, like a line break. For example, if you type
--> A = [1 2 3; 4 5 6]
--> B = [1 2; 3 4; 5 6]
You will now have a 2x3 matrix A and a 3x2 matrix B.
Freemat can now do matrix arithmetic with these variables. For example, to multiply the
matrices we can type:
--> A*B
Freemat responds with the answer
ans = 22 28
49 64
If we wanted to store this answer in another matrix C, we would instead type:
--> C=A*B
Freemat will give an error message when the quantity is undefined. For example, if we try to
add the matrices A and B, we would get:
--> A+B
Error: size mismatch in arguments to binary operator
To access an individual element of a matrix, we look it up as (row,column). For example, we
could change the entry in the 2nd row, 3rd column of A from 6 to 42 by
--> A(2,3)=42
A= 1 2 3
4 5 42
Some useful commands
Matrix dimensions: size(A)
Scalar Multiplication: 4*A
Matrix Multiplication: A*B
Matrix Powers: A^4
Transpose: A'
NxN Identity Matrix: eye(N)
Pointwise Multiplication: A.*B
Pointwise Exponent: A.^3
Determinant: det(A)
Rank: rank(A)
Inverse: inv(A)
Trace: trace(A)
Row Reduction: rref(A)
Eigenvalues: eig(A)
Eigenvectors: [v,d]=eig(A)
Singular Value Decomposition: [U,S,V]=svd(A)