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  Matrices and Matrix Operations in Matlab

Matrices and Matrix Operations in Matlab:

Matrix operations:

Recollect how to multiply a matrix A times a vector v:

1137_matrix.jpg

This is a particular case of matrix multiplication. To multiply two matrices A as well as B you proceed as follows:

1426_matrix multiplication.jpg

Here both A as well as B are 2 × 2 matrices. Matrices are able to be multiplied together in this way provided that the number of columns of A matches the number of rows of B. We forever list the size of a matrix by rows then columns therefore a 3 × 5 matrix would have 3 rows and 5 columns. Consequently if A is m × n and B is p × q then we is able to multiply AB if and only if n = p. A column vector is able to be thought of as a p×1 matrix and a row vector as a 1×q matrix. Unless or else specified we will presume a vector v to be a column vector and so Av makes nous as long as the number of columns of A matches the number of entries in v.

Printing matrices on the screen takes up a large number of space so you may want to utilize:

>format compact

Enter a matrix into Mat lab with the following syntax

> A = [ 1 3 -2 5 ; -1 -1 5 4 ; 0 1 -9 0]

As well enter a vector u

> u = [ 1 2 3 4]’

To multiply a matrix times a vector Au utilize *:

> A*u

Since A is 3 by 4 also u is 4 by 1 this multiplication is valid as well as the result is a 3 by 1 vector.

At present enter another matrix B using

> B = [3 2 1; 7 6 5; 4 3 2]

You are able to multiply B times A:

> B*A

But A times B isn’t defined and

> A*B

Will result in an error message

You are able to multiply a matrix by a scalar

> 2*A

Adding matrices A + A will give the similar result

> A + A

You are able to even add a number to a matrix

> A + 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . This must add 3 to every entry of A.

Component-wise operations:

Merely as for vectors, adding a ’.’ prior to ‘*’, ‘/’, or ‘^’ produces entry-wise multiplication division as well as exponentiation. If you enter

> B*B

the result will be BB that is matrix multiplication of B times itself. However if you enter

> B.*B

The entries of the resulting matrix will holds the squares of the same entries of B. Likewise if you wish for B multiplied by itself 3 times then enter

> B^3

However if you want to cube all the entries of B then enter

> B.^3

Note that B*B and B^3 merely make sense if B is square however B.*B and B.^3 make intelligence for any size matrix.

The identity matrix as well as the inverse of a matrix.

The n × n identity matrix is a square matrix with ones on the diagonal as well as zeros everywhere else.

It is called the identity for the reason that it plays the same role that 1 plays in multiplication that is

AI = A, IA = A, Iv = v

For any matrix A or vector v where the sizes match. An identity matrix in Mat lab is produced by the command:

> I = eye(3)

A square matrix A is able to have an inverse which is denoted by A−1. The description of the inverse is that

AA−1 = I and A−1A = I.

In theory an inverse is extremely important because if you have an equation:

Ax = b

Where A as well as b is known and x is unknown (since we will see such problems are very common and significant) then the theoretical solution is

x = A−1b.

We will observe later that this is not a practical way to solve an equation, and A−1 is only important for the purpose of derivations.

In Malta we are able to calculate a matrix’s inverse very conveniently:

> C = randn(5,5)
>inv(C)

Nevertheless not all square matrices have inverses:

> D = ones(5,5)
>inv(D)

The ‘Norm’ of a matrix:

For a vector the ‘norm’ means the same thing as the length. One more way to think of it is how far the vector is from being the zero vector. We desire to measure a matrix in much the same way and the norm is such a quantity. The typical definition of the norm of a matrix is the following:

Definition 1 assumes A is am × n matrix. The norm of A is:

|A|≡ Max^|A|=1|Av|

The maximum in the definition is taken over every vectors with length 1 (unit vectors) thus the definition means the largest factor that the matrix stretches (or shrinks) a unit vector. This definition appears cumbersome at first but it turns out to be the best one.

For illustration with this definition we have the following inequality for any vector v

|Av| ≤ |A||v|.

In Mat lab the norm of the matrix is acquired by the command

>norm(A)

For illustration the norm of an identity matrix is 1:

>norm(eye(100))

And the norm of a zero matrix is 0:

>norm(zeros(50,50))

For a matrix the norm defined above as well as calculated by Mat lab isn’t the square root of the sum of the square of its entries. That amount is called the Froebenius norm which is as well sometimes useful however we will not need it.

A few other useful commands:

Try out the following:

> C = rand(5,5) *******************arbitrary matrix with uniform distribution in [0, 1].
>size(C) ***********************************. Provides the dimensions (m × n) of C.
>det(C) ******************************************* the determinant of the matrix.
> max(C****************************************** the maximum of each column.
>min(C) ***************************************** the minimum in each column.
> sum(C)**************************************************sums each column.
> mean(C) ******************************************* the average of each column.
>diag(C) ********************************************  just the diagonal elements.
> C’ *******************************************************tranpose the matrix.

Additionally to zeros, eye, ones, rand and randn Mat lab has numerous other commands that automatically produce special matrices

>hilb(6)
>pascal(5)

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