MATRIX OPERATIONS

• The usual matrix operations such as addition "+" , subtraction "-" and multiplication "*" require that matrices be conformable for the operation.
  For addition and subtraction A and B must have the same size.
  Multiplication requires that the number of columns of the first matrix be the same as the number of rows of the second.
• To multiply a number times a matrix use the symbol "*" which multiples every entry by the constant.

A=[1 2;-4 5]
3*A 

Here is another example

A=floor(10*rand(3)); 
B=floor(10*rand(3)); 
C=floor(10*rand(3));
D=A*(2*B-3*C)

• Every number in Matlab is automatically treated as a complex number. If you enter an integer it is displayed as an integer. Similarly, a real number entered with a nonzero decimal part is displayed in a floating point format with no imaginary part. If, for example

 
A=[2 3 -1;5 8 2]
B=[9 5 1;2.1 5 8]

then

C=A+i*B

gives the matrix with entries [A(i,j)+sqrt(-1)*B(i,j)].

• If A is a matrix and r is a number, then A+r is the matrix with r added to each element of A. Note the differences in the following results.

A=[2 1;0 -2]
eye(2)
B=A+eye(2)
C=A+1

• You can also take positive integer powers of a square matrix A by typing A^n.

A=[2 3 -1;5 8 2;0 1 -2]
B=A^3

• You know that, in general, division of matrices makes no sense. Even the existence of the inverse of a matrix is not simple to decide. The matrix must be nonsingular to have an inverse (it must be square and have a nonzero determinant). To compute the determinant in Matlab simply type

det(A)

If A is nonsingular , then the inverse in Matlab is given by inv(A) or A^(-1). More generally, in the nonsingular case, you can compute A^(-r).

• Combining the above we can build polynomials of matrices. Here is an example.

A=[2 1;0 -2]
B=A^2+2*A+eye(2)

Suppose A is an n by n square matrix, c is an n component row vector and b is an n component column vector, then the following calculation gives a scalar f=c*(A^3+A^2+A)*b

A=[2 1;0 -2]
c=[1 2]
b=[1 -1]'
f=c*(A^3+A^2+A)*b

On the other hand, it is much more efficient to write

g=c*(A*(A*(A*b+b)+b))

To see this redo the two examples using the flops command

A=[2 1;0 -2]
c=[1 2]
b=[1 -1]'
flops(0)
f=c*(A^3+A^2+A)*b
f_flops=flops;
flops(0)
g=c*(A*(A*(A*b+b)+b))
g_flops=flops;
disp([f_flops g_flops])

The basis for this difference is Horner's method (or synthetic division).

• Another useful set of operations on matrices are the ``hadamard'' operations or elementwise operations:

 
C=A.*B  has entries c(i,j)=a(i,j)b(i,j)
C=A./B  has entries c(i,j)=a(i,j)/b(i,j)
C=A.\B  has entries c(i,j)=b(i,j)/a(i,j)
C=A.^B  has entries c(i,j)=a(i,j)^{b(i,j)}
C=A.^r  has entries c(i,j)=a(i,j)^r where r is a number.
C=r.^A" has entries c(i,j)=r^{a(i,j)}

For example,

A=[2 1;3 -2]
B=[-4 1;3 2]
C=A./B
D=A.^B
E=A.^2
F=2.^B

• Built-in functions in Matlab can be applied to any matrix (recall that a number is just a one by one matrix). The result might not be what you expect.

sin(A*pi/2)
exp(B*pi*i)
log(B)

As you can see these operations also take place componentwise; that is, the function is applied to each entry. This is very useful for numerical programming.