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1 Matrix Operations

1.1 Multiplication

In Linear Algebra 2 matrices can be multiplied by each only if the inner dimensions are equal.

$LaTeX: A=BC$

where

$LaTeX: B$ is a matrix of n rows and m columns

$LaTeX: C$ is a matrix of m rows and o columns

then $LaTeX: A$ must be a matrix of n rows and o columns.

Notice that C has the same number of rows (m) and B does columns - the inner dimensions of the product. This also implies that the order of multiplication is important.

To multiply 2 matrices in MATLAB use * operator

 >> a = [1, 2 ; 3, 4]
      ans = 
        1     2
        3     4
 >> b = [3, 4 ; 5, 6];
 >> a * b
      ans =
        13    16
        29    36
 >> c = [3, 4, 5 ; 6, 7, 8];
 >> a * c
      ans =
        15    18    21
        33    40    47
        

1.2 Division

Division is more constrained than multiplication. In Algebra division is simple

$LaTeX: a=bc$

$LaTeX: \frac{a}{c}=b$

In Linear Algebra the order of the multiplication must be preserved. So

$LaTeX: A=BC$

$LaTeX: AC^{-1}=BCC^{-1} \ne C^{-1}BC$

and

$LaTeX: B^{-1}A=B^{-1}BC=IC=C$

where

$LaTeX: I$ is the identity matrix.

In order to do division the matrix "in the denominator" must be invertable. Not all matrices are invertable. First and foremost the matrix must be square since $LaTeX: A^{-1}A=I$ For those matrices that are not invertable the pseudoinverse can be used. In MATLAB the pseudoinverse can be found using the pinv command

 >> a = [1, 2 ; 3, 4];
 >> a^-1
      ans =
       -2.0000    1.0000
        1.5000   -0.5000
 >> pinv(a)
      ans =
       -2.0000    1.0000
        1.5000   -0.5000

These 2 operations lead to the same results since a is invertable. However, c is not

 >> c = [3, 4, 5 ; 6, 7, 8];
 >> c^-1
      ??? Error using ==> mpower
      Matrix must be square.
 >> pinv(c)
      ans =
       -1.2778    0.7778
       -0.1111    0.1111
        1.0556   -0.5556

See this sie on pseudoinverses for more information.

$LaTeX: A^+=\left( A^T A \right)^{-1} A^T$

'Pseudoinverse'

1.3 Squaring a matrix

Remember that squaring any value is the same as

$LaTeX: X^2=X*X$

In order to square a matrix in Linear Algebra the matrix must be square (same number of rows and columns) so that the inner dimensions are equal. Given a square matrix you can then take that matrix to some power in MATLAB using the ^ operator. For example

 >> a = [1, 2 ; 3, 4];
 >> a^2
      ans =
         7    10
        15    22

If, however, you wish to square the value in each element then use the . and ^ operators together

 >> a.^2
      ans =
        1     4
        9    16

The period before the ^ operator tells MATLAB to perform the operation element by element. The . operator works with division, multiplication, and other operators.

1.4 Determinant

To get the determinant of a matrix use the det command on that matrix, as follows

 >> a = [1, 2 ; 3, 4];
 >> det(a)
        ans = -2

1.5 Transpose

The transpose of a matrix is that matrix with the rows and columns swapped. To get the transpose of a matrix use the transpose command or the ' operator.

 >> a = [1, 2 ; 3, 4]
        ans = 
          1     2
          3     4 
 >> transpose(a)
        ans =
          1     3
          2     4

or

 >> a'
        ans =
          1     3
          2     4