Matrix Operations

Each of these operators works pairs of matrices with appropriate dimensions or matrices and scalars.

Plus +

If both operands are matrices of the same dimensions, the matrix entires are added elemement wise.

>> [1 2; 3 4] + [2 3; 5 7]

ans =

     3     5
     8    11

If either operand is a scalar, that scalar is added to each entry element in the other matrix:

>> [1 2; 3 4] + 5

ans =

     6     7
     8     9

In order to add a scalar r to the diagonal elements of a matrix A, use A + r*eye(size(A)).

Minus -

If both operands are matrices of the same dimensions, the matrix entires are added elemement wise.

>> [1 2; 3 4] - [2 3; 5 7]

ans =

    -1    -1
    -2    -3

If the first operand is a scalar, each entry in the second matrix is subtracted from that scalar. If the second operand is a scalar, that scalar is subtracted from each element in the first matrix.

>> 5 - [1 2; 3 4]

ans =

     4     3
     2     1

>> [1 2; 3 4] - 5

ans =

    -4    -3
    -2    -1

In order to subtract a scalar r from the diagonal elements of a matrix A, use A - r*eye(size(A)).

Matrix Times *

You can take the prodcut of two matrices A and B if the column dimension of the first matrix equals the row dimension of the second. That is, size( A, 2 ) == size( B, 1 ).

Matrix multiplication is defined such that given a column vector v with length equal to the row dimension of B, then we define A*B so that A*(B*v) == (A*B)*v.

>> [1 2; 3 4] * [2 3; 5 7]

ans =

    12    17
    26    37

>> [1 2; 3 4] * [2 3]'

ans =

     8
    18

If either operand is a scalar, each entry in the other matrix is multiplied by that scalar.

Matrix Power ^

At least one operand must be a scalar, and any matrix must be square. I have not yet found a use for raising a scalar to a matrix.

>> [1 2; 3 4]^3

ans =

    37    54
    81   118

>> [1 2; 3 4]^1.1

ans =

   1.2742 - 0.0793i   2.3245 + 0.0363i
   3.4867 + 0.0544i   4.7609 - 0.0249i

Matrix Right Divide /

A/B is equivalent to (B'\A')'. The column dimensions must be equal, that is, size( A, 2 ) == size( B, 2 ).

>> [1 2; 3 4] / [2 3]

ans =

    0.6154
    1.3846

>> [1 2; 3 4] / [2 3; 5 7]

ans =

    3.0000   -1.0000
   -1.0000    1.0000

Matrix Left Divide \

A\B is equivalent to inv(A)*B. The row dimensions must be equal, that is, size( A, 1 ) == size( B, 1 ).

>> [1 2; 3 4] \ [2; 3]

ans =

   -1.0000
    1.5000

>> [1 2; 3 4] \ [2 3; 5 7]

ans =

    1.0000    1.0000
    0.5000    1.0000

Complex Transpose '

Transpose .'

The transpose operator .' takes an m × n matrix A and creates an n × m matrix B such that A(i, j) = B(j, i).

>> A = [1 2; 3 4; 5 6]

ans =

     1     2
     3     4
     5     6

>> A.'

ans =

     1     3     5
     2     4     6

Element-wise Operations

Each of these operators works element wise. If either operand is a scalar r, that scalar is interpreted as the matrix r*ones(m, n) where m and n are the dimensions of the other operand.

Times .*

If both operands are matrices of the same dimensions, the matrix entires are multiplied elemement wise.

>> [1 2; 3 4] .* [2 3; 5 7]

ans =

     2     6
    15    28

Power .^

If both operands are matrices of the same dimensions, the elements in the first matrix are raised to the power of the corresponding elements in the second matrix.

>> [1 2; 3 4] .^ [2 3; 5 7]

ans =

           1           8
         243       16384

>> [1 2; 3 4] .^ 2

ans =

     1     4
     9    16

>> 2 .^ [2 3; 5 7]

ans =

     4     8
    32   128

Right Divide ./

If both operands are matrices of the same dimensions, the elements in the first matrix are divided by the corresponding elements in the second matrix.

>> [1 2; 3 4] ./ [2 3; 5 7]

ans =

    0.5000    0.6667
    0.6000    0.5714

Left Divide .\

If both operands are matrices of the same dimensions, the elements in the second matrix are divided by the corresponding elements in the first matrix.

>> [1 2; 3 4] .\ [2 3; 5 7]

ans =

    2.0000    1.5000
    1.6667    1.7500