7.02 Matrix operations

When we add or subtract two matrices A and B, we add or subtract the corresponding elements. So we can only add or subtract matrices that have the same dimensions. In general, the sum and difference of two matrices can be represented by the following identities. \begin{bmatrix} a&b\\ c&d \end{bmatrix} + \begin{bmatrix} e&f\\ g&h \end{bmatrix} = \begin{bmatrix} a+e&b+f\\ c+g&d+h \end{bmatrix} and\begin{bmatrix} a&b\\ c&d \end{bmatrix} - \begin{bmatrix} e&f\\ g&h \end{bmatrix} = \begin{bmatrix} a-e&b-f\\ c-g&d-h \end{bmatrix}\\

Examples

A scalar is a quantity, or a magnitude. When we multiply a matrix by a scalar, we multiply each element in the matrix by the given scalar. In general, the multiplication of a matrix by a scalar will appear similar the following: \lambda \begin{bmatrix} a&b\\ c&d \end{bmatrix}= \begin{bmatrix} \lambda a & \lambda b\\ \lambda c & \lambda d\\ \end{bmatrix}

As an example, the multiplication of a matrix with numerical elements by a scalar will look like the following: 3 \begin{bmatrix} 2&-2\\ 6&0 \end{bmatrix}= \begin{bmatrix} 6 & -6\\ 18 & 0\\ \end{bmatrix} We can see that each element was multiplied by 3.

Examples

The transpose of a matrix is made by interchanging the rows and columns. The transpose of matrix A is written as A^{T}. For example, the first row of matrix A becomes the first column of A^{T}, the second row of A becomes the second column of A^{T}, and so on. If a_{mn} is an element of matrix A, then this is equal to a^{T}_{nm} of the matrix A^{T}. This can be represented by the following identity.

\begin{bmatrix} a & b & c \\ d & e & f \end{bmatrix}^{T} = \begin{bmatrix} a & d \\ b & e \\ c & f \end{bmatrix}

As an example, the transpose of a matrix with numerical elements will look like the following:

\begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{bmatrix}^{T} = \begin{bmatrix} 1 & 4 \\ 2 & 5 \\ 3 & 6 \end{bmatrix}

It's also easy to verify that the transpose of the matrix A^T returns the original matrix A. For instance, if we take the transpose of the resulting matrix above, we get the same matrix we started with.

\begin{bmatrix} 1 & 4 \\ 2 & 5 \\ 3 & 6 \end{bmatrix}^T = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{bmatrix}

If a matrix is symmetric, then the transpose of the matrix returns itself. This is because the element in the mth row and the nth column is equal to the element in the nth row and the mth column.

A= \begin{bmatrix} 1 & 2 & 3 \\ 2 & 0 & -4 \\ 3 & -4 & 9 \end{bmatrix} = A^T

Lastly the transpose of a matrix is distributive, meaning that given two matrices A and B, then we have the following identity:

\left(A+B\right)^T=A^T+B^T

To illustrate this, consider the following matrices A and B and their transpose.

A= \begin{bmatrix} 1 & 2 & 5 \\ 2 & 1 & -4 \\ 4 & 0 & 9 \end{bmatrix}, B= \begin{bmatrix} -1 & 2 & -2 \\ 1 & 0 & 0 \\ 3 & 3 & 5 \end{bmatrix}

A^T= \begin{bmatrix} 1 & 2 & 4 \\ 2 & 1 & 0 \\ 5 & -4 & 9 \end{bmatrix}, B^T= \begin{bmatrix} -1 & 1 & 3 \\ 2 & 0 & 3 \\ -2 & 0 & 5 \end{bmatrix}

If we add each pair of matrices, that is, if we compute A+B and A^T+B^T then we can see that they are transpose of one another.

A + B= \begin{bmatrix} 0 & 4 & 3 \\ 3 & 1 & -4 \\ 7 & 3 & 14 \end{bmatrix}, A^T+B^T= \begin{bmatrix} 0 & 3 & 7 \\ 4 & 1 & 3 \\ 3 & -4 & 14 \end{bmatrix}

The distributive property also works for subtraction, i.e. \left(A-B\right)^T=A^T-B^T.

Examples