6.01 Types of matrices

This table of information can also be represented as a matrix. In mathematics, a matrix is a particular method of displaying information. It is any rectangular array of numbers, symbols, or expressions arranged in rows and columns. So the table above would be represented by a matrix, which we can call A and is shown below.A=\begin{bmatrix} 13 & 6 \\ 3 & 10 \\ 8 & 9 \end{bmatrix}

We refer to the dimensions or order of a matrix as a reference to the number of rows and number of columns.

A matrix with dimensions m\times n has m rows and n columns. For instance, the following matrix has dimensions 3\times 4:\begin{bmatrix} 3 & 4 &5&3 \\ 1 & 0 & 7&0 \\ 6 & 0 &4 &0 \end{bmatrix}

Elements are the individual entries of a matrix. An element can be identified by its position (that is, its row and column) in the matrix. For the following matrix B, the elements in the second row and third column is 7, where we use the following notation b_{23}=7.B=\begin{bmatrix} 1&3&0&1 \\ 4&5&7&8 \end{bmatrix}

Generally, we may represent any matrix with m rows and n columns as shown:A=\begin{bmatrix} a_{11}&a_{12}&a_{13}&a_{14}&...&a_{1n} \\ a_{21}&a_{22}&a_{23}&a_{24}&...&a_{2n} \\ .&.&.&.&...&. \\ .&.&.&.&...&. \\ .&.&.&.&...&. \\ a_{m1}&a_{m2}&a_{m3}&a_{m4}&...&a_{mn} \end{bmatrix}

Examples

A row matrix or row vector has just a single row. The following matrix T is an example of a row matrix.T=\begin{bmatrix} 3&4&5 \end{bmatrix}

A column matrix or column vector has just a single column. The following matrix M is an example of a column matrix.M=\begin{bmatrix} 5\\3\\0\\11 \end{bmatrix}

A square matrix has an equal number of rows and columns. The matrices G and J are examples of square matrices.G=\begin{bmatrix} 1&5\\ 5&19 \end{bmatrix}, \quad J=\begin{bmatrix} 2&0&-1\\ 16&\sqrt{2}&5\\ 0.1&3&-7 \end{bmatrix}

A diagonal matrix is a square matrix with zero elements off of the leading diagonal or main diagonal. The leading diagonal represents the elements along the diagonal starting at the top-left to the bottom-right. The matrix C is an example of a diagonal matrix.G=\begin{bmatrix} 6&0&0\\ 0&-7&0\\ 0&0&\frac{4}{7} \end{bmatrix}

The identity matrix is a special type of diagonal matrix where all the elements on the main diagonal are ones. For example: N=\begin{bmatrix} 1&0&0\\ 0&1&0\\ 0&0&1 \end{bmatrix}

Matrix N is also called a binary matrix as it consists entirely of 1's and 0's.

A zero matrix is a matrix of any dimension where all of the elements are zero. For example: \begin{bmatrix} 0&0\\ 0&0\\ \end{bmatrix}.

Examples