Special Kinds of Matrices

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There are many matrices that have special forms and hence have special names — which we now list.

A square matrix is a matrix with the same number of rows and columns; that is, a square matrix is an $n\times n$ matrix.
A diagonal matrix is a square matrix whose only nonzero entries are along the main diagonal; that is, $a_{ij}=0$ if $i\neq j$ . The following is a $3\times 3$ diagonal matrix $\left (\begin {array}{ccc} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 3 \end {array} \right ).$ There is a shorthand in MATLAB for entering diagonal matrices. To enter this $3\times 3$ matrix, type diag([1 2 3]).
The identity matrix is the diagonal matrix all of whose diagonal entries equal $1$ . The $n\times n$ identity matrix is denoted by $I_n$ . This identity matrix is entered in MATLAB by typing eye(n).
A zero matrix is a matrix all of whose entries are $0$ . A zero matrix is denoted by $0$ . This notation is ambiguous since there is a zero $m\times n$ matrix for every $m$ and $n$ . Nevertheless, this ambiguity rarely causes any difficulty. In MATLAB, to define an $m\times n$ matrix $A$ whose entries all equal $0$ , just type A = zeros(m,n). To define an $n\times n$ zero matrix $B$ , type B = zeros(n).
The transpose of an matrix is the matrix obtained from by interchanging rows and columns. Thus the transpose of the matrix is the matrix Suppose that you enter this matrix into MATLAB by typing
```
     A = [2 1; -1 2; 3 -4; 5 7]
     
```
The transpose of a matrix is denoted by . To compute the transpose of in MATLAB, just type A.
A symmetric matrix is a square matrix whose entries are symmetric about the main diagonal; that is $a_{ij}=a_{ji}$ . Note that a symmetric matrix is a square matrix $A$ for which $A^t=A$ .
An upper triangular matrix is a square matrix all of whose entries below the main diagonal are $0$ ; that is, $a_{ij}=0$ if $i>j$ . A strictly upper triangular matrix is an upper triangular matrix whose diagonal entries are also equal to $0$ . Similar definitions hold for lower triangular and strictly lower triangular matrices. The following four $3\times 3$ matrices are examples of upper triangular, strictly upper triangular, lower triangular, and strictly lower triangular matrices: $\left (\begin {array}{rrr} 1 & 2 & 3\\ 0 & 2 & 4\\ 0 & 0 & 6\end {array}\right ) \quad \left (\begin {array}{rrr} 0 & 2 & 3\\ 0 & 0 & 4\\ 0 & 0 & 0\end {array}\right )$ $\left (\begin {array}{rrr} 7 & 0 & 0\\ 5 & 2 & 0\\ -4 & 1 & -3\end {array}\right ) \quad \left (\begin {array}{rrr} 0 & 0 & 0\\ 5 & 0 & 0\\ 10 & 1 & 0\end {array}\right ).$
A square matrix $A$ is block diagonal if $A = \left (\begin {array}{cccc} B_1 & 0 & \cdots & 0 \\ 0 & B_2 & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & B_k \end {array} \right )$ where each $B_j$ is itself a square matrix. An example of a $5\times 5$ block diagonal matrix with one $2\times 2$ block and one $3\times 3$ block is: $\left (\begin {array}{ccccc} 2 & 3 & 0 & 0 & 0\\ 4 & 1 & 0 & 0 & 0\\ 0 & 0 & 1 & 2 & 3\\ 0 & 0 & 3 & 2 & 4\\ 0 & 0 & 1 & 1 & 5\end {array}\right ).$