Special Matrices

Zero Matrices

The $m \times n$ zero matrix is denoted by $0_{mn}$ and has all entries equal to zero. For example, the $3 \times 4$ zero matrix is $$0_{34} = \begin{bmatrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{bmatrix}.$$

Identity Matrices

The $n \times n$ identity matrix is denoted by $I_n$ and has all entries equal to zero except for the diagonal, which is all 1. For example, the $4 \times 4$ identity matrix is $$I_4 = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}.$$

Diagonal Matrices

A $n \times n$ diagonal matrix has all entries equal to zero except for the diagonal entries. We typically use $D$ for diagonal matrices. For ecample, $4 \times 4$ diagonal matrices have the form: $$\begin{bmatrix} d_{11} & 0 & 0 & 0 \\ 0 & d_{22} & 0 & 0 \\ 0 & 0 & d_{33} & 0 \\ 0 & 0 & 0 & d_{44} \end{bmatrix}.$$

Triangular Matrices

A lower-triangular matrix is a square matrix that is entirely zero above the diagonal. We typically use $L$ for lower-triangular matrices. For example, $4 \times 4$ lower-triangular matrices have the form: $$L = \begin{bmatrix} \ell_{11} & 0 & 0 & 0 \\ \ell_{21} & \ell_{22} & 0 & 0 \\ \ell_{31} & \ell_{32} & \ell_{33} & 0 \\ \ell_{41} & \ell_{42} & \ell_{43} & \ell_{44} \end{bmatrix}.$$

An upper triangular matrix is a square matrix that is entirely zero below the diagonal. We typically use $U$ for upper-triangular matrices. For example, $4 \times 4$ upper-triangular matrices have the form: $$U = \begin{bmatrix} u_{11} & u_{12} & u_{13} & u_{14} \\ 0 & u_{22} & u_{23} & u_{24} \\ 0 & 0 & u_{33} & u_{34} \\ 0 & 0 & 0 & u_{44} \end{bmatrix}.$$

Properties of triagular matrices:

1. An $n \times n$ triangular matrix has $n(n-1)/2$ entries that must be zero, and $n(n+1)/2$ entries that are allowed to be non-zero.
2. Zero matrices, identity matrices, and diagonal matrices are all both lower triangular and upper triangular.

Permutation Matrices

A permuation matrix is a square matrix that is all zero, except for a single entry in each row and each column which is 1. We typically use $P$ for permutation matrices. An example of a $4 \times 4$ permutation matrix is $$P = \begin{bmatrix} 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{bmatrix}.$$

The properties of a permutation matrix are:

1. Exactly $n$ entries are non-zero.
2. Multiplying a vector with a permutation matrix permutes (rearranges) the order of the entries in the vector. For example, using $P$ above and $x = [1, 2, 3, 4]^T$, the product is $Px = [2, 4, 1, 3]^T$.
3. If $P_{ij} = 1$ then $(Px)_i = x_j$.
4. The inverse of a permutation matrix is its transpose, so $PP^T = P^TP = I$.

Block Form

A matrix in block form is a matrix partitioned into blocks. A block is simply a submatrix. For example, consider $$M = \begin{bmatrix} A & B \\ C & D \end{bmatrix}$$ where $A$, $B$, $C$, and $D$ are submatrices.

There are special matrices in block form as well. For instance, a block diagonal matrix is a block matrix whose off-diagonal blocks are zero matrices.