Back to References


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.