Special Matrices

Identity Matrix

A square matrix whose all elements are zeros except the diagonal elements which are all 1.

  • $AI = IA = A$
  • $I^{n} = I$
  • $I^{-1} = I = I^{T}$
  • $\detm{I} = 1$
  • The columns or rows form the basis of $\real^{n}$
  • All eigenvalues are same and equal to 1

Identity matrix of size $n \times n$ is denoted by $I_{n}$. \begin{align} I_{3} = \begin{bmatrix} 1 &0 &0\newline 0 &1 &0\newline 0 &0 &1 \end{bmatrix} \end{align}

Null Matrix

Any matrix with all elements 0 is called a null matrix (it need not be square). A square null matrix of size $n$ is denoted by $O_{n}$

  • $\detm{O_{n}} = 0$
  • A + O = O + A = A

Upper Triangular Matrix

A square matrix in which all the elements below the main diagonal are 0, i.e., $a_{ij} = 0 : \forall i > j$.

  • $\detm{A}$ = product of all the diagonal elements
  • eigenvalues(A) are the diagonal elements

Lower Triangular Matrix

A square matrix in which all the elements above the main diagonal are 0, i.e., $a_{ij} = 0 : \forall i < j$

  • $\detm{A}$ = product of all the diagonal elements
  • eigenvalues(A) are the diagonal elements

Idempotent Matrix

Any square matrix which satisfies $A^{2} = A$. Null matrix and identity matrix are examples of such matrices.

Involuntary Matrix

A square matrix which satisfies $A^{2} = I$.

Nilpotent Matrix

A matrix $A$ is said to be nilpotent of class $x$ if $x$ is the smallest index such that $A^{x} = O$ and $A^{x-1} \neq O$.

Singular Matrix

A square matrix is singular if its determinant is 0. Equivalently, a square matrix is non-singular if its determinant is non-zero. A singular matrix is not invertible.

Symmetrix Matrix

A square matrix is symmetric if $a_{ij} = a_{ji} \quad \forall i,j$. Or simply, if $A^{T} = A$.

For any matrix A

  • $AA^{T}$ is always symmetric
  • $(A + A^{T})/2$ is always symmetric
  • If A and B are symmetric, A + B and A - B are also symmetric

Positive Definite

A symmetric matrix is positive definite ($A > 0$) if all its eigenvalues are positive. Further, for such a matrix $x^{T}Ax > 0$ for all vectors $x$.

It is positive semi-definite ($A \geq 0$) if all its eigenvalues are non-negative. Further, for such a matrix $x^{T}Ax \geq 0$ for all vectors $x$.

If A is positive definite (semi-definite), then there exists $A^{1/2}$ that is also positive definite (semi-definite) such that $A^{1/2} A^{1/2} = A$. This follows from the fact that the eigenvectors of the matrix $A$ are orthogonal.

Skew Symmetric Matrix

A square matrix is skew symmetric if $a_{ij} = a_{ji} : \forall i,j$. This is equivalent to saying $A^{T} = -A$.

  • A skew symmetric matrix must have all zeros in the diagonal $(A = A^{T} = O)$
  • $(A - A^{T})/2$ is always skew symmetric

Any square matrix can be expressed as a sum of a symmetric and a skew symmetric matrix \begin{align} A = \frac{A + A^{T}}{2} + \frac{A - A^{T}}{2} \end{align}

Orthogonal Matrix

Orthogonal or orthonormal matrix is a matrix whose rows and columns are orthonormal vectors. An orthogonal matrix $Q$ will satisfy \begin{align} QQ^{T} &= Q^{T}Q = I\newline Q^{-1} &= Q^{T}\newline det(Q) &= \pm 1 \end{align}

The last one follows from the fact that \begin{align} 1 = det(I) = det(QQ^{T}) = det(Q)^{2} \end{align}

  • Orthogonal matrices play an important role in QR decomposition and SVD.
  • If A and B are orthogonal matrices, then AB and BA are also orthogonal
  • If matrix A is orthogonal, then $\detm{A} = \pm 1$, but the converse is not always true