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 \(I_{n}\).

\[ \begin{aligned} I_{3} = \begin{bmatrix} 1 &0 &0\newline 0 &1 &0\newline 0 &0 &1 \end{bmatrix} \end{aligned} \]

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

\[ \begin{aligned} A = \frac{A + A^{T}}{2} + \frac{A - A^{T}}{2} \end{aligned} \]

Orthogonal Matrix¶

Orthogonal or orthonormal matrix is a matrix whose rows and columns are orthonormal vectors. An orthogonal matrix \(Q\) will satisfy

\[ \begin{aligned} QQ^{T} &= Q^{T}Q = I\newline Q^{-1} &= Q^{T}\newline det(Q) &= \pm 1 \end{aligned} \]

The last one follows from the fact that

\[ \begin{aligned} 1 = det(I) = det(QQ^{T}) = det(Q)^{2} \end{aligned} \]

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