Skip to content

Algebraic Properties of Matrices

Trace

For any square matrix, the trace is defined as the sum of all the diagonal elements. Specifically, for a matrix \(A\) of size \(n\), \(trace(A) = \sum_{i=1}^{n} a_{ii}\). Trace is also denoted by \(Tr()\). * Tr(\(\lambda\)A) = \(\lambda\)Tr(A) * Tr(A + B) = Tr(A) + Tr(B) * Tr(AB) = Tr(BA)

The last statement can be proved by writing out the product.

Transpose

Let A be a \(m \times n\) matrix. Then the \(n \times m\) matrix obtained by switching the rows and columns is called the transpose and is denoted by \(A^{T}\).

\[ \begin{aligned} A &= \begin{bmatrix} 1 &2\newline 3 &4\newline 5 &6\newline \end{bmatrix}\newline A^{T} &= \begin{bmatrix} 1 &3 &5\newline 2 &4 &6 \end{bmatrix} \end{aligned} \]
  • \((A^{T})^{T}\) = A
  • \((A + B)^{T}\) = \(A^{T} + B^{T}\)
  • \((AB)^{T}\) = \(B^{T}A^{T}\)
  • \((ABC)^{T}\) = \(C^{T}B^{T}A^{T}\)
  • \((kA)^{T}\) = \(kA^{T}\) (k being any complex number)

Conjugate Matrix

The matrix obtained by taking complex conjugate of all the elements of the given matrix A is called its conjugate matrix and is denoted by \(\bar{A}\).

The real numbers remain unchanged, but for the complex numbers, the sign of the imaginary part is reversed (\(2 + 3i\) becomes \(2 - 3i\)).

  • \(\bar{(\bar{A})} = A\)
  • \(\detm{\bar{A}} = \bar{\detm{A}}\)
  • \(\bar{(A + B)} = \bar{A} + \bar{B}\)
  • \(\bar{(kA)} = \bar{k}\bar{A}\)
  • \(\bar{(AB)} = \bar{A}\bar{B}\)
  • \(\bar{A} = A\) if A is real

Transposed conjugate of a matrix is defined as \((\bar{A})^{T}\) and is denoted by \(A^{\theta}\) or \(A^{\*}\).

  • \((A^{\theta})^{\theta} = A\)
  • \(\detm{A^{\theta}} = \detm{A}\)
  • \((A + B)^{\theta} = A^{\theta} + B^{\theta}\)
  • \((kA)^{\theta} = \bar{k}A^{\theta}\)
  • \((AB)^{\theta} = B^{\theta}A^{\theta}\) (order is reversed because of the transpose)

Complex Matrices

  • Hermitian Matrix: \(A^{\theta} = A\)
  • Skew-Hermitian Matrix: \(A^{\theta} = -A\)
  • Unitary Matrix: \(A^{\theta} = A^{-1}\) or \(AA^{\theta} = I\)

Properties * \((A + B)^{\theta} = A^{\theta} + B^{\theta}\) * \((AB)^{\theta} = B^{\theta}A^{\theta}\) * The diagonal elements of a Hermitian matrix are necessarily real (otherwise the conjugate will change the values) * Every square matrix A can be written a sum of Hermitian and Skew-Hermitian matrix

\[ \begin{aligned} A = \frac{A + A^{\theta}}{2} + \frac{A - A^{\theta}}{2} \end{aligned} \]
  • Absolute value of the determinant of a unitary matrix is 1
\[ \begin{aligned} AA^{\theta} &= I\newline \detm{AA^{\theta}} &= \detm{A} \detm{A^{\theta}} = \detm{A}^{2} = 1 \end{aligned} \]