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{align} 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{align}

  • $(A^{T})^{T}$ = 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{align} A = \frac{A + A^{\theta}}{2} + \frac{A - A^{\theta}}{2} \end{align}
  • Absolute value of the determinant of a unitary matrix is 1 \begin{align} AA^{\theta} &= I\newline \detm{AA^{\theta}} &= \detm{A} \detm{A^{\theta}} = \detm{A}^{2} = 1 \end{align}