$\require{cancel} \newcommand{\Ket}[1]{\left|{#1}\right\rangle} \newcommand{\Bra}[1]{\left\langle{#1}\right|} \newcommand{\Braket}[1]{\left\langle{#1}\right\rangle} \newcommand{\Rsr}[1]{\frac{1}{\sqrt{#1}}} \newcommand{\RSR}[1]{1/\sqrt{#1}} \newcommand{\Verti}{\rvert} \newcommand{\HAT}[1]{\hat{\,#1~}} \DeclareMathOperator{\Tr}{Tr}$

Matrix Types and Properties

^{First created in March 2019}

Symbols

In this context, the type of matrices is implied in the symbol:

• $N$ is normal matrix ($NN^\dagger=N^\dagger N$.)
  
  

• $U$ is unitary matrix ($U^\dagger=U^{-1}$, so $UU^\dagger=U^\dagger U=I$ and therefore also normal.)
  
  

• $H$ is Hermitian matrix ($H^\dagger=H$, so $HH^\dagger=H^\dagger H$ and therefore also normal.)
  
  

• $P$ is invertible matrix (full rank with independent column vectors.)
  
  

• $D$ is invertible diagonal matrix (i.e. diagonal elements are non-zero.) $D_0$ is general diagonal matrix (zero diagonal elements allowed.)
  
  

• $\Lambda$ is invertible diagonal matrix with (non-zero) eigenvalues of the subject matrix on its diagonal ($\Lambda_{ii}=\lambda_i$). $\Lambda_0$ allows zero eigenvalues. Both $\Lambda$ and $\Lambda_0$ allow degeneracy (nondistinct eigenvalues).
  
  

• $M$ is idempotent matrix ($M^2=M~\Rightarrow M^n=M$)
  
  

• $S$ is involutory matrix ($S=S^{-1}$, i.e. $\sqrt{I}$)

When we use the lowercase symbol, it means a column vector of the corresponding matrix.
e.g. $\Ket{p_i}$ is the $i^\mathrm{th}$ column vector of $P$. That is $\Bra{p_i}$ is the $i^\mathrm{th}$ row of $P^\dagger$.

When we say basis $B$, it means a basis formed by $B$'s column vectors.

When we say $A\in\{X\}$, it means $A$ belongs to class of $X$ as implied by the simbol. e.g. $A\in\{N\}$ means $A$ is normal.

When the converse is true, i.e. the result is sufficient for the condition, we write (suf.)

Properties

Here matrix $A$ is said to be:

• Diagonalisable: $A=PDP^{-1}$
  
  

• Unitarily diagonalisable: $A=UDU^{-1}$
  
  

• Unit-length: $\lvert z\rvert=1,~z\in\mathbb{C}$
  
  

• isometry or length-preserving: $\big\lVert A\Ket\psi\big\rVert=\big\lVert\Ket\psi\big\rVert$

Unitary Matrix

$UU^\dagger=U^\dagger U=I.$

• $\{\Ket{u_i}\}$ orthonormal (suf.)

• $\{\Bra{u_i}\}$ orthonormal (suf.)

• Isometry (suf.)

• Eigenvalues unit-length

• Normal - $U\in\{N\}.$

Normal Matrix

$NN^\dagger=N^\dagger N.$

• Unitarily diagonalisable

• Eigenvectors orthogonal

• $\sum_{i,j}\lvert a_{ij}\rvert^2=\sum_k\lvert\lambda_k\rvert^2.$

Properties	$N$	$U$	$H$
Is also		$N$	$N$
Diagonisable	?Y	?Y	?Y
Unitarily Diagonisable		?Y	?Y
Invertible
Columns of unit-length
Columns orthogonal
Rows of unit-length
Rows orthogonal
Isometry (preserve length)
Eigenvectors orthogonal
Eigenvalues norm 1
Eigenvalues real