Rank, trace, determinant, transpose, and inverse of matrices

Let ${\bf A}$ be an $m\times n$ square matrix:

$${\bf A}=[{\bf a}_1,\cdots,{\bf a}_n]=\left[ \begin{array}{c... ...\ a_{m1} & \cdots & a_{mn} \end{array} \right]_{m\times n}$$

where

$${\bf a}_j=\left[ \begin{array}{c} a_{1j} \vdots a_{mj} \end{array}\right], \;\;\;\;\;\;\;\;(j=1,\cdots,n)$$

is the jth column vector and $[a_{i1}\;\cdots\;a_{in}]$ is the ith row vector ($i=1,\cdots,m$). If $m=n$, ${\bf A}$ is a square matrix. In particular, if all entries of a square matrix are zero except those along the diagonal, it is a diagonal matrix. Moreover, if the diagonal entries of a diagonal matrix are all one, it is the identity matrix:

$${\bf I}=\left[\begin{array}{cccc} 1 & 0 & \cdots & 0 0 & ... ...\vdots & & \ddots & 0 0 & \cdots & 0 & 1 \end{array}\right]$$

• Rank

  The $m$ row vectors span the row space of ${\bf A}$ and the $n$ columns vectors span the column space of ${\bf A}$. The rank of each space is its dimension, the number of independent vectors in the space. The row and column spaces have the same rank, which is also the rank of matrix ${\bf A}$, i.e.:
  

  $$R=rank({\bf A}) \leq \min(m,n)$$
  
  

• Transpose

  The transpose of a matrix ${\bf A}$, denoted by ${\bf A}^T$, is obtained by switching the positions of elements $a_{ij}$ and $a_{ji}$ for all $i\in \{1,\cdots,m\}$ and $j\in\{1,\cdots,n\}$. In other words, the ith column of ${\bf A}$ becomes the ith row of ${\bf A}^T$, or equivalently, the ith row of ${\bf A}$ becomes the ith column of ${\bf A}^T$:
  

  $${\bf A}^T=[{\bf a}_1 \cdots {\bf a}_n]^T= \left[ \begin{array}{c} {\bf a}_1^T \vdots {\bf a}_n^T \end{array} \right]$$
  
  
  where vector ${\bf a}_i$ is the ith column of ${\bf A}$ and its transpose ${\bf a}_i^T$ is the ith row of ${\bf A}^T$.

  The properties of the transpose

  • $({\bf A}^T)^T={\bf A}$
  • $({\bf A}+{\bf B})^T={\bf A}^T+{\bf B}^T$
  • $({\bf A}{\bf B})^T={\bf B}^T{\bf A}^T$

  If ${\bf A}={\bf A}^T$, it is a symmetric matrix.

• Trace

  The trace of a square matrix $A$ is the sum of its diagonal elements:
  

  $$tr({\bf A}) = \sum_{i=1}^n a_{ii}$$
  
  
  The properties of the trace:
  • $tr(c{\bf A})=c\;tr({\bf A})$
  • $tr({\bf A}^T)=tr({\bf A})$
  • $tr({\bf A}+{\bf B})=tr({\bf B}+{\bf A})$
  • $tr({\bf A}{\bf B})=tr({\bf B}{\bf A})$
  • $tr({\bf A}^T{\bf B})=tr({\bf A}{\bf B}^T)$
  • $tr({\bf R}^{-1}{\bf A}{\bf R})=tr({\bf R}^{-1}({\bf A}{\bf R})) =tr(({\bf A}{\bf R}){\bf R}^{-1})=tr({\bf A})$

• Determinant

  The determinant of a square matrix $A$ is denoted by $det({\bf A}) = \vert {\bf A} \vert$, and $det({\bf A})\ne 0$ if and only if it is full rank, i.e., $rank({\bf A}) = n$.

  The properties of the determinant:

  • $det{\bf I}=1$
  • $det({\bf A}^T)=det({\bf A})$
  • $det({\bf A}^{-1})=det({\bf A})^{-1}$
  • $det({\bf A}{\bf B})=det({\bf A})\;det({\bf B})$
  • $det(c{\bf A})=c^n det({\bf A})$

• Inverse

  If ${\bf AB}={\bf BA}={\bf I}$, then ${\bf B}={\bf A}^{-1}$ is the inverse of ${\bf A}$. ${\bf A}^{-1}$ exists if and only if $det({\bf A}) \neq 0$, i.e., $rank({\bf A}) = n$.

  The properties of the inverse:

  • $({\bf A}^{-1})^{-1}={\bf A}$
  • $({\bf A}^T)^{-1}=({\bf A}^{-1})^T$
  • $det({\bf A}^{-1})=det({\bf A})^{-1}$
  • $({\bf AB})^{-1}={\bf B}^{-1}{\bf A}^{-1}$