📂Matrix Algebra

Transpose Matrix

Definition¹

Let’s consider a matrix of size $m\times n$ as $A$. The matrix obtained by swapping the rows and columns of $A$ is called the transpose of $A$ and is denoted by $A^{\mathsf{T}}$ or $A^{T}$, $A^{t}$.

Description

Following the definition, if $A$ is a $m \times n$ matrix then $A^{\mathsf{T}}$ will be a $n \times m$ matrix. Also, the $i$th row of $A$ is the same as the $i$th column of $A^{\mathsf{T}}$ and vice versa.

$$A=\begin{bmatrix} 10 & 0 & 3 \\ 0 & 8 & 22 \end{bmatrix} ,\quad A^{\mathsf{T}} = \begin{bmatrix} 10 & 0 \\ 0 & 8 \\ 3 & 22 \end{bmatrix}$$

It can be considered as being symmetric with respect to the main diagonal.

Properties

Let’s assume $r,s\in \mathbb{R}$ and $A,B$ are sizes that make the matrix operations well-defined in each case. Then, the following hold:

(a) Linearity: $$\left( rA + sB\right)^{\mathsf{T}}=r A^{\mathsf{T}} + s B^{\mathsf{T}}$$

(b) The transpose of a product is equal to the product of the transposes in reverse order.

$$(AB)^{\mathsf{T}}=B^{\mathsf{T}}A^{\mathsf{T}}$$

(b’) The transpose of the product of several matrices is equal to the product of the transposes of those matrices in reverse order.

$$\left( A_{1} A_{2}\cdots A_{n} \right)^{\mathsf{T}} = A_{n}^{\mathsf{T}} \cdots A_{2}^{\mathsf{T}} A_{1}^{\mathsf{T}}$$

Proof

(b)

For the $m\times n$ matrix $A$ and the $n\times k$ matrix $C$

$$\begin{align*} \left[ { \left( AC \right) }^{\mathsf{T}}\right] _{ km } &= \sum _{ i=1 }^{ n }{ [A] _{ m i } { [C] } _{ i k } } \\ &= \sum _{ i=1 }^{ n }{ { \left[ { A }^{\mathsf{T}}\right] } _{ i m } { \left[ { C }^{\mathsf{T}}\right] } _{ k i } } \\ &= \sum _{ i=1 }^{ n }{ { \left[ { C }^{\mathsf{T}}\right] } _{ k i }{ \left[ { A }^{\mathsf{T}}\right] } _{ i m } } \\ &= { \left[ { C }^{\mathsf{T}}{ A }^{\mathsf{T}}\right] } _{ km } \end{align*}$$

Therefore, if each element is the same, then the matrices are equal and thus the following equation holds.

$$\left( AC \right) ^{\mathsf{T}}= { C }^{\mathsf{T}}{ A }^{\mathsf{T}}$$

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(b')

This is a corollary to (b).

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