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Define $T : M_{n×n}(\mathbb{R}) → M_{n×n}(\mathbb{R})$ by $T(A) := A^t$.

I know this transformation is linear and just takes a matrix and spits out it's transpose. I also know that the transpose is just a matrix with it's columns and rows swapped; however, I don't know how to form a matrix representation of this transformation for arbitrary $n$.

Any help to get me started would be appreciated!

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  • $\begingroup$ Linear transformations only operate on vector spaces Not on a matrix .right? $\endgroup$ Commented May 26, 2015 at 18:49
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    $\begingroup$ @ShubhamUgare $M_{n \times n}(\Bbb R)$ is the vector space of real $(n \times n)$-matrices. $\endgroup$
    – aexl
    Commented May 26, 2015 at 18:50
  • $\begingroup$ I hesitate to call this a duplicate, but this answer is certainly relevant. $\endgroup$ Commented May 26, 2015 at 18:52
  • $\begingroup$ I saw this post also but it didn't answer my question exactly. $\endgroup$ Commented May 26, 2015 at 19:09

5 Answers 5

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$\newcommand{\R}{\mathbb R}$ Just write what you would do for any linear transformation: write a basis for $M(n,\R)$ then find the "effect" of the transformation on its elements. In this case you'll formally have to go through an isomorphism between $M(n,\R)$ and $\R^{n^2}$: you want to represent it as a matrix, so it has to multiply some vectors. These vectors will represent the matrix in a given basis.

I'll make an example just in $M(2,\R)$, for the sake of simplicity. A general matrix in this space is $$ M= \begin{pmatrix} a & b \\ c & d \end{pmatrix}. $$ Now, let's choose a basis for the vector space: the obvious (but not the best*) choice is the standard basis $\mathcal S=\{E_{11},E_{12},E_{21},E_{22}\}$ given by $(E_{\mu\nu})_{ij}=\delta_{i\mu}\delta_{j\nu}$, so $$ S=\left\{ \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}, \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}, \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix}, \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix} \right\}. $$ In this basis, the matrix $M$ can be written as $aE_{11}+bE_{12}+cE_{21}+dE_{22}$ so it's easily represented as the vector in $\R^{2^2}=\R^4$ $$ v= \begin{pmatrix} a \\ b \\ c \\ d \end{pmatrix}. $$ Now, the transposition (I'll call it $T$) acts on the elements of the basis $\mathcal S$ as follows: $$ T(E_{11})=E_{11},\ T(E_{12})=E_{21},\ T(E_{21})=E_{12},\ T(E_{22})=E_{22}. $$ Therefore the matrix associated with $T$ is just $$ \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix}. $$


* The calculations were simple anyway, but there's a nicer way to do it (according to me, of course); it just requires a bit of knowledge a priori. The matrix vector space $M(n,\R)$ is a direct sum of the space $S(n,\R)$ of symmetric matrices and $A(n,\R)$ of antisymmetric ones: $$ M(n,\R)=S(n,\R)\oplus A(n,\R) $$ so a basis of $M(n,\R)$ can be found from these two subspaces. Moreover, $\dim A(n,\R)=\frac12(n-1)n$ and $\dim S(n,\R)=\frac12n(n+1)$.

Now, if $x\in A(n,\R)$ then $T(x)=-x$, and if $y\in S(n,\R)$ then $T(y)=y$, and this is obviously true for the elements of their bases too. So, we form a basis of $M(n,\R)$ consisting of $\frac12n(n+1)$ linearly independent elements from $S(n,\R)$ and $\frac12n(n-1)$ elements from $A(n,\R)$.

The matrix representing $T$ is diagonal in this basis, since symmetric and antisymmetric matrices are all "eigenvectors" of $T$ (and we have a basis of them). So $M$ will have 1 as the first $\frac12n(n+1)$ elements on the diagonal, and $-1$ on the remaining $\frac12n(n-1)$ ones.

I like this method because it is already general for any order $n$ of the matrices, without doing any calculations.

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  • $\begingroup$ What would you say is the best choice for the basis? $\endgroup$ Commented May 26, 2015 at 19:10
  • $\begingroup$ I'd say a basis in which the matrix is already diagonal. I'm editing the answer right now to include this ;-) $\endgroup$
    – yellon
    Commented May 26, 2015 at 19:14
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    $\begingroup$ Why is the transformation $4 \times 4$ if the range of the transformation is supposed to be $M_{2\times 2}$? $\endgroup$
    – mavavilj
    Commented Feb 28, 2016 at 19:40
  • $\begingroup$ The vector space of 2x2 matrix has four dimensions (roughly, "one for each component of the matrices), so the transposition is an endomorphisms of a 4-dimensionale vector space, and is represented by a 4x4 matrix. $\endgroup$
    – yellon
    Commented Feb 29, 2016 at 15:23
  • $\begingroup$ But if you were to apply T(x) by T(x) = Ax then you would do (4x4)*(2x2) right? $\endgroup$ Commented Sep 5, 2018 at 17:34
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Let $e_{ij}$ be the matrix with a value of $1$ at entry $(i,j)$ and zero elsewhere. This is a basis for the space $M_{n \times n}$. Then you can define your transpose operation on that space as follows:

$$T(e_{ij}) = e_{ji}$$

If you want to display this as a matrix you will need to come up with an arbitrary ordering of $\{e_{ij}\}$. Then you can use the above definition to find out which entries are $0$ or $1$.

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  • $\begingroup$ Will this transformation matrix change depending upon A ? $\endgroup$ Commented May 26, 2015 at 19:03
  • $\begingroup$ It will not. $A$ is an element of the space on which it acts. $\endgroup$
    – muaddib
    Commented May 26, 2015 at 19:04
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Here is another way to think about it.

Consider a square matrix $\mathbf{A}$ of dimension $d$ for simplicity. Then consider the standard basis of $d$ column unit vectors $\mathbf{e}_i$ with elements $(\mathbf{e}_i)_j = \delta_{ij}$. These are just column vectors with a single element set to $1$ and all other elements set to zero.

A matrix $\mathbf{A}$ with elements $a_{ij}$ can always be decomposed using the basis vectors as: $$ \mathbf{A} = \sum_{ij} a_{ij}\,\mathbf{e}_i \mathbf{e}_j^T. $$ In particular, the matrix $\mathbf{A}^T$ may be written as $$ \mathbf{A}^T = \sum_{ij} a_{ji}\,\mathbf{e}_i \mathbf{e}_j^T. $$ The elements $a_{ji}$ of $\mathbf{A}^T$ can also be expressed in terms of the unit vectors: $$ a_{ji} = \mathbf{e}_j^T \mathbf{A} \mathbf{e}_i. $$ Therefore, $$ \mathbf{A}^T = \sum_{ij} \left(\mathbf{e}_j^T \mathbf{A} \mathbf{e}_i\right)\,\mathbf{e}_i \mathbf{e}_j^T = \sum_{ij} \left(\mathbf{e}_i \mathbf{e}_j^T\right) \mathbf{A} \left(\mathbf{e}_i \mathbf{e}_j^T \right). $$ If we relabel the matrices $\mathbf{e}_i \mathbf{e}_j^T$ as $\mathbf{u}_k$, with $k = 1,\dots,d^2$, then we can write $$ \mathbf{A}^T = \sum_{k} \mathbf{u}_k \mathbf{A} \mathbf{u}_k. $$ The matrices $\mathbf{u}_k$ are just all matrices containing only one non-zero element equal to $1$.

The above expression is the expression for the linear transposition map in the original matrix space. Here, we don't need to reshape the matrix in vector form. We can simply multiply the matrix by the unit matrices $\mathbf{u}_k$ on the left and on the right.

It can then be shown that if the elements of the matrix $\mathbf{A}$ are ordered by column in standard order, $\mathcal{A} = \left(a_{11},a_{21},a_{31},\dots,a_{12},a_{22},a_{32},\dots\right)^T$, the linear transposition map takes the form $$ \mathcal{A}^T = \mathcal{T}\mathcal{A}, $$ with $$ \mathcal{T} = \sum_k \mathbf{u}_k^T \otimes \mathbf{u}_k. $$ Here, $\otimes$ is the Kronecker product as defined by, e.g., Matlab's kron( ) function. For $d=2$, we have $$ \begin{align} \mathcal{T} = \left( \begin{array}{cccc} 1 & 0 & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 1 \end{array} \right) \end{align} $$ For $d = 3$, we have $$ \begin{align} \mathcal{T} = \left( \begin{array}{ccccccccc} 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0\\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0\\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\\ \end{array} \right). \end{align} $$ For $d=4$, we have $$ \begin{align} \mathcal{T} = \left( \begin{array}{cccccccccccccccc} 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ \end{array} \right), \end{align} $$ and so on.

Notice that the transposition map is a permutation of the columns (or rows) of the identity matrix. The permutation is the permutation that groups the columns by jumps of $d$. The first four columns are columns $1,d+1,2d+1,\dots, (d-1)d+1$ of the identity matrix. The next four columns are columns $2,d+2,2d+2,\dots, (d-1)d+2$ of the identity matrix. And it goes like this in groups of $d$.

Here is the Matlab code that will give you the matrix for any dimension:

% Set the dimension of A
d = 4;
% Calculate the permutation of the columns of the identity operator
perm = (1:d^2).';
perm = reshape(perm,d,d).';
perm = perm(:).';
% Calculate the linear transposition map
T = eye(d^2);
T = T(:,perm);
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    $\begingroup$ Your $\mathcal{A}$ is just $\operatorname{vec}(\mathbf A)$, no? $\endgroup$ Commented Jul 1, 2021 at 4:59
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    $\begingroup$ It's just all columns of $\mathbf{A}$ stacked on top of each other to form a column vector, yes. As in the Matlab command mathcalA = mathbfA(:), if that helps. $\endgroup$
    – Ben
    Commented Jul 1, 2021 at 6:10
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Let's consider the case $n = 2$. We have the linear transformation $$ \begin{align*}T: M_{2 \times 2}(\Bbb R) &\to M_{2 \times 2}( \Bbb R) \\ A &\mapsto A^{\mathrm T} \end{align*} \; $$ Now we can identify the vector space $M_{2 \times 2}(\Bbb R)$ with $\Bbb R^{4} = \Bbb R^{2 \cdot 2}$ by identifiying $$ \begin{bmatrix} a & b \\ c & d \end{bmatrix} \in M_{2 \times 2}(\Bbb R) \quad \text{with} \quad \begin{bmatrix} a \\ b \\ c \\ d \end{bmatrix} \in \Bbb R^4 \; .$$ Now, we just have to find the images of the standard basis of $\Bbb R^4$: $$ \begin{bmatrix} 1 \\ 0 \\ 0 \\ 0 \end{bmatrix} \mapsto \begin{bmatrix} 1 \\ 0 \\ 0 \\ 0 \end{bmatrix} \, , \quad \begin{bmatrix} 0 \\ 1 \\ 0 \\ 0 \end{bmatrix} \mapsto \begin{bmatrix} 0 \\ 0 \\ 1 \\ 0 \end{bmatrix} \, , \quad \begin{bmatrix} 0 \\ 0 \\ 1 \\ 0 \end{bmatrix} \mapsto \begin{bmatrix} 0 \\ 1 \\ 0 \\ 0 \end{bmatrix} \, , \quad \begin{bmatrix} 0 \\ 0 \\ 0 \\ 1 \end{bmatrix} \mapsto \begin{bmatrix} 0 \\ 0 \\ 0 \\ 1 \end{bmatrix} \; .$$

So now we can represent the linear trasformation $T$ with the representation matrix $$ M_T := \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} \; .$$

So for $A = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \in M_{2 \times 2}(\Bbb R)$, we get the image of $A$ under $T$ by using our tranformation matrix $A$ by calculating $$ \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} \cdot \begin{bmatrix} a \\ b \\ c \\ d \end{bmatrix} = \begin{bmatrix} a \\ c \\ b \\ d \end{bmatrix} \; ,$$ and this vector is identified with $\begin{bmatrix} a & c \\ b & d \end{bmatrix} = A^{\mathrm T}$.

You can generalize this method for an arbitrary $n \in \Bbb N$.

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The transformation can as you said be written as a linear transformation, but in the vector representation of the matrix: $$ T(\mathrm{vec}(\mathbf(A))) = \mathbf{P}\mathrm{vec}(\mathbf(A)) $$ where $\mathbf{P}\in\mathbb{R}^{n^2\times n^2}$. The matrix $\mathbf{P}$ is a permutation matrix known as a stride permutation or a perfect shuffle matrix. It's a bit more complicated for $\mathbf{A}\in\mathbb{R}^{m\times n}$, but for the $2\times 2$ case this matrix is $$ \begin{pmatrix} 1 & 0 & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 1 \end{pmatrix}. $$ In general the matrix can be written as $\mathbf{I}_m\boxtimes \mathbf{I}_n$ where $\boxtimes$ is the box-product that for $\mathbf{C}\in\mathbb{R}^{m_1\times n_1}$ and $\mathbf{D}\in\mathbb{R}^{m_2\times n_2}$ is defined by $$ \mathbf{\mathbf{C}\boxtimes\mathbf{D}}_{(i-1)m_2+j,(k-1)n_1+l}=c_{il}d_{jk}. $$ The box-product is very similar to the Kronecker product.

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  • $\begingroup$ @ Peder , you want to write $vec(T(A))=P(vec(A))$. Your remark about the box-product is very interesting; yet , the index $l$ does not appear in the LHS. $\endgroup$
    – user91684
    Commented May 26, 2015 at 19:59
  • $\begingroup$ Sorry, my mistake. I had mistakenly typed $(k-1)n_1+i$ instead of $(k-1)n_1+l$. The entries on the LHS $(i-1)m_2+j$ can be thought of as the index of $X_{ij}$ in $\mathrm{vec}(X_{i,j})$, and perhaps a better way to write the LHS is $[\mathbf{C}\boxtimes\mathbf{D}]_{(ij),(kl)}$. I hope that makes it clearer. $\endgroup$
    – Peder
    Commented May 30, 2015 at 3:54

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