# Swap elements of a diagonal matrix

Suppose you have a diagonal matrix, e.g.,

$$\begin{bmatrix} 3 & 0 & 0 & 0 \\ 0 & 2 & 0 & 0 \\ 0 & 0 & 9 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}$$

What linear transformation (or any method, really) could you apply to get a different matrix, e.g.,

$$\begin{bmatrix} 2 & 0 & 0 & 0 \\ 0 & 3 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 9 \end{bmatrix}$$

Where we swap every odd indexed diagonal with the element at (that index + 1)

I think this problem is equivalent to rearranging the eigenvalues and eigenvectors of a matrix with each other. For example moving the eigenvalue $$3$$ from the vector $$\begin{bmatrix} 1 \\ 0 \\ 0 \\ 0\end{bmatrix}$$ to $$\begin{bmatrix} 0 \\ 1 \\ 0 \\ 0\end{bmatrix}$$

This seems like a trivial question, but I can't find a solution myself.

• Hint: permutation matrix. Oct 28, 2022 at 21:51
• To permute rows and columns you can multiply on the left and right by a permutation matrix. You can model all position swaps that way for any matrix, even rectangular. Oct 29, 2022 at 0:56

To swap the $$i$$th and $$j$$th entires along the diagonal of a diagonal matrix: Using $$E_{ij}$$ to denote the matrix with $$1$$ in the $$i$$th row and $$j$$th column and $$0$$ in all other positions, multiply your matrix on both sides by $$I - E_{ii} - E_{jj} + E_{ij} + E_{ji}$$. (Since this matrix is involutory, this can be viewed as conjugation.)
The matrix for swapping adjacent entries in pairs would look like $$\begin{pmatrix} A & 0 & \cdots & 0 \\ 0 & A & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & A \end{pmatrix}$$ in block form, where $$A = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$$.