Consider the matrix

$$A_2:= \begin{pmatrix} a & b_1 \\ b_2 & a\end{pmatrix}.$$

Let $\sigma_2 = \begin{pmatrix} 0 & i \\ -i & 0 \end{pmatrix}$, then

$$\sigma_2 A_2 \sigma_2 = \begin{pmatrix} a & -b_2 \\ -b_1 & a\end{pmatrix}.$$

I wonder if I have the matrix

$$A_3:= \begin{pmatrix} a & b_1 & 0 \\ b_2 & a & c_1 \\ 0 & c_2 & a \end{pmatrix}$$ if there is an analogous matrix

$\sigma_3$ that works for all possible choices of coefficients in $A_3$ such that

$$\sigma_3 A_3 \sigma_3^* = \begin{pmatrix} a & -b_2 & 0 \\ -b_1 & a & -c_2 \\ 0 & -c_1 & a \end{pmatrix}.$$

That there exists one such matrix for each set of coefficients is clear, since the eigenvalues of the two $3x3$ matrices are the same. I am looking for one that works for all choices.

Consider the matrix

$$A_2:= \begin{pmatrix} a & b_1 \\ b_2 & a\end{pmatrix}.$$

Let $\sigma_2 = \begin{pmatrix} 0 & i \\ -i & 0 \end{pmatrix}$, then

$$\sigma_2 A_2 \sigma_2 = \begin{pmatrix} a & -b_2 \\ -b_1 & a\end{pmatrix}.$$

I wonder if I have the matrix

$$A_3:= \begin{pmatrix} a & b_1 & 0 \\ b_2 & a & c_1 \\ 0 & c_2 & a \end{pmatrix}$$ if there is an analogous matrix

$\sigma_3$ that works for all possible choices of coefficients in $A_3$ such that

$$\sigma_3 A_3 \sigma_3^{-1} = \begin{pmatrix} a & -b_2 & 0 \\ -b_1 & a & -c_2 \\ 0 & -c_1 & a \end{pmatrix}.$$

That there exists one such matrix for each set of coefficients is clear, since the eigenvalues of the two $3x3$ matrices are the same. I am looking for one that works for all choices.