How would you prove that $\begin{bmatrix} a & b \\ c & d \end{bmatrix}$ and $\begin{bmatrix} d & c \\ b & a \end{bmatrix}$ are similar? Their characteristic polynomials are identical, but this isn't sufficient, and without actually calculating the eigenvectors and the dimensions of the eigenspaces it seem we can't use the Jordan form.

In general, how would you determine two matrices are similar without numerical calculations, simply based on such relations between their entries?

Thanks in advance.

  • $\begingroup$ I don't think that there is any way to determine without calculation. $\endgroup$ – Chiranjeev_Kumar Aug 15 '15 at 5:11

It's enough to exhibit a matrix similarity.

Let $P=\begin{bmatrix} 0&1\\1&0\end{bmatrix}$. Note that $P=P^{-1}$.

Consider $P\begin{bmatrix} a&b\\c&d\end{bmatrix}P=\begin{bmatrix}d&c\\b&a\end{bmatrix}.$ Hence, your two matrices are similar.

In general, determining similarity between two matrices involves calculations in one form or another. If you don't want to use JCF or RCF, you should take notice of structure/pattern in the entries of the matrices you are comparing. One may exploit such patterns. In particular, in this problem, I observed that the entries of one matrix just get tossed around in new positions and no adding across rows or columns. I worked my way around that observation.


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