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I gave a detailed answer to a fuzzy in the link below:

(But compared to this question, the 39515 question qualifies as a detailed exposition.)

The steps would be:

• check the characteristic polynoials, if there is no match, we can stop here.
• build the eigenvalues, sort them somehow, if there is no match, stop here.
• build the jordan decomposition and match the eigenvectors, allowing rescaling on the one side, and permutations of rows on the other side. A rough match can be done by first rescaling in the two Jordan base change matrices, so that the maximal absolute value occurs for the entry $1$ in each column.

I gave a detailed answer to a fuzzy question by Anonymous in the link below:

• build the jordan decomposition and match the eigenvectors, allowing rescaling on the one side, and permutations of rows on the other side. A rough match can be done by first rescaling in the two Jordan base change matrices, so that the maximal absolute value occurs for the entry $1$ in each column. column.