# Questions tagged [similar-matrices]

Two $n\times n$ matrices $A,B$ are similar if there exists some non-singular matrix $P$ such that $A=PBP^{-1}$. Do NOT use this tag when referring to similarity between matrices based on distance or another norm. Use this tag when the question involves similarity between matrices, or conjugacy in the General Linear Group of invertible matrices.

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### In what sense are similar matrices "the same," and how can this be generalized?

I sort of intuitively see why we care about similar matrices, i.e., when $A=S^{-1}BS$ for some invertible matrix $S$. But I want to make this intuition more precise and abstract. Matrices: First of ...
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### Exercise 12, Section 7.2 of Hoffman’s Linear Algebra

Let $F$ be a subfield of the field of complex numbers, and let $A$ and $B$ be $n\times n$ matrices over $F$. Prove that if $A$ and $B$ are similar over the field of complex numbers, then they are ...
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Prove that if $A$ and $B$ are $3\times 3$ matrices over the field $F$, a necessary and sufficient condition that $A$ and $B$ be similar over $F$ is that they have the same characteristic polynomial ...