# 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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### Ambiguity in similarity definition

I am struggling to understand similarity of two matrices. I got that similar matrices represent the same linear map for different bases, but I have a question about it which I can't answer. Suppose I ...
1 vote
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### "Simple" proof that an n x n matrix is similar to its transpose

At the risk of getting voted down for duplicating A matrix is similar to its transpose or Proving an $n\times n$ matrix is similar to its transpose, or such, I didn't feel comfortable "answering&...
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### matrix similarity over a field shows matrix similarity over subfield

$M_n(F)$ is set of all $n\times n$ matrices over field $F$. Suppose we have $A,B\in M_n(\mathbb{R})$ and invertible matrix $P\in M_n(\mathbb{C})$ such that $P^{-1}AP=B$. Show that an invertible matrix ...
1 vote
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### Similar matrices and their column space

Let $A$ be similar to $B$ (so $A=PBP^{-1}$). If we have a vector $v$ that's in the column space of $A$, then how do we show that $P^{-1}v$ is in the column space of $B$? I'm trying to figure out how ...
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1 vote
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### Prove that A $\in M_3(\mathbb{R})$ is similar to $\begin{bmatrix} \cos(x) & -\sin(x) & 0 \\ \sin(x) & \cos(x) & 0 \\ 0 & 0 & \epsilon \end{bmatrix}$

Show that every orthogonal matrix A $\in M_3(\mathbb{R})$ is similar to a matrix $$\begin{bmatrix} cos(x) & -sin(x) & 0 \\\\ sin(x) & cos(x) & 0 \\\\ 0 & 0 & y\end{bmatrix}$$ ...
1 vote
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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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### Exercise 11, Section 7.2 of Hoffman’s Linear Algebra

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 ...
1 vote
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### Theorem 5, Section 7.2 of Hoffman’s Linear Algebra

Definition: $A\in M_{n\times n}(F)$ is in rational form if $$A=\begin{bmatrix} A_1& & \\ & \ddots & \\ & & A_r\\ \end{bmatrix}$$ where $A_i$ is companion matrix of non scalar ...
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### Block-diagonalization with unitary similarity transformations: ($A \rightarrow U B U^\dagger$, $B$ block-diagonal)

The problem Given a matrix $A$, find a unitary matrix $U$ such that $U^\dagger A U=B$, where $B$ is approximately block-diagonal (when possible). Explanation In other words, assume I'm given a matrix ...
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### If two matrices have different characteristic polynomials are they not similar?

This question is very basic, but I couldn't find a definitive answer. I know that if two matrices are similar, they have the same characteristic polynomial. However, if two matrices don't have the ...
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### If $A,B \in M(\mathbb C)$ are two invertible $2 \times 2$ matrices such that $ABA^{-1} = B^5$, then all the eigenvalues of $B$ are 24th roots of unity

Prove that if $A,B \in M(\mathbb C)$ are two $2$ by $2$ invertible matrices such that $ABA^{-1} = B^5$, then all the eigenvalues of $B$ are $24$-th roots of unity. If $\lambda$ is an eigenvalue of $B$...
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### What is the number of equivalence classes of $n\times n$ similar matrices over $\mathbb{Z}/p$?

Two $n\times n$ matrices $A$ and $B$ with entries in a field $\mathbb{F}$ are said to be similar if there exists $P\in GL_{n}(\mathbb{F})$ such that $B=PAP^{-1}$. When $\mathbb{F}$ is an infinite ...
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### Prove a non-invertible matrix $A$ is similar to certain matrices

I’ve encountered the following question, which I’m having trouble trying to solve: Given a non-invertible matrix $A$, prove it is similar to: A matrix with a zero column A matrix with a zero row I ...
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1 vote
### Compute $PSP^{-1} y$ without computing matrix inverse
Let $P$ (symmetric) and $S$ be two matrices of size $N \times N$, and $y$ be a vector of size $N \times 1$. I would like to compute $PSP^{-1} y$ without computing $P^{-1}$ because, in my case, $P$ is ...