# 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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29 views

### Relationship Between Similar Matrices A and B and their Inverses

This question came up in my linear algebra class, and I'm having trouble answering it. I found this same question on Math Stack Exchange, but it was never properly answered (although marked as ...
28 views

### Are unitary matrices still unitary under similarity transformations?

$\newcommand\dag\dagger$ I would assume that the property of being unitary is invariant under similarity transformations since similarity transformations are just a change of basis of a linear map, ...
37 views

### What do you call the upside down V character when it refers to a Jacobian Matrix or similar matrix?

What do you call the upside down V character when it refers to a Jacobian Matrix? Not looking for the wedge operator... specifically, this is what it looks like: $$\Lambda$$
23 views

### If conjugation by a matrix preserves the matrix norm then the matrix must be conformal?

Let $A$ be an $n \times n$ real invertible matrix, and suppose that $\| X\|^2=\| AXA^{-1}\|^2$ for every $n \times n$ real matrix $X$. Is it true that $A$ must be conformal? (It is easy to see that ...
26 views

### Can matrix similarity be extended to include pseudoinverses instead of inverse?

We know two matrices $A \in \mathcal{R}^{n \times n}$ and $B \in \mathcal{R}^{n \times n}$ are said to be similar, iff there exists a matrix $P$ such that $A= PB P^{-1}$. Can this be extended to ...
20 views

### Similarity of matrices and Smith normal form

It is stated in the wikipedia article https://en.wikipedia.org/wiki/Smith_normal_form that $A \sim B$ iff SNF($xI-A$)=SNF($xI-B$). How do you prove that?
46 views

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### Are products and sums of similar matrices similar?

Assume that $A$ and $B$ are similar matrices. Which of the following is true? a) Matrices $AB$ and $BA$ can't be similar. b) Matrices $A + B$ and $B + A$ can't be similar. c) Matrices $AA$ and $B$ ...
53 views

### How many similar matrices does one given matrix have? [closed]

I know that $I$ has only one similar matrix, so does $O$, and I also know that any matrix with finite dimensions has a Jordan form. But my question is: How to find out all the similar matrices of ...
33 views

### Does a Householder Matrix commute with a unitary matrix?

If $A$ is a unitary matrix and $P=I-2\alpha\alpha^H(0\neq\alpha\in\mathbb F^{n\times 1},\lVert\alpha\rVert=1)$, then does $PA$ equals to $AP$?
19 views

### Similarity and Binary Matrices

Suppose that $\bf A$ and $\bf B$ are two $n \times n$ matrices over $\mathbb{F}_2$ such that the characteristic polynomial of $\bf A$ and $\bf B$ over $\mathbb{F}_2$ are the same and is equal to an ...
### $\exp(A)=\exp(B)\Rightarrow A,B$ similar
Let $A,B\in\mathbb{C}^{n\times n}$. I want to show the following: If $\exp(A)=\exp(B)$ then $A$ and $B$ are similar. Here \begin{align} \exp:\mathbb{C}^{n\times n}\rightarrow \mathbb{C}^{n\times n}, ...