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.

For some field $F$, matrices $A,B\in M_n(F)$ are similar if there exists some $P\in GL_n(F)$ such that $$A=PBP^{-1}$$ Similar matrices represent the same linear operator under two (possibly) different bases, with $P$ being the change of basis matrix.

Given $A,B\in GL_n(F)$, similarity is equivalent to conjugacy, and the conjugation map $c_P:A\mapsto PAP^{-1}$ is also known as a similarity transformation.

Similarity is a useful concept as it is an equivalence relation on $M_n(F)$ that preserves many key invariants, such as

  • Rank
  • Characteristic polynomial, and attributes that can be derived from it:
    • Determinant
    • Trace
    • Eigenvalues, and their algebraic multiplicities
  • Geometric multiplicities of eigenvalues (but not the eigenspaces, which are transformed according to the base change matrix P used).
  • Minimal polynomial
  • Frobenius normal form
  • Jordan normal form, up to a permutation of the Jordan blocks
  • Index of nilpotence
  • Elementary divisors, which form a complete set of invariants for similarity of matrices over a principal ideal domain

Because of this, it is often useful to find a similar matrix that is "simpler" than the matrix of study to analyze. Over any algebraically closed field $F$, every matrix is similar to a matrix in Jordan form.

Use this tag if your question involves computing, conjugating, normalizing, or simply using similar matrices.

188 questions
93
votes
3 answers

Why did no student correctly find a pair of $2\times 2$ matrices with the same determinant and trace that are not similar?

I gave the following problem to students: Two $n\times n$ matrices $A$ and $B$ are similar if there exists a nonsingular matrix $P$ such that $A=P^{-1}BP$. Prove that if $A$ and $B$ are two similar $n\times n$ matrices, then they have the same…
69
votes
6 answers

How do I tell if matrices are similar?

I have two $2\times 2$ matrices, $A$ and $B$, with the same determinant. I want to know if they are similar or not. I solved this by using a matrix called $S$: $$\left(\begin{array}{cc} a& b\\ c& d \end{array}\right)$$ and its inverse in terms…
user4681
  • 755
  • 1
  • 6
  • 6
33
votes
1 answer

Elegant proofs that similar matrices have the same characteristic polynomial?

It's a simple exercise to show that two similar matrices has the same eigenvalues and eigenvectors (my favorite way is noting that they represent the same linear transformation in different bases). However, to show that two matrices has the same…
Gadi A
  • 18,225
  • 6
  • 73
  • 116
27
votes
2 answers

Is every matrix conjugate to its transpose in a continuous way?

It is well-known that every square matrix is conjugate to its transpose. This means (in the case of real matrices) that, for each $n\times n$ matrix $M$ with real entries, there is a matrix $S_M\in GL(n,\mathbb{R})$ such that ${S_M}^{-1}MS_M=M^T$.…
José Carlos Santos
  • 397,636
  • 215
  • 245
  • 423
11
votes
2 answers

On two special kind of invertible similar matrices with rational entries

Let $A,B \in GL(n, \mathbb Q)$ be two similar matrices i.e. there exists $X \in GL(n, \mathbb Q)$ with $XAX^{-1}=B.$ If there is an integer $s$ such that $A^{s+1}B=BA^s$, then how to prove that $A,B$ are identity matrices?
9
votes
4 answers

The existence of an algebra homomorphism between $\mathcal{M}_n({\mathbb{K}})$ and $\mathcal{M}_s(\mathbb{K})$ implies $n | s$

Let $n,s \geq 1$ be integers and $\mathbb{K}$ a field. We assume there exist $\Phi : \mathcal{M}_n(\mathbb{K}) \rightarrow \mathcal{M}_s(\mathbb{K})$ an unital algebra homomorphism ($\Phi(I_n)=I_s$). See here for the definition. We must show that…
9
votes
2 answers

why similarity over $\bar{\mathbb{F}}$ of $A,B\in M_n(\mathbb{F})$ implies similarity over $\mathbb{F}$?

A classic problem in linear algebra is to determine if two matrices $A,B\in M_n(\mathbb{F})$ are similar one to another. When $\mathbb{F}=\bar{\mathbb{F}}$, we know that $A,B$ are similar if and only if they have the same Jordan form. What about the…
7
votes
0 answers

What can we say about two graphs if they have similar adjacency matrices?

Suppose we have two (finite, simple, undirected) graphs, what can we say about these graphs if they have similar adjacency matrices? Observations to begin with: If $G_1$ and $G_2$ are isomorphic, then they have similar adjacency matrices, $A_1$ and…
7
votes
2 answers

Showing two matrix blocks are similar

Let $A \in M_n$ and $B,C \in M_m$. Prove that if $$H= \begin{bmatrix} A&0 \\ 0 & B \end{bmatrix}$$ is similar to $$K = \begin{bmatrix} A&0 \\ 0 & C \end{bmatrix}$$ then $B$ is similar to $C$. I am not sure how I would do this…
6
votes
2 answers

Similar matrices over $\mathbb{Z}/2\mathbb{Z}$

Given the following matrices $P=\left( \begin{array}{rrr} 1 & -1 & 0 \\ 0 & 2 & 5 \\ 0 & 0 & 3 \\ \end{array}\right), Q=\left( \begin{array}{rrr} 1 & 0 & 0 \\ -1 & 4 & 0 \\ 0 & 3 & 7 \\ \end{array}\right)$, such that $P,Q \in M(3\times3,…
6
votes
3 answers

How do I find out that the following two matrices are similar?

How do I find out that the following two matrices are similar? $N = \begin{pmatrix} 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{pmatrix}$ and $M= \begin{pmatrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 &…
MPB94
  • 613
  • 4
  • 10
5
votes
1 answer

What are norms of sub-matrices invariant under a block diagonal similarity transformation of a block matrix?

Say $M := \begin{pmatrix} A & B\\ C & D \end{pmatrix}$ is a block matrix with $A, D$ being square matrices and this $B$ and $C^T$ having the same shape. Is there any norm characterizing the collection of $B$ and $C$ that is invariant under all block…
4
votes
1 answer

Quick way of showing an $n\times n$ Jordan block associated to $1$ is similar to the companion matrix of $(x-1)^n$

Is there a quick, clean way of proving that the $n\times n$ Jordan block with $1$'s on the diagonal and the Frobenius companion matrix corresponding to the polynomial $(x-1)^n$ are similar matrices? Apparently, the (triangular) Pascal matrix is the…
4
votes
2 answers

Finding the normalizer of $\left\{ \left(\begin{matrix} x &0 \\0 & y \end{matrix}\right) : x,y\in \mathbb R\setminus\{0\} \right\}$

I'm having some trouble with the following question: Let $G=\text{GL}_2(\mathbb R)$. What are the elements of the set: $$N_G \left( \underbrace{\left\{ \left(\begin{matrix} x &0 \\0 & y \end{matrix}\right) : x,y\in \mathbb R\setminus\{0\}…
4
votes
0 answers

similarity symmetric block-tridiagonal matrix

I am solving a problem, in the middle of which I reached the following symmetric block-tridiagonal matrix. ‎\begin{bmatrix}‎ 0 & B & &0 \\‎ B^T & 0‎ & ‎\ddots‎ \\‎ ‎&‎\ddots‎& ‎\ddots‎&B \\‎ 0 ‎& & B^T & 0 ‎\end{bmatrix}‎ Where $B$ is an…
1
2 3
12 13