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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.

284 questions
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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…
Taladris
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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
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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
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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
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Why do similar matrices have the same rank?

I have seen some proofs on the Internet, which make use of the transformation map. However, I couldn't understand the methods since what I learned about the transformation map is so superficial. Can you use a simple way to explain it? And for now,…
JimfyWinsy
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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?
uno
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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…
Contestosis
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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…
Ofir Schnabel
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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…
Rebecca J. Stones
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Counting the number of similar matrices over finite fields

Among $3 \times 3$ invertible matrices with entries from the field $ \mathbb{Z/3Z}$, how many matrices are similar to the following matrix? \begin{pmatrix} 2 & 0 &0 \\ 0&2 &0 \\ 0&0 &1 \end{pmatrix} Things I'm familiar with: I know similar…
Dots_and_Arrows
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Sufficient condition for a matrix to be diagonalizable and similar matrices

my question is about diagonalizable matrices and similar matrices. I have a trouble proving a matrix is diagonalizable. I know some options to do that: Matrix $A$ $(n \times n)$, is diagonalizable if: Number of eigenvectors equals to number of…
D.Rotnemer
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Prove there exist matrices $S$ and $T$ such that $B=S(A+T)S^{-1} -T$ iff $\text{tr}(A)=\text{tr}(B)$.

So this problem appeared on a list called "Matrix Problems", meant to be like training for competition-level maths. The whole problem goes like this: Let $A$ and $B$ be $n\times n$ matrices with complex entries. Prove there exists an invertible…
Kendall Rodriguez
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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…
Fernando Martinez
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If two polynomial matrices are similar, prove that the similarity transform matrix can be chosen to be a constant matrix

Consider two $n\times n$ real symmetric polynomial matrices, $P(x)$ and $Q(x)$ in a single indeterminate $x$. That is, $$P(x)=\sum_{i=0}^m P_i x^i ~~~~~ Q(x)=\sum_{i=0}^m Q_i x^i ~~~~~ P_i,Q_i\in S_n(\mathbb{R}) ~~~ m
Troy Van Voorhis
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Gershgorin circle theorem and similarity transformations

Consider the following problem, that was part of an old exam I am studying for: Let $$ A = \begin{pmatrix} 4 & 0 & 2\\ -2 & 8 & 2\\ 0 & 2 & -4 \end{pmatrix}$$ Using the Gershgorin circle theorem, show that $A$ has exactly one eigenvalue with a…
Katharina
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