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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.

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Upper bound on the condition number of the similarity transformation matrix

In the context of square matrices, we have been given $$A = T\Lambda T^{-1}$$ where $\Lambda$ is a known diagonal matrix. It is also known that the condition number of A is bounded above. Say, $$\...
Manish Kumar's user avatar
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Showing a matrix is not similar to its transpose over $\mathbb Z$

This answer claims the matrix $A = \begin{bmatrix}8 & 2 \\ 0 & 1\end{bmatrix}$ is not similar over $\mathbb Z$ to its transpose, without telling why. By taking a generic $S = \begin{bmatrix}x &...
Carla_'s user avatar
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Similar matrices with entries in $p$-adic numbers

It is known that Given a rational matrix $Q$, if the characteristic equation of $Q$ has integral coefficients, then $Q$  is similar ( over $ \mathbb Q$ ) to a matrix with integral entries. Do we ...
ghc1997's user avatar
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Is there a matrix with rational entries similar to a given matrix?

I am working on the following problem. Is there a matrix $A$ with rational entries similar to the matrix $\begin{pmatrix} \sqrt{2} & 0\\0 & \sqrt{2}\end{pmatrix}$? What about $\begin{pmatrix} \...
user123456's user avatar
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Finding a similar matrix with a zero block to a given matrix

I have an $n\times n$ matrix $A$. I have $r$ eigenvalues of $A$ and their corresponding eigenvectors. I would like to construct a matrix $B$ similar to $A$ of the form $\begin{bmatrix} S_{r\...
Ali N's user avatar
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3 answers
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Classify, up to similarity, the $3$ by $3$ matrices with coefficients in $\mathbb{Q}$ that satisfy $A^6=I$.

I am working on the following question in review for my algebra final: Classify, up to similarity, the $3$ by $3$ matrices with coefficients in $\mathbb{Q}$ that satisfy $A^6=I$. My work: As $A^6 - ...
Clyde Kertzer's user avatar
1 vote
1 answer
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about change of basis matrix and matix similarity

so lets say A,B are similar matrices and P is a change of basis matrix from A to B. so by matrix similarity $A = PAP^{-1}$ is the order of P and P^-1 important? ($P^{-1}AP = PAP^{-1}$) if yes what is ...
tensai's user avatar
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Is it possible for two diagonal matrices over a commutative ring with no nontrivial idempotents to be similar without being permutation-similar?

The definition of eigenvalues and eigenvectors could be generalized to matrices over commutative rings that are not necessarily fields. That is, if $A \in M_{n \times n}(R)$ and $0 \ne v \in M_{n \...
Geoffrey Trang's user avatar
3 votes
1 answer
50 views

Proving Similarity of 2 Matrices

I am given two matrices $U$ and $V$. I'm also given that $U=JK$ and $V=KJ$, where $J$ is an invertible matrix. I am supposed to prove similarity but I'm unsure if my proof is sufficient. $$V=KJ$$ $$VJ^...
Grey's user avatar
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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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Is this basis for complex matrices necessarily a unitary basis?

Let us consider the vector space of complex $n \times n$ matrices. Let $\{ V_i \}_{i=1,2,\cdots,n^2}$ be a trace-orthogonal basis of matrices, i.e., $$ \forall i,j \in \{1,2,\cdots,n^2\} : \quad \...
Ruben Verresen's user avatar
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Any singular matrix is similar to a matrix of form $\begin{pmatrix}B&\\&0\end{pmatrix}$

Any singular matrix $A$ is similar to a matrix of form $\begin{pmatrix}B&\\&0\end{pmatrix}$, where $B$ is a matrix with rank $r$? Is this right? that is, is there exists an invertible matrix $...
xldd's user avatar
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Find the change of basis matrix so that the following is in Jordan Normal Form

Let the following matrix be given. Note that we are in the field consisting of five integers: F = (0, 1, 2, 3, 4) $A = \begin{bmatrix} 1 & 2 & 0\\ 3 & 2 & 1\\ 0 & 2 & 2 \end{...
Newbie1000's user avatar
2 votes
4 answers
158 views

Are two diagonals with equal spectra similar to each other?

Let $A,B$ be two diagonal $n \times n$ matrices over some field $R$. And let $A$ and $B$ have the same eigenvalues (equal spectra) (in other words, each diagonal entry in $A$ is a diagonal entry in $B$...
mathlover's user avatar
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$A,B,A_1,B_1$ are $n\times n$ matrices, $A\sim A_1, B\sim B_1$, $n$ is odd, $AB=0$, show that at least one of $A+A_1,B+B_1$ is singular.

Question : $A , B , A_1 , B_1$ are $n \times n$ matrices , $A \sim A_1 , B \sim B_1$ , $n$ is odd , $AB = 0$ Show that at least one of $A+A_1 , B+B_1$ is singular. My attempt : Similarity indicates ...
xldd's user avatar
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If characteristic polynomial of $A$ is $(x-1)^n$, then show $A$ and $A^{-1}$ are similar

I have a problem about matrix similarity: Let $A$ be an $n\times n$ complex matrix whose characteristic polynomial is $(x-1)^n$. I want to show that $A$ and $A^{-1}$ are similar matrices. I know ...
user avatar
1 vote
1 answer
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Does similarity of matrices preserve sum of principal minors?

I am new to linear algebra; someone on Quora posted this "shortcut" method to find the characteristic equation of a $3\times 3$ matrix. Though they demonstrated it through an example, here's ...
Nothing special's user avatar
5 votes
2 answers
205 views

Show that if $A \in M_{2 \times 2}(\mathbb{R})$ then $A, A^T$ are similar.

Show that if $A \in M_{2 \times 2}(\mathbb{R})$ then $A, A^T$ are similar. We say that two matrices $A, B \in M_{n}$ are similar if there exists an invertible matrix $P \in M_{n}$ s.t $A = P^{-1}BP$ ...
X4J's user avatar
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2 votes
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141 views

If $A$ and $B$ have the same eigenvalues...

I'm currently studying for my Linear Algebra final tomorrow, and this question irked me: If $A$ and $B$ have the same eigenvalues, then (a) $A$ and $B$ must be similar matrices. (b) $A$ and $B$ must ...
Samuel Lee's user avatar
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Does similar matrices have same column space or row space

I am reading similar matrices and I found they share many properties like char. Polynomial, trace, eigenvalues, etc. I am curious to know whether they also have same column space or row space? My ...
Maths's user avatar
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Similar matrix whose square is a Hessenberg matrix

I have a known matrix $\mathbf{A}\in \mathbb{R}^{n \times n}$. I want to find a way to obtain a matrix $\mathbf{B}$ that is similar to $\mathbf{A}$ such that $\mathbf{B^2}$ is a Hessenberg matrix. ...
Mathisfreedom's user avatar
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How are two matrices similar?

I am aware that this question has been asked before, but I would like to see if anyone can come up with a better explanation. So what I have been stuck on is, what are the requirements for two ...
N G's user avatar
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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 ...
Dmitry_IT_03's user avatar
2 votes
1 answer
177 views

"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&...
Blue Ghost's user avatar
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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 ...
Mason Rashford's user avatar
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1 answer
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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 ...
user1244767's user avatar
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33 views

If $A,B$ are $n\times n$ complex matrices with $A^2=A, B^2=B$, then show that $A$ and $B$ are similar if and only if they are equivalent [duplicate]

Here equivalent means that there exist invertible matrices $P,Q$ such that $A = PBQ$. I've think I've got the forwards direction: If $A,B$ are similar then by definition there is an invertible matrix $...
rosemary 2.0's user avatar
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52 views

Proving $A$ similar to $B$, unable to find a valid $P$.

Let $D$ denote the set of all $2\times2$ diagonal matrices. Consider an ordered basis $$β=\left\{\begin{bmatrix}1&0\\0&0\end{bmatrix}, \begin{bmatrix}0&0\\0&2\end{bmatrix}\right\}.$$ $...
Bhavya Gupta's user avatar
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106 views

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 $$\begin{bmatrix} cos(x) & -sin(x) & 0 \\\\ sin(x) & cos(x) & 0 \\\\ 0 & 0 & y\end{bmatrix}$$ ...
Mathstudent123's user avatar
1 vote
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39 views

Similarties and Preserving the characteristic polynomial and rank of a nilpotent matrix

Let $M_n$ be the algebra of all complex $n \times n$ matrices. The theorem states that Let $\phi : M_n \to M_n$ be a surjective mapping satisfying $\det(A+\lambda B)=\det(\phi(A)+\lambda\phi(B)),\...
sasa's user avatar
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0 answers
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Simultaneous similarity of pairs of $2\times2$-matrices over algebraically closed fields

This problem should be simple enough to solve by hand, but I don't have any good approach. Let $K$ be an algebraically closed field. Consider matrices $A_1, A_2, B_1, B_2 \in M_2(K)$. Find a ...
Gargantuar's user avatar
1 vote
1 answer
333 views

What are the differences between a "dissimilarity matrix" and a "distance matrix"?

QUESTION. What are the differences between a "dissimilarity matrix" and a "distance matrix"? Do they share the same properties, or not? In case, can you show an example? If there ...
Ommo's user avatar
  • 349
2 votes
1 answer
89 views

Show that two matrices are similar when matrix is not diagonalizable

I have two matrices: $$ A = \begin{bmatrix} 1 & 2 & -1 \\\ 0 & 1 & 0 \\\ 1 & 0 & 3 \end{bmatrix}, B = \begin{bmatrix} 1 & 0 & 0 \\\ 0 & 2 & 1 \\\ 0 & 0 &...
Sonamu's user avatar
  • 299
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0 answers
27 views

How to judge permutation-similarity from eigenvalues?

I need to judge whether two $N\times N$ matrices $A$ and $B$ are permutation-similar or not; if we have $A_{ij}=B_{\sigma(i)\sigma(j)}$ where $\sigma(i)$ denote a permutation, we say $A\sim B$. Here $\...
Takao Kotani's user avatar
2 votes
1 answer
177 views

Cube roots of the identity matrix

Describe the set of cube roots of the identity matrix in $M_{n \times n}(\mathbb{R})$. My attempt: Let $I_n$ denote the identity matrix of dimension $n \times n$. To find the cube roots of $I_n$, one ...
Partim's user avatar
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0 answers
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Rank and similarity transformation of an idempotent matrix

Above is an excerpt from page 437 of Chapter 11 about Kalman filters in the book "Control System Design" written by Bernard Friedland. The symbol $I$ denotes the identity matrix, $K$ denotes ...
8cold8hot's user avatar
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2 votes
1 answer
59 views

How to compute quickly the Jordan normal form of a product of the form $B A \operatorname{adj} B$? [duplicate]

Compute the Jordan normal form of the product $$BA\operatorname{adj}(B),$$ where $$A = \begin{pmatrix} 1 & 3 & -1 \\ -1 & 4 & 0 \\ 0 & -1 & 4 \\ \end{pmatrix}, \qquad B = \...
Tymofii256's user avatar
1 vote
0 answers
82 views

Checking a proof about the structure of the matrix of a linear transformation

Artin's "Algebra" book (1st ed, page 114, Proposition 2.9b) claims that given any $m\times n$ matrix A, there're matrices $Q \in GL_m(F)$ and $P \in GL_n(F)$ so that $QAP^{-1}$ has the form: ...
abcd's user avatar
  • 279
1 vote
1 answer
80 views

Orthogonal solution of matrix equation $ {\bf X} {\bf A} {\bf X}^\top = {\bf B} $

Given symmetric positive definite (SPD) matrices $\bf A$ and $\bf B$, define the following matrix equation in ${\bf X} \in \mathrm{O}(n)$. $$ {\bf X} {\bf A} {\bf X}^\top = {\bf B},$$ Given any SPD ...
gsoldier's user avatar
  • 175
4 votes
3 answers
255 views

Are $\begin{bmatrix} 1 & 0 \\ 1 & 0 \end{bmatrix}$ and $\begin{bmatrix} 1 & 1 \\ 0 & 0 \end{bmatrix}$ similar?

I have two matrices: $$ A =\begin{bmatrix} 1 & 0 \\ 1 & 0 \end{bmatrix}~~~~~~~ B = \begin{bmatrix} 1 & 1 \\ 0 & 0 \end{bmatrix} $$ I am to show they are similar or not similar. I set ...
user129393192's user avatar
1 vote
2 answers
226 views

If the matrices $A$ and $B$ are similar, then are $AB$ and $BA$ similar?

Let $A, B \in M_n(F)$. Suppose that $n \geq 2$. Prove or disprove: If $A$ and $B$ are similar, then $AB$ and $BA$ are similar. I think that this statement is false. I tied to find A,B which are ...
Try_hard's user avatar
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0 answers
135 views

Some clarification over matrix congruence, matrix similarity, and the (finite-dim) spectral theorem

In linear algebra, we have the following well-known result. Proposition. Every real symmetric matrix $A$ is congruent to a diagonal matrix with real eigenvalues on the diagonal. That is, $A=P^T \...
user760's user avatar
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283 views

Conjecture: Any two matrices of size $n×n$ with same characteristic and minimal polynomial are similar implies $n\le 3$.

Notations: $\mathcal{M}_n(\Bbb{R}) $: the set of all $n×n$ matrices over $\Bbb{R}$ $\chi_A(x)$: Characteristic polynomial of $A$ $m_A(x)$ : Minimal polynomial of $A$ $A\sim B$ : $\exists P\in Gl_n(\...
Ussesjskskns's user avatar
0 votes
1 answer
90 views

If two matrices have the same eigenvalues and dimensions of eigenspaces, are they similar?

My attempt at an informal proof for this would be: if they have the same eigenvalues and dimensions of eigenspaces, their generalised eigenspaces also have the same dimensions. This means they will ...
thebasqueinterdisciplinarian's user avatar
1 vote
0 answers
91 views

Finding block triangular matrix similar to "almost symmetric" block matrix

I have a block matrix $$M=\begin{bmatrix}A & B \\ -B & 0\end{bmatrix}$$ where $A$ is negative definite ($A \prec 0$) and $B$ is positive semidefinite ($B \succeq 0$). It might not matter, but $...
Spencer Kraisler's user avatar
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1 answer
233 views

How to find the similarity transformation matrix?

Here's the question I got stuck on: Suppose $S$ is a matrix that performs a similarity transformation. Applying $S$ to the point $(0, 0)$ takes it to the point $(3, 2)$. Applying $S$ to the point $(1,...
Heavenguard.01's user avatar
2 votes
1 answer
87 views

How to transform the Arbitrary Rotation Matrix A to a coordinate system where the Z axis lies along the axis of rotation by Similarity Transformation? [closed]

In Chapter 4 of the book Classical Mechanics by Goldstein, it was written that "By means of some similarity transformation, it is always possible to transform the matrix A to a system of ...
Lusypher's user avatar
  • 101
2 votes
1 answer
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Exercise 13, Section 7.2 of Hoffman’s Linear Algebra

Let $A$ be an $n\times n$ matrix with complex entries. Prove that if every characteristic value of $A$ is real, then $A$ is similar to a matrix with real entries. My attempt: Suppose $A\in M_n(\Bbb{C}...
user264745's user avatar
  • 4,249
1 vote
0 answers
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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 ...
user264745's user avatar
  • 4,249
0 votes
1 answer
118 views

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 ...
user264745's user avatar
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