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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Unitarity similarity transformation

Prove that $R$ and $R^{\dagger}$ can be diagonalized by a common unitary similarity transformation if $R^{\dagger}$ is commutable with $R$. Let $R = SMS^{-1}$, where $M$ is diagonal and $S$ is ...
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36 views

Is there a pair of two $2 \times 2$ complex matrices that are never simultaneously conjugate to real matrices

The problem is as in the title. Let $A$ and $B$ be $2 \times 2$ complex matrices. Are there $A,B$ such that for any $2 \times 2$ matrix $C$, either $CAC^{-1}$ or $CBC^{-1}$ is never real. A ...
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1answer
14 views

A and B have identical singular-values and identical eigenvalues, are they unitary similar?

I've come across this question while taking a matrix analysis course. Given: $A,B \in\mathbb{C}^{3 \times 3}$ with $\lambda_1(A) = \lambda_1(B) ,\,\lambda_2(A) = \lambda_2(B),\,\lambda_3(A) = \...
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On the proof of existence of upper triangular form of complex operators

I am reading through the proof by Sheldon Axler on his Linear Algebra Done Right of the existence of upper triangular matrix form for complex operators, while I don't quite get why he uses induction ...
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66 views

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{...
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1answer
95 views

About a very elementary method which determines if $A$ is similar to $B$ or not. Please tell me a linear algebra book which includes this method.

I am reading "LINEAR ALGEBRA WITH Mathematica" (in Japanese) by Yoshiharu Taniguchi and Kiyokazu Nagatomo. The following elementary method which determines if $A$ is similar to $B$ is in ...
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23 views

Kernel PCA similarity matrix analogy

The standard explanation to linear PCA begins with the covariance matrix. That is, for a dataset $D$ of dimension $N \times d$, the covariance matrix is given as $\sum = \frac{D^{T}D}{N}$ where the ...
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39 views

Proving that every symmetric matrix is similar to a symmetric tridiagonal matrix

If we consider $A$ to be a symmetric $n\times n$ matrix, then how can we show that there exists a unitary matrix $G$ such that : $$ A=GTG^{*} $$ where $T$ is a tridiagonal symmetric matrix. I am not ...
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33 views

Determine if complex matrices are similar

Consider the complex matrices $$ \begin{array}{c} A=\left(\begin{array}{ccc} -2 & 4 & 3 \\ 0 & 10 & 9 \\ 0 & -16 & -14 \end{array}\right) \qquad \text{ and }\qquad B=\left(\...
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237 views

Let A be an n × n orthogonal matrix. Either prove or disprove that any matrix similar to A is also orthogonal.

This is the question I have, I think the statement is false but am at a loss at how to proceed disproving it. I am struggling to come up with a nice counter example and have no idea how else to ...
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53 views

Similarity transformation of an orthogonal matrix

I'm self studying linear algebra and I got stuck with one doubt. I've a transformation $T$ represented by an orthogonal matrix $A$ , so $A^TA=I$. This transformation leaves norm unchanged. I do a ...
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64 views

What's the name of the transformation from a matrix $M$ to $A M A^T$?

Suppose I have a square matrix $M$. If $P$ is another square, invertible matrix, then $PMP^{-1}$ is said to be similar to $M$. Now consider a transformation of the form $AMA^T$, where $A$ can be ...
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29 views

finding the matrix for this linear transformation: "x-y, -x+3y" (in R2; it's an endomorphism)

I was trying to solve this linear algebra exercise. it's about finding the matrix of this linear transformation: matrix with respect to canonical basis of R2 (it's equal to the linear transformation &...
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1answer
49 views

The result of two change of basis is the same. Then the bases must be the same?

Let $A,P,Q$ be square matrices where $P$ and $Q$ are invertible and $A \neq I$. Suppose $$ Q^{-1} A Q = P^{-1} A P \neq 0. $$ Can we conclude that $P = Q$? My intuition says that this is correct, ...
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52 views

Prove the existence of a similar matrix

$A$, $B$ are two matrices such that $A\ge0$ and $B\ge0$ and either $A>0$ or $B>0$. I am trying to show that matrix $BA$ is similar to a matrix with non-negative diagonal elements. Here; $A$ and $...
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1answer
35 views

When do similarity transformations preserve Hermitianity?

Let $H$ be a Hermitian matrix ($H=H^\dagger = (H^T)^*$) and $S$ be an invertible matrix. Denote $\tilde{H} = S^{-1}HS$. Questions: (a) What conditions on $S$ ensure that the matrix $\tilde{H}$ is ...
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85 views

Nilpotent matrix and similar matrix

If $A$ is a complex matrix of order $n$ and $A$ is nilpotent, that is, there exists a positive integer $s$ such that $A^s=0$. Let's say that $e^A = \sum_{k=0}^{\infty} \frac{A^k}{k!}$. Prove that $e^A$...
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If A and B are both diagonalizable to another matrix C, is A similar to B? [duplicate]

Since A and B are diagonalizable to C so PAP^-1 =C and so as B(QBQ-1= C) Which is PAP^-1 = QBQ-1= C
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48 views

A question related to similarity of a Complex matrix that is not scalar multiple of $I_n$

This question was asked in a masters exam for which I am preparing and I was unable to solve it. Let $A$ be an $n\times n$ complex matrix that is not the scalar multiple of $I_n$. Then show that $A$ ...
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How to find matrix similarity score for two non square matrices?

I have two m x n matrices and I need to find the similarity/distance score between these two matrices. How can we find matrix similarity score for two non square ...
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1answer
173 views

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&...
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When is an invertible $n \times n$ matrix (over an arbitrary ring) similar to its inverse?

I want to find necessary and sufficient conditions for an invertible $n\times n$ matrix (over an arbitrary ring) similar to its inverse. Two $n\times n$ matrices $A$ and $B$ are called similar if ...
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24 views

Deriving Similarity Transformations for SU(2) Algebra

I'm working on a project that requires similarity transformations for certain spin operators, but the book I'm working with Mathematical Methods of Quantum Optics doesn't include all the ones I need. ...
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75 views

If $A\in M_n(\mathbb{C})$ is a nilpotent matrix then $A$ is similar to $2A$

If $A\in M_n(\mathbb{C})$ is a nilpotent matrix then $A$ is similar to $2A$ I am trying to prove this property but the truth is I cannot find how to express the matrix $ P $ such that $$ A=P^{-1}2AP \...
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Is combination of squared Euclidean and normalized cosine distance follow Bregman-divergence?

I know that squared euclidean distance satisfy the property of Bregman-divergence. I wanted to do some experiments using combination of various distance metric. So I am curious to know if I add ...
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1answer
25 views

Question about finding 2 non-similar matrices

I need to find two 8x8 matrices $A,B$ with the same minimal & characteristic polynomials and same algebraic multiplicity for every eigenvalue. I was thinking about something like $A=J_3(0), J_2(0),...
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$\begin{pmatrix}A_1&A_{12}\\0&A_2\end{pmatrix}$ is similar to $\begin{pmatrix}A_1&0\\0&A_2\end{pmatrix}$

Let $A_1$ be an $m \times m$ matrix and $A_2$ be an $n \times n$ matrix. Let the respective characteristic polynomials $f_1(x)$ and $f_2(x)$ be relative prime, i.e., $(f_1, f_2) = 1$. Show that $$\...
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30 views

Matrix is orthogonally similar to a diagonal matrix

I have a problem understanding the following problem: Let $U$ be a non-zero vector of $\mathcal{M}_{n, 1}(\mathbb {R})$, of components $u_1,...,u_n$. We set $M = U^TU$. The eigenvalues and the ...
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104 views

What matrix functionals are invariant under change of basis?

Fix some integer $n$, and consider the linear space $M(n,\mathbb F)$ of square $n\times n$ matrices in some field $\mathbb F$. Let $f:M(n,\mathbb F)\to\mathbb F$ be a functional that is invariant ...
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55 views

How to prove that the sign of the all eigenvalues of $AD$ are always the same as that of $BD$? [closed]

$A$ is a similar matrix of $B$, all eigenvalues of $A$ and $B$ are positive, $D$ is a positive semi-definite diagonal matrix. How to prove that the sign of all eigenvalues of $AD$ is always the same ...
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Is there a special name for a matrix of the form $A^TBA$?

Let $P\in\mathbb{R}^{m\times n}$ and $A\in\mathbb{R}^{m\times m}$ be arbitrary matrices. I would like to understand matrices of the form $B=P^TAP$. I know that if $P\in\mathbb{R}^{n\times n}$ is ...
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160 views

Is a symmetric real matrix similar to a diagonal matrix through an orthogonal matrix?

Definition Two matrices $A$ and $B$ are said similar if there exist an inverible matrix $P$ such that $$ B=PAP^{-1} $$ Definition A square matrix $A$ is said orthogonal if it is invertible and its ...
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Columns of a Linear Transformation matrix

Gilbert Strang, Linear Algebra and it's applications Pg 129 Suppose the vectors $x_{1}, \ldots, x_{n}$ are a basis for the space $\mathbf{V}$, and vectors $y_{1}, \ldots, y_{m}$ are a basis for $\...
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Proof that a matrix is similar to one the following matrices. [duplicate]

I am required to prove the following statement: Let $A \in \Bbb M_2(\Bbb R)$ . Prove that if $A$ has one eigenvalue λ, then A is similar to $ \begin{bmatrix} λ & 0 \\ 0 & λ \\ ...
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proving than every $2\times 2$ is similar to a united form

I'm having troble proving this: Let $A\in M_n(R). $ If $n=2$ and $A$ has exactly one eigenvalue $\lambda$ then $A$ is similar to $\begin{bmatrix} \lambda & 1\\ 0 & \lambda \\ \end{bmatrix}$ ...
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42 views

A question regarding Similar Matrix

(image link) Suppose there exist a $3 \times 3$ matrix $A$ and a $3$-dimensional column-vector $x$ such that the set of vectors $x,Ax,A^2x$ are linearly independent, and $$ A^3x = 3Ax - 2A^2 x $$ ...
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Similarity transform for matrix product with determinant equal to zero.

Context: Consider matrices $A \in \mathbb{R}^{m \times n}$ and $B \in \mathbb{R}^{m \times m}$ where $B$ is invertible. Let where $A$ have $k$ zero columns, so $A$ has the following form: $$A = \begin{...
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1answer
92 views

Matrix similar to its square

I am trying to find the matrices $ M \in M_n (\mathbb{R})$ such that $M$ is similar to $M^2$. I tried to use the fact that if these matrices are similar, then they have the same eigenvalues, but I ...
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1answer
63 views

Classify all real $4 \times 4$ matrices satisfying $A^3 + A = A^2 + I$

This is an old prelim problem. The question is to classify all conjugacy classes of real $4 \times 4$ matrices satisfying $A^3 + A = A^2 + I$. Factoring this gives $(A-I)^2(A+I) = 0$. I can also see ...
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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 $A,B$ ...
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44 views

Showing a set of non-zero matrices that are similar to $A \in M_{n\times n}(\mathbb C)$ are a subspace of $M_{n\times n}(\mathbb C)$

I'm currently studying linear algebra and came across this question that asks to state whether or not if $A \in M_{n\times n}(\mathbb C)$, then the set $S$ containing all $n\times n$ matrices that are ...
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71 views

How do I prove - If matrix A is similar to matrix B and matrix C is similar to matrix D, then matrix A*C is similar to matrix B*D? [closed]

I am generally looking for solving two pairs of n*n matrices. Also, if the nth row of matrix A is in the mth row of matrix C then the nth row of matrix B will also be in the mth row of matrix D. I am ...
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Transformation rules of a linear operator $S\mathbf y = A \mathbf x \implies \mathbf y' = AS \mathbf x'$

Given are the equations $$ \mathbf x = S \mathbf x' \tag{8.91} $$ $$ \mathbf y = A \mathbf x, \quad \mathbf y' = A'\mathbf x' \tag{8.93} $$ It (Riley Hobson Bence, 3rd) then says, But using $(8.91)$...
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A real matrix is orthogonally similar to a real lower Hessenberg matrix via plane rotations

In the Matrix Analysis book by Horn, Problem 2.2.P2, it says that a real matrix $A$ is orthogonally similar to a real lower Hessenberg matrix $H$ via a sequence of plane rotations. First annihilate ...
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1answer
60 views

What quantities of real symmetric matrices are invariant under matrix congruence?

Two real symmetric matrices $A$ and $B$ are called congruent if there exists an invertible matrix $P$ such that $$P^TAP=B$$ I am aware that the number of positive, negative, and zero eigenvalues is an ...
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50 views

Complex matrices are similar over $R$ iff they and thier conjugates are similar over $C$.

Prove that there is a nonsingular $T \in M_n(R)$ such that $A = TBT^{−1}$ if and only if there is a nonsingular $S \in M_n$ such that both $A = SBS^{−1}$ and $\bar{A} = S\bar{B}S^{−1}$. (Here $M_n$ is ...
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58 views

Diagonalization of a Matrices Represent Change of Basis.

In our examination of Mathematical Physics course, a question came which had a matrix written in the standard basis for a 2 by 2 matrix. Now he changed the basis of the given matrix and told us to ...
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58 views

Does the transformation $(A\otimes A)B(A\otimes A)^{-1}$ have a name?

If $B$ is a $n^2\times n^2$ complex valued matrix and $A$ is a nonsingular $n\times n$ complex valued matrix then define the transformation: $$B_{new}=(A\otimes A)B(A\otimes A)^{-1}$$ I'm wondering if ...
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36 views

Orthogonal transformations and matrices having similar eigenvalues

If there is an orthogonal similarity between symmetric matrices $A$ and $B$ by having $B=OAO'$ for an orthogonal matrix $O$ ($'$ is transpose) we infer $A$ and $B$ are having identical eigenvalues (...
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Isomorphic graphs definition, understanding of the application of permutation matrix

I was thinking about the definition for two graphs being isomorphic: Basically considering two graphs $\mathcal{G}, \, \mathcal{S}$ they are known to be isomorphic if they are essentially identical in ...