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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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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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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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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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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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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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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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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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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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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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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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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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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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: $$A = \begin{... 1answer 92 views Matrix similar to its square I am trying to find the matrices  M \in M_n (\mathbb{R}) such that M is similar to M^2. I tried to use the fact that if these matrices are similar, then they have the same eigenvalues, but I ... 1answer 63 views Classify all real 4 \times 4 matrices satisfying A^3 + A = A^2 + I This is an old prelim problem. The question is to classify all conjugacy classes of real 4 \times 4 matrices satisfying A^3 + A = A^2 + I. Factoring this gives (A-I)^2(A+I) = 0. I can also see ... 2answers 239 views On two special kind of invertible similar matrices with rational entries Let A,B \in GL(n, \mathbb Q) be two similar matrices i.e. there exists X \in GL(n, \mathbb Q) with XAX^{-1}=B. If there is an integer s such that A^{s+1}B=BA^s, then how to prove that A,B ... 2answers 44 views Showing a set of non-zero matrices that are similar to A \in M_{n\times n}(\mathbb C) are a subspace of M_{n\times n}(\mathbb C) I'm currently studying linear algebra and came across this question that asks to state whether or not if A \in M_{n\times n}(\mathbb C), then the set S containing all n\times n matrices that are ... 2answers 71 views How do I prove - If matrix A is similar to matrix B and matrix C is similar to matrix D, then matrix A*C is similar to matrix B*D? [closed] I am generally looking for solving two pairs of n*n matrices. Also, if the nth row of matrix A is in the mth row of matrix C then the nth row of matrix B will also be in the mth row of matrix D. I am ... 1answer 23 views Transformation rules of a linear operator S\mathbf y = A \mathbf x \implies \mathbf y' = AS \mathbf x' Given are the equations$$ \mathbf x = S \mathbf x' \tag{8.91}  \mathbf y = A \mathbf x, \quad \mathbf y' = A'\mathbf x' \tag{8.93} $$It (Riley Hobson Bence, 3rd) then says, But using (8.91)... 1answer 33 views A real matrix is orthogonally similar to a real lower Hessenberg matrix via plane rotations In the Matrix Analysis book by Horn, Problem 2.2.P2, it says that a real matrix A is orthogonally similar to a real lower Hessenberg matrix H via a sequence of plane rotations. First annihilate ... 1answer 60 views What quantities of real symmetric matrices are invariant under matrix congruence? Two real symmetric matrices A and B are called congruent if there exists an invertible matrix P such that$$P^TAP=B$$I am aware that the number of positive, negative, and zero eigenvalues is an ... 1answer 50 views Complex matrices are similar over R iff they and thier conjugates are similar over C. Prove that there is a nonsingular T \in M_n(R) such that A = TBT^{−1} if and only if there is a nonsingular S \in M_n such that both A = SBS^{−1} and \bar{A} = S\bar{B}S^{−1}. (Here M_n is ... 1answer 58 views Diagonalization of a Matrices Represent Change of Basis. In our examination of Mathematical Physics course, a question came which had a matrix written in the standard basis for a 2 by 2 matrix. Now he changed the basis of the given matrix and told us to ... 0answers 58 views Does the transformation (A\otimes A)B(A\otimes A)^{-1} have a name? If B is a n^2\times n^2 complex valued matrix and A is a nonsingular n\times n complex valued matrix then define the transformation:$$B_{new}=(A\otimes A)B(A\otimes A)^{-1} I'm wondering if ...
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 (...
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 ...