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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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 ...
1
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1
answer
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"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&...
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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 ...
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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 ...
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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 $...
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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 $$β=\left\{\begin{bmatrix}1&0\\0&0\end{bmatrix}, \begin{bmatrix}0&0\\0&2\end{bmatrix}\right\}.$$
$...
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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 $$\begin{bmatrix} cos(x) & -sin(x) & 0 \\\\ sin(x) & cos(x) & 0 \\\\ 0 & 0 & y\end{bmatrix}$$ ...
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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)),\...
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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 ...
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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 ...
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1
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Show that two matrices are similar when matrix is not diagonalizable
I have two matrices:
$$
A = \begin{bmatrix} 1 & 2 & -1 \\\ 0 & 1 & 0 \\\ 1 & 0 & 3 \end{bmatrix}, B = \begin{bmatrix} 1 & 0 & 0 \\\ 0 & 2 & 1 \\\ 0 & 0 &...
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How to judge permutation-similarity from eigenvalues?
I need to judge whether two $N\times N$ matrices $A$ and $B$ are permutation-similar or not; if we have $A_{ij}=B_{\sigma(i)\sigma(j)}$ where $\sigma(i)$ denote a permutation, we say $A\sim B$. Here $\...
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1
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Cube roots of the identity matrix
Describe the set of cube roots of the identity matrix in $M_{n \times n}(\mathbb{R})$.
My attempt:
Let $I_n$ denote the identity matrix of dimension $n \times n$. To find the cube roots of $I_n$, one ...
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Rank and similarity transformation of an idempotent matrix
Above is an excerpt from page 437 of Chapter 11 about Kalman filters in the book "Control System Design" written by Bernard Friedland. The symbol $I$ denotes the identity matrix, $K$ denotes ...
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How to compute quickly the Jordan normal form of a product of the form $B A \operatorname{adj} B$? [duplicate]
Compute the Jordan normal form of the product $$BA\operatorname{adj}(B),$$ where
$$A = \begin{pmatrix}
1 & 3 & -1 \\
-1 & 4 & 0 \\
0 & -1 & 4 \\
\end{pmatrix}, \qquad
B = \...
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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:
...
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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 ...
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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 ...
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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 ...
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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 \...
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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(\...
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1
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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 ...
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0
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70
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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 $...
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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,...
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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 ...
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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}...
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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 ...
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Finding unitary similarity transformation that maps one set of matrices to another, given that it exists
All the matrices I'm considering have the same dimensions. We're handed two sets of $N$ unitary matrices $\{A_i\}$ and $\{B_i\}$, and we're told that there is at least one unitary $U$ such that
$U^\...
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Block diagonal unitary similarity
Let $A\in M_4(\mathbb{C})$ be an unitarily irreducible matrix. Suppose $A$ is unitarily similar to $H+iK$ where $H,K$ are Hermitian matrix. If $H$ is a diagonal matrix then,
$K=\begin{bmatrix}
K_1 &...
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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 ...
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Theorem 5, Section 7.2 of Hoffman’s Linear Algebra
Definition: $A\in M_{n\times n}(F)$ is in rational form if $$A=\begin{bmatrix} A_1& & \\ & \ddots & \\ & & A_r\\ \end{bmatrix}$$ where $A_i$ is companion matrix of non scalar ...
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Block-diagonalization with unitary similarity transformations: ($A \rightarrow U B U^\dagger $, $B$ block-diagonal)
The problem
Given a matrix $A$, find a unitary matrix $U$ such that $U^\dagger A U=B$, where $B$ is approximately block-diagonal (when possible).
Explanation
In other words, assume I'm given a matrix ...
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If two matrices have different characteristic polynomials are they not similar?
This question is very basic, but I couldn't find a definitive answer.
I know that if two matrices are similar, they have the same characteristic polynomial. However, if two matrices don't have the ...
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If $A,B \in M(\mathbb C)$ are two invertible $2 \times 2$ matrices such that $ABA^{-1} = B^5$, then all the eigenvalues of $B$ are 24th roots of unity
Prove that if $A,B \in M(\mathbb C)$ are two $2$ by $2$ invertible matrices such that $ABA^{-1} = B^5$, then all the eigenvalues of $B$ are $24$-th roots of unity.
If $\lambda$ is an eigenvalue of $B$...
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What is the number of equivalence classes of $n\times n$ similar matrices over $\mathbb{Z}/p$?
Two $n\times n$ matrices $A$ and $B$ with entries in a field $\mathbb{F}$ are said to be similar if there exists $P\in GL_{n}(\mathbb{F})$ such that $B=PAP^{-1}$.
When $\mathbb{F}$ is an infinite ...
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Prove a non-invertible matrix $A$ is similar to certain matrices
I’ve encountered the following question, which I’m having trouble trying to solve:
Given a non-invertible matrix $A$, prove it is similar to:
A matrix with a zero column
A matrix with a zero row
I ...
- 336
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2
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Similarity of $3\times 3$ matrices
I am considering $3\times 3 $ matrices, one of them is diagonalizable i.e
$$
A=\left[\begin{array}{ccc}
3&0&4\\
0&-1&0\\
-2&0&-3
\end{array}\right]= \left[\begin{array}{ccc}
-1&...
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2
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Linear algebra — similarity of reflection matrices
The question is as follows:
Let $A,B$ be $2\times2$ reflection matrices.
Are $A$ and $B$ similar?
What I’ve tried:
It did seem like a proof to me:
I have calculated the characteristic polynomial and ...
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Exercise 4, Section 6.3 of Hoffman’s Linear Algebra
Is the matrix $A=\begin{bmatrix} 1&1&0&0\\ -1&-1&0&0\\ -2&-2&2&1\\ 1&1&-1&0\\ \end{bmatrix}$ similar over the field of complex numbers to a diagonal ...
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Simultaneous similarity of two complex matrices and their hermitian conjugates implies their unitary similarity
Let $A$ and $B$ be two similar complex $n \times n$ matrices, i.e., there exists $P \in GL(\mathbb{C},n)$ such that $A=P B P^{-1}$. Furthermore, suppose that $A^{*} = P B^{*} P^{-1}$, where $M^{*}$ ...
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Find change-of-basis matrix for similar rotation matrices
I am trying to find the change-of-basis matrix $P$ that converts a known 3d rotation $R_A$ in one coordinate system into an also known rotation $R_B$ in another another coordinate system:
$$R_B = P^{-...
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Compute $PSP^{-1} y$ without computing matrix inverse
Let $P$ (symmetric) and $S$ be two matrices of size $N \times N$, and $y$ be a vector of size $N \times 1$. I would like to compute $PSP^{-1} y$ without computing $P^{-1}$ because, in my case, $P$ is ...
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0
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Range space of Similar matrices
Do similar matrices share the same column space $Colsp$ and null space $Nullsp$?
What I think !: If $A$ and $B$ are two similar matrices then they must have same range space and null space i.e. $...
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votes
1
answer
39
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How can I derive this bound for the spectral radius of $A$?
Suppose that $A \in \mathbb C^{n \times n}$, such that $\rho(A)$ is the spectral radius of $A$ (its largest eigenvalue in magnitude) and $\bar\sigma(A)$ is the largest singular value of $A$. How can I ...
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0
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views
Let $T : \mathbb C^2 \to \mathbb C^2 $ such that $T^2 = T \circ T = 0$. Show that: a)$Im(T) \subseteq Ker(T)$...
Let $T : \mathbb C^2 \to \mathbb C^2 $ such that $T^2 = T \circ T = 0$. Show that:
a)$Im(T) \subseteq Ker(T)$;
b)If $T \ne 0$, then there is a basis $\cal{B}$ over $\mathbb C^2$ that $[T]_{\cal{B}} = $...
3
votes
0
answers
60
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quotient space of similar matrices is not Hausdorff?
This is a problem of my midterm test...
Denote invertible $2\times 2$ matrices on $\mathbb{C}$ by ${\rm GL}(2,\mathbb{C})$, and define the conjugation action of ${\rm GL}(2,\mathbb{C})$ on itself by ...
- 61
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votes
0
answers
43
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Similarity of Jordan cannonical forms
I'm struggling with this question:
Determine all possible Jordan canonical forms (up to the ordering of the Jordan blocks) for a 6 ×6 matrix
A, if A has an eigenvalue 2 with algebraic multiplicity 6, ...
- 3
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votes
1
answer
52
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Every complex square matrix is unitarily similar to a lower triangular matrix?
Is every complex square matrix is unitarily similar to a lower triangular matrix?
I know that by Schur's Lemma, upper triangular matrices would suffice, but what about lower. Intuitively, I think it ...
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0
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57
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Could we tell whether two matrices are congruent only by its eigenvalues? if not, How many conditions do we need?
When I was learning in my junior high, I am fascinated by the fact that There are more than one way like SSS, SAS etc to tell whether two triangles are congruent or not. Thus I am wondering, Is there ...
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1
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For 2 matrices $A$ and $B$, does $A^2$ similar to $B^2$ imply $A$ is similar to $B$?
My doubt is: if $A^2$ is similar to $B^2$, then $A$ is similar to $B$. Are the following steps correct?
$\implies B^2 = P^{-1}A^2P$
$\implies B^2B^{-1} = P^{-1}A^2PB^{-1} $
$\implies B=P^{-1}A^2PB^{-...
