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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29 views

Relationship Between Similar Matrices A and B and their Inverses

This question came up in my linear algebra class, and I'm having trouble answering it. I found this same question on Math Stack Exchange, but it was never properly answered (although marked as ...
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1answer
28 views

Are unitary matrices still unitary under similarity transformations?

$\newcommand\dag\dagger$ I would assume that the property of being unitary is invariant under similarity transformations since similarity transformations are just a change of basis of a linear map, ...
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37 views

What do you call the upside down V character when it refers to a Jacobian Matrix or similar matrix?

What do you call the upside down V character when it refers to a Jacobian Matrix? Not looking for the wedge operator... specifically, this is what it looks like: $$\Lambda$$
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If conjugation by a matrix preserves the matrix norm then the matrix must be conformal?

Let $A$ be an $n \times n$ real invertible matrix, and suppose that $\| X\|^2=\| AXA^{-1}\|^2$ for every $n \times n$ real matrix $X$. Is it true that $A$ must be conformal? (It is easy to see that ...
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Can matrix similarity be extended to include pseudoinverses instead of inverse?

We know two matrices $A \in \mathcal{R}^{n \times n}$ and $B \in \mathcal{R}^{n \times n}$ are said to be similar, iff there exists a matrix $P$ such that $A= PB P^{-1}$. Can this be extended to ...
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20 views

Similarity of matrices and Smith normal form

It is stated in the wikipedia article https://en.wikipedia.org/wiki/Smith_normal_form that $A \sim B$ iff SNF($xI-A$)=SNF($xI-B$). How do you prove that?
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46 views

How to prove matrix similarity?

I couldn't find the answer to this but, is it possible to prove 2 matrices are similar by their properties ("if and only if" - works both way)? i.e – how do u prove those matrices are similar $$A = \...
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82 views

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 ...
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40 views

Naming conventions in matrix diagonalization $ D = S^{-1} \cdot A \cdot S $

The process of matrix diagonalzation is often summarized as $$ D = S^{-1} \cdot A \cdot S $$ I understand that we often choose the first letter of the alphabet for the original matrix $A$ and $D$ ...
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30 views

Similar Diagonal matrices

$$A= \begin{pmatrix} a & 0 & 0 \\ 0 & b & 0 \\ 0 & 0 & c \\ \end{pmatrix} $$ $$B= \begin{pmatrix} c & 0 & 0 \\ 0 & a & 0 \\ ...
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Are products and sums of similar matrices similar?

Assume that $A$ and $B$ are similar matrices. Which of the following is true? a) Matrices $AB$ and $BA$ can't be similar. b) Matrices $A + B$ and $B + A$ can't be similar. c) Matrices $AA$ and $B$ ...
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How many similar matrices does one given matrix have? [closed]

I know that $I$ has only one similar matrix, so does $O$, and I also know that any matrix with finite dimensions has a Jordan form. But my question is: How to find out all the similar matrices of ...
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33 views

Does a Householder Matrix commute with a unitary matrix?

If $A$ is a unitary matrix and $P=I-2\alpha\alpha^H(0\neq\alpha\in\mathbb F^{n\times 1},\lVert\alpha\rVert=1)$, then does $PA$ equals to $AP$?
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Similarity and Binary Matrices

Suppose that $\bf A$ and $\bf B$ are two $n \times n$ matrices over $\mathbb{F}_2$ such that the characteristic polynomial of $\bf A$ and $\bf B$ over $\mathbb{F}_2$ are the same and is equal to an ...
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$\exp(A)=\exp(B)\Rightarrow A,B$ similar

Let $A,B\in\mathbb{C}^{n\times n}$. I want to show the following: If $\exp(A)=\exp(B)$ then $A$ and $B$ are similar. Here \begin{align} \exp:\mathbb{C}^{n\times n}\rightarrow \mathbb{C}^{n\times n}, ...
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Jordan normal form of $A=\left(\begin{smallmatrix} 0 & 4 & 2 \\ -3 & 8 & 3 \\ 4 & -8 & -2 \\ \end{smallmatrix}\right)$

Find a matrix in Jordan normal form that is similar to $$A=\begin{pmatrix} 0 & 4 & 2 \\ -3 & 8 & 3 \\ 4 & -8 & -2 \\ \end{pmatrix}$$ The characteristic equation of $A$ is $(...
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1answer
34 views

Spectral norm of similar matrices [closed]

In the table it says that if $$A=P B P^{-1}$$ then spectral norm is the same for Similar matrices. Unitary similar matrices. Is the first statement (1) true? If $A=P B P^{-1}$ (where $P$ is not a ...
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53 views

Two complex matrices with same rank are similar?

If two matrices with pure nonzero complex entries in $M_n( \Bbb{C}) $ are of the same rank, then can we say that both matrices are similar? Edit: Where the term 'pure nonzero complex entries' ...
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Curious self-similarity hiding in special case of Sherman-Morrison inversion. What does it imply?

Trying to help deriving a special case of Sherman Morrison formula, I found something peculiar. I made the ansatz $$(I+cd^T)^{-1} = I + A$$ Which lead me to two different ways to build identity ...
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1answer
49 views

Endomorphism of $S_n(\mathbf{R})$ such that $L(O^T S O) = O^T L(S) O$

Let $n \in \mathbb{N}$ such that $n \geq 2$. Let $L$ be an endomorphism of $S_n(\mathbf{R})$ such that: $$ \forall O \in O_n(\mathbf{R}), \forall S \in S_n(\mathbf{R}), L(O^T S O) = O^T L(S) O $$ I ...
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Which $2\times 2$ matrices with entries from finite field are similar to upper triangular matrix?

I have a $2\times 2$ matrix with entrices from a finite field. I am wondering when the matrix is similar to an upper triangular matrix. My thoughts so far: If the matrix does not have any eigenvalues,...
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1answer
53 views

Show that the following matrices in $\mathbb C^{2\times2}$ are similar over the field $\mathbb C$.

How to find a matrix $P$ invertible such that $PA=BP$ where, $$A= \begin{pmatrix} cos\theta & -sin\theta \\ sin\theta & cos\theta \\ \end{pmatrix} $$ and $$B= \begin{...
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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 ($\Phi(I_n)=I_s$). ...
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1answer
48 views

Can we say that there is $P$ such that $P^T AP=B?$

Consider the quadratic form $Q(v)=v^{t}Av,v=(x,y,z,w)$ where matrix $A$ is given by \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\...
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Symmetrization of pentadiagonal matrices

Unsymmetric tridiagonal matrices $T_3$ can easily be symmetrized via a (diagonal) similarity transformation $D=\text{diag}(d_1, \dots, d_n)$ (i.e. see Wikipedia) $$ J_3=D^{-1} T_3 D \,. $$ Is there a ...
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26 views

Understanding an example in Golan on pg.309 [duplicate]

The example is given below: My questions are: 1-How do we get the matrix $B$? 2- can not the diagonal elements be arranged as follows instead 2,2,3,3,3? 3- From where the 1 below number 3 ...
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1answer
26 views

Permuting tensor indices and relations to rearranging factors in a general-size kronecker product operators on same space?

Given the following conjecture, we can start considering larger than $2$ factor Kronecker products. Let us say we define: $$R_1\otimes R_2 \otimes \cdots \otimes R_N$$ And then the operation "...
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Can permutation similarities change order of Kronecker products?

Consider the Kronecker products $${\bf R_1 \otimes R_2} \text{ and } {\bf R_2 \otimes R_1}$$ We can show in the special case they both reside in $\mathbb R^{2\times 2}$ that we can find permutation ...
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How do I eliminate the repeating cases?

Let $K$ be the field with exactly $7$ elements. Let $\mathscr M$ be the set of all $2×2$ matrices with entries in $K$. How many elements of $\mathscr M$ are similar to the following matrix? $ \begin{...
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119 views

Does every polynomial with a Perron root have a primitive matrix representation?

Let $p(x)=x^6-13x^4-20x^3+x^2-x+2$ and $C$ be the companion matrix of $p(x)$. How can I find a primitive matrix similar to $C$ ? Is there a general method to transform the companion matrix with a ...
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1answer
77 views

Show an $n\times n$ matrix $A$ is sim. to the companion matrix for $p_A(t)\iff \exists$ a vector $x$ such that $x,Ax,\ldots,A^{n-1}x$ is a basis

Show that an $n\times n$ matrix $A$ is similar to the companion matrix for $p_A(t)$ if and only if there exists a vector $x$ such that $$x, Ax, \ldots, A^{n-1}x$$ is a basis for $\mathbb C^n$. The ...
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1answer
288 views

Matrix similar to a companion matrix

I am currently intensively reading my linear algebra notes under dim light and was wondering whether it is true, that a an endomorphism whose minimal polynomial has the same degree as the dimension of ...
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1answer
943 views

When is a matrix similar to the companion matrix of its characteristic polynomial?

Let $A$ be a complex matrix and $A_c$ the companion matrix of its characteristic polynomial. From what I have read, I believe the following two statements to be true: not every $A$ is similar to $...
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1answer
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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). ...
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1answer
477 views

Quick way of showing an $n\times n$ Jordan block associated to $1$ is similar to the companion matrix of $(x-1)^n$

Is there a quick, clean way of proving that the $n\times n$ Jordan block with $1$'s on the diagonal and the Frobenius companion matrix corresponding to the polynomial $(x-1)^n$ are similar matrices? ...
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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\\ ...