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Questions tagged [diagonalization]

For questions about matrix diagonalization. Diagonalization is the process of finding a corresponding diagonal matrix for a diagonalizable matrix or linear map. This tag is NOT for diagonalization arguments common to logic and set theory.

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block diagonalizing a 4x4 Hermitian matrix with vanishing anti diagonal elements

I need help diagonalizing the following 4x4 matrix that has a vanishing anti-diagonal: \begin{align} H = \begin{pmatrix} a & b & c & 0 \\ b^* & d & 0 & e \\ c & 0 & f &...
elfarhan's user avatar
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How to show a matrix DAD has distinct eigenvalues, where D is a diagonal matrix and A is a highly structured matrix

If D is a positive diagonal matrix with well-separated diagonal entries (in particular, $(1 + k) |D_{i - 1, i - 1} < D_{i, i} < (1 - k) D_{i + 1, i + 1}$, where $k$ is a constant and the ...
Stephen Jiang's user avatar
1 vote
1 answer
48 views

Eigenvectors of two commuting diagonalizable matrices when the eigenspaces need not have dimension one

Let $A,B$ be commuting diagonalizable $n\times n$ matrices over $\Bbb C$. Suppose that the eigenvalues of $A$'s are all distinct (so the eigenspaces have dimension one), and the same for $B$. Then any ...
user302934's user avatar
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Why is there not a test for diagonalizability of a matrix

Let $A$ be square.This question is a bit opinion based, unless there is a technical answer.I think it is helpful tho. Also this question is closely related to this question : quick way to check if a ...
Mahammad Yusifov's user avatar
2 votes
0 answers
33 views

Eigenvalues of product of diagonal matrices and Sylvester-Hadamard matrices

Set $n=2^k$ (for some integer $k$) and let $D={\rm diag}(d_1,d_2,\cdots,d_n)$ and $D' = {\rm diag}(d_1', d_2 ,\cdots, d_n')$ be two diagonal matrices in $\mathbb C^{n \times n}$. Let us also presume ...
Ruben Verresen's user avatar
2 votes
4 answers
258 views

Verify that a quadratic form is NOT positive definite

Verify that the quadratic form $$q(x_1,x_2,x_3)=x_1^2+4x_1x_2+3x_2^2+2x_2x_3+6x_3^2$$ is NOT positive definite and find a vector in $v\in\mathbb{R}^3$ such that $q(v)<0$ . I have made several ...
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Reasoning for reduced SVD factorization

I am aware that for any $m \times n$ matrix $A$, we can write: Known 1: $A = U\Sigma V^T$ where $U$ is orthogonal and $m \times m$, $V$ is orthogonal and $n\times n$, and $\Sigma$ is diagonal and $m \...
doctorpigeonhole's user avatar
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1 answer
78 views

Exam question about JNF and matrix diagonalization

Q: Let $G$ be a finite group, $n>0$ a positive integer and let $\mathbb C$ be an algebraically closed field. Recall that $$\operatorname{GL}_{n}(\mathbb{C})=\left\{M \in \operatorname{Mat}_{n, n}(\...
Jason Xu's user avatar
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High order finite difference schemes for boundary value problems on a finite interval

I have some questions. I'm going to assume everything is in 1d with a Laplacian operator. If I discretize the Laplacian operator using $p = 2a+1$ grid points with periodic boundary conditions, I ...
Cuhrazatee's user avatar
1 vote
2 answers
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Congruent diagonalization using row and column operations

Let $$A=\begin{pmatrix} 1 & 2 & 3\\ 2 & 4 & 6\\ 3 & 6 & 9 \end{pmatrix}.$$ Find an invertible matrix $P$ such that $P^tAP$ is diagonal. Let me start by saying that I already ...
user926356's user avatar
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Why doesn't this diagonal argument work?

I have a question about the standard rules for computing p.r. terms (see below). It seems pretty clear that these rules could be used to define a p.r. operation that evaluates any p.r. term of the ...
nontology's user avatar
3 votes
1 answer
146 views

Find a base in which both of these forms are diagonal. Provide this form for both forms.

I need to find a base in which both of these forms are diagonal. Provide this form for both forms. $f(x,y,z)=z^2+2xz+3x^2+y^2-2xy$ $g(x,y,z)=5x^2-2xy+y^2+4xz+2z^2$ I know how to bind a basis, where $f$...
zaba12's user avatar
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If the number of intersection of two conics is an odd number, the quadratic forms are not simultaneous diagonalizable

I'm trying to do Exercise 3.6 at the end of this pdf (in $\Bbb CP^2$): Show that the two quadratic forms $$x^2+y^2-z^2, \quad x^2+y^2-y z$$ cannot be simultaneously diagonalized. Attempt 1: Their ...
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Eigenvectors of Johnson Graph $J(N,2)$

I am having trouble finding an orthogonal basis of eigenvectors of the Johnson Graph $J(N,k)$ with k=2 in an explicit form. In the paper "On the reconstruction of eigenfunctions of Johnson graphs&...
Alessio Catanzaro's user avatar
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What do I do when the first column of my characteristic equation for finding eigenvectors are zero?

$$A=\begin{pmatrix} 2 &4 &1\\ 6 &5 &2\\ 3& 1& 0\end{pmatrix}$$ I got the eigenvalues to be $-2,3,-1$ and I've gotten the eigenvectors of the first $2$ eigenvalues. I'...
Glasstablegirl's user avatar
1 vote
3 answers
89 views

Inequality involving matrix trace and diagonalisable matrices

Given two real PSD matrices diagonalisable by orthogonal matrices: $A=UDU^T$ and $B=VEV^T$, prove that $$tr(A+B-2(A^{1/2}BA^{1/2})^{1/2})\geq0.$$ We can rewrite the inequality as $$tr(D)+tr(E)\geq tr((...
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Spectral Theorem for normal maps in complex scalar spaces

I'm revisiting my first year lecture notes, but embarrassingly cannot follow a crucial step in a proof and would like to have some opinions on this: Let $V$ be a finite dimensional complex-scalar ...
Taleofwoe's user avatar
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Check whether the statement is true or false for diagonal matrices.

Check whether the statement is true or false. Let $||.||:\mathbb C^{n,n}\to \mathbb R$ be a consistent matrix norm. Let $\Lambda=\text{diag}(\lambda_1,\lambda_2,...,\lambda_n)$ be a diagonal matrix. ...
Unknown x's user avatar
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1 vote
1 answer
56 views

How to find eigenvectors, finding a base f in which linear operator has diagonal matrix and determining the operator in that new basis ([L]_f)

Original question where all details are available: How to find a matrix, characteristic and minimal polynomial of a linear operator? Here I will only copy the results from that question. $$L(g(X))=g(0)...
Danilo Jonić's user avatar
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3 answers
83 views

Limit of $\mathrm{tr}(P(\alpha QP + I)^{-1})$ as $\alpha \to \infty$?

Say that $P, Q$ are two real, symmetric positive semidefinite, possibly singular matrices. What is the limit $$ \lim_{\alpha \to \infty} \mathrm{tr}(P(\alpha QP + I)^{-1})? $$ Simultaneously ...
Drew Brady's user avatar
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0 votes
2 answers
80 views

Is this special BLOCK upper triangular matrix diagonalizable?

Let $A$ be a block upper triangular matrix: $$A = \begin{bmatrix} A_{1,1}&A_{1,2}\\ 0&A_{2,2} \end{bmatrix}$$ where $A_{1,1} ∈ {\mathbb{R}}^{p \times p}$, $A_{2,2} ∈ {\mathbb{R}}^{(q) \times (...
Manish Kumar's user avatar
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0 answers
27 views

Diagonalization of quadratic forms and partial derivatives

I have the quadratic form V= $x^2 + y^2 + xy -a(2x+y)$, which is the potential of a quantum particle. I have to find a change of variables such that I get a harmonic oscillator. I tried originally to ...
Jaime Yepes de Paz's user avatar
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23 views

How to compute Sylvester form of a matrix representing a symmetric bilinear form?

Can somebody state a step-by-step algorithm to, given a symmetric n x n-matrix A, (congruently) diagonalize A such that the entries of the diagonal are 1, -1 and 0 corresponding to the signature of A ...
romanson's user avatar
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0 answers
37 views

Inverse of a matrix sum and difference in terms of known inverses

I have been working on a problem related to the inverses of matrices and would appreciate any insights or solutions. The problem is as follows: Given two invertible matrices $A$ and $B$ with known ...
triple_tactic's user avatar
1 vote
1 answer
103 views

General lorentz transformation as exponential

I am looking at Lorentz transformations and how the Dirac equation transforms under them. A Lorentz transformation is an element of the matrix group $$O(1,3) := \left\{\Lambda\in\mathbb{R}^{4\times4} \...
Rasmus's user avatar
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1 answer
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Proving rank of $A-\lambda vv^t $ is $r-1$.

Let $A$ be an $m\times m$ symmetric real matrix of rank $r$ s.t. $r\ne m$.If $\lambda$ nonzero is an eigenvalue of $A$ with corresponding unit column vector $v$ s.t. $Av=\lambda v$.Then prove the ...
user avatar
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24 views

Matrix of a module over a complex polynomial ring

I'm having trouble with the following question: Given a $\mathbb{C}[x]$ module of the form $M=\bigoplus_1^n\mathbb{C}[x]/(x-\lambda_i)^{n_i}$. what condition on the $\lambda_i$ ensures that $M$ is ...
Ben Carpenter's user avatar
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1 answer
58 views

Diagonalizability of two matrices.

Let $M_n(\Bbb R)$ be the ring of $n\times n$ matrices over $\Bbb R$ ,where $n ≥ 2$. If $A,B\in M_n (\Bbb R)$ and $AB = BA$. Then $A$ is diagonalisable over $\Bbb R$ if and only if $B$ is ...
user avatar
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1 answer
71 views

Diagonalization using pauli matricies [closed]

In our class, the professor easily diagonalizes a $2\times2$ matrix, claiming he "sees" it using a linear combination of Pauli matrices to represent the original matrix. He couldn't explain ...
Denis  Streltsovski's user avatar
1 vote
1 answer
32 views

Co-block-diagonalization of unitary matrices

I have the following open problem. For $n\in\mathbb{N}$, consider two complex squared matrices of the same size verifying $A^n=1$ and $A^{\dagger}=A^{n-1}$. Can you find an integer k such that the two ...
Hunfail Karta Hunfail505's user avatar
1 vote
0 answers
27 views

Converting between diagonalizations of a linear transformation with respect to different basis

I was originally asked to find a transition matrix $T$ and a diagonal matrix $D$ such that $D=T^{-1}[\theta]_{e,e}T$ for matrix representation of a linear transformation $\theta:\mathbb R^3 \to \...
Jason Xu's user avatar
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1 vote
1 answer
52 views

Determining the signature of a symmetric billinear form

Relevant Question: Let $V:=\mathbb{R}^3$ and $s:V \times V \rightarrow \mathbb{R}$ the symmetric bilinear form: $$s(x,y):=x_1y_2+x_2y_1+x_2y_3+x_3y_2+x_3y_3$$ for $x=(x_1,x_2,x_3), y=(y_1,y_2,y_3) \in ...
Nayirus's user avatar
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5 votes
0 answers
117 views

If $B^3=B$, is $B$ diagonalizable?

Let $B\in M_n(\mathbb{F})$ such that $B^3=B$. Is $B$ diagonalizable? If $B^3=B$, then $B^3-B=0$. Consider the polynomial $p(x)=x^3-x$. If $p(B)= B^3-B=0$. Since we know that the minimal polynomial of ...
user926356's user avatar
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4 votes
1 answer
84 views

How to approximate $\frac{(2n+1)}{2^n} \int_0^1 \left( 1-x^2+\sqrt{1+2x^2-3x^4}\right)^ndx$

Let $$ f(x) = \frac{1-x^2+\sqrt{1+2x^2-3x^4}}{2} $$ How to approximate the integral $$ I_n = (2n+1)\int_0^1 f(x)^n dx? $$ Experiments seem to indicate that it is something like $cn^{0.75}+1$ where $c$ ...
ploosu2's user avatar
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2 votes
1 answer
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Matrix of an Inner Product and Spectral Theorem

My linear algebra has become very rusty and now I've confused myself entirely. Let $V$ be an inner product space over an $n$-dimensional real vector space $V$. Moreover, let the set of vectors $$\...
Algebro1000's user avatar
1 vote
1 answer
54 views

Diagonalizability of a special matrix involving a parameter k

I'm taking a linear algebra course this semester and we just got our second assignment. I've been working on one particular problem and I think I've made some progress, but I want to make sure my ...
Arihant Tripathy's user avatar
0 votes
3 answers
57 views

Diagonal matrix two-sided multiplication

If $D$ is a diagonal matrix, and $D=ADB$, where $A,B$ are invertible matrices. Can we conclude $A=B^T$? All matrices are real. $D$ is nonzero but not necessarily invertible. If that's true, suppose $D$...
user760's user avatar
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3 votes
1 answer
81 views

Find values of $a,b$. such that a matrix is diagonalizable

Find all values of $a,b\in\mathbb{R}$ such that $A$ is diagonalizable. $$A=\begin{pmatrix} -1 & a & b\\ 0 & 1 & 2\\ 0 & 2 & 1\\ \end{pmatrix}.$$ So far, I have that: $$det(A-\...
user926356's user avatar
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1 vote
0 answers
51 views

Determining if a Matrix is Diagonalizable and the Geometric and Algebraic Multiplicity of Eigenvalues

I am provided the matrix $$ A = \begin{bmatrix} 3 & 0 & 0 \\ 2 & 1 & 1 \\ -2 & -2 & 4 \end{bmatrix} $$ and asked to determine the algebraic and geometric multiplicity of the ...
Grey's user avatar
  • 745
3 votes
2 answers
277 views

Is Johnson Graph J(N, 2) circulant?

I have stumbled upon the problem of diagonalizing the matrix of a Johnson graph $(N,k)$ with $k=2$. From Wikipedia and several other references I found the explicit form for the eigenvalues https://en....
Alessio Catanzaro's user avatar
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0 answers
10 views

The positive/negative sign in inner product involve diagonal matrix and basis transformation matrix.

The expression of one inner product is $<\pi PDEP^{-1}, \pi PD^2EP^{-1}>$. Here, $\pi$ is a row vector that has strictly non-negative terms, D is a diagonal matrix with non-positive diagonal ...
Puning Wang's user avatar
1 vote
1 answer
65 views

Proving a matrix is diagonalizable

Consider the matrix $M\in M_3(\mathbb{R})$ $$M=\begin{pmatrix} 5 & -6 & -6 \\ -1 & 4 & 2\\ 3 &-6&-4 \end{pmatrix}.$$ I need to show that $M$ is diagonalizable, and find a ...
user926356's user avatar
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1 vote
1 answer
73 views

Diagonalizability of a matrix $A$.

Let $A=\begin{bmatrix} 0&0&0&0\\ 0&0&0&0\\ 0&0&1&1\\ 0&0&0&1\\ \end{bmatrix}$. Then is A diagonalizable? My attempt: Matrix A is a Jordan matrix ...
Gggg's user avatar
  • 618
2 votes
2 answers
242 views

Matrix representation of linear transformation w.r.t a basis

Let $T\in L(\mathbb{C}^2)$ defined by $T(x,y)=(y,-x)$. Then $P_T(x)=x^2+1=(x-i)(x+i)$. Then $-i,i$ are eigenvalues of $T$ and $T$ is diagonalizable. Now, we try to find the corresponding eigenvectors: ...
user926356's user avatar
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0 votes
0 answers
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problem about diagonalizable operator [duplicate]

So here is the problem I'm working on with. ''Let $V$ be a finite dimensional vector space over $\mathbb{C}$ and let $T: V \rightarrow V$ be a linear operator. Suppose that for every subspace $U$ of $...
jackie chan fan's user avatar
4 votes
0 answers
108 views

Can't seem to get the correct result by diagonalizing a matrix.

I am trying to understand parts of the authors solution given to the following question: The generators of $\mathrm{SO}(3)$ can be chosen as $t^1=\begin{pmatrix}0 & 0 & 0\\ 0 & 0 & -i ...
digital's user avatar
  • 185
7 votes
3 answers
401 views

If $f(X)=AX-XA$ is diagonalizable, show that $A$ is diagonalizable

Let $f:M_n(F)\rightarrow M_n(F), X\mapsto AX-XA$. If $f$ is diagonalizable, I want to show that $A$ is diagonalizable. I'd prefer to avoid Jordan Blocks. I know that $f$ is diagonalizable if and only ...
RainField's user avatar
  • 436
-3 votes
1 answer
46 views

Finding characteristic polynomial of a square matrix and how to proving that matrix is diagonalizable [closed]

Let $A$ be a square matrix of order $n$ such that $|A + I| = |A − 3I| = 0$ and also $\operatorname{rank}(A)= 2$. I need to find characteristic polynomial of $A$ and have prove that $A$ is ...
Denis Lutsenko's user avatar
1 vote
0 answers
23 views

What is the class of matrices diagonalizable by "generalized FFT"?

As a well-known fact, a matrix is circulant if and only if it has a representation of the form $F^{-1}DF$ where $D$ is a diagonal matrix and $F$ is a discrete Fourier transformation. In other words, ...
nalzok's user avatar
  • 806
0 votes
1 answer
72 views

$f$ is diagonalizable iff its minimal polynomial is "free from squares" (proof)

Let $f \in End(V)$; then $f$ is diagonalizable iff its minimal polynomial is "free from squares", as in, all of its terms are all raised to the first power and (edit:) all irreducible ...
bocceri's user avatar
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