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Questions tagged [diagonalization]

For questions about matrix diagonalization, that is, writing a matrix, a bilinear form or an operator into a "basis" making this one diagonal. This tag is **NOT** for diagonalization arguments from logic and set theory.

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35 views

How come a matrix not diagonalizable if the geometric multiplicity is less than the algebraic multiplicity?

I understand that the algebraic multiplicity is the number of times an eigenvalue appears as a root of the characteristic polynomial. I also understand that the geometric multiplicity is the dimension ...
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0answers
53 views

If zero is an eigenvalue are dimensions lost?

This is likely a silly question so sorry in advance. However, I am wondering if I am right in thinking that if zero is an eigenvalue, then some dimension must be lost. My understanding is that ...
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2answers
56 views

Representations of $\mathbb{G}_m$

I know that the multiplicative affine group scheme $\mathbb{G}_m$ is diagonalizable, since the algebra that represents it is $k[X,X^{-1}]$, which is isomorphic to the group algebra $k[\mathbb{Z}]$. ...
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51 views

Properties of $n \times n$ complex matrix with $A^m = I$

If $A^m = I_n$ then what can we say about the eigenvalues and diagonalizablity of $A$? The equation given above is an annihilating polynomial of $A$ and therefore minimal polynomial divides it. Since ...
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1answer
19 views

Alternative definition of diagonalisable transformation

Supposedly a transformation $T: V \to V$ is diagonalisable iff there exists a basis of $V$ consisting only of eigenvectors of $T$. Can someone show me why this is true? I don't really know where to ...
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1answer
66 views

Eigenvalues, diagonalization and convergence of matrices

I am trying to wrap my head around some basic results in Linear Algebra. I am trying to avoid more abstract concepts like Rank-Nullity, and stay in simple properties at an introductory level. (I've ...
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1answer
44 views

If $A$ is in $\mathbb R^{n \times n}$, then is $A=-A^*$ diagonalizable? [closed]

If $A$ is in $\mathbb R^{n \times n}$, then is $A=-A^*$ diagonalizable?
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0answers
23 views

show that $C(C[u])=K[u]$

Let $E$ be a $K$-vector space of dimension $n$, and $u$ be a diagonalizable linear map from $E$ to $E$. Let $C[u]$ be the set of linear maps from $E$ to $E$ which commute with $u$. In other words,...
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0answers
10 views

What would be the restrictions to diagonalization of this type transformation of Laplacian matrix?

This question is a little specific. I have read about graph theory and saw in one place the following transformation of an Laplacian matrix $L$: \begin{eqnarray} ULW=\begin{bmatrix} -(l_{11}-l_{22})&...
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3answers
35 views

Find 3x3 matrix by determinant and 2 eigenvalues/-vectors

I have two eigenvectors: $(2, 1, -1)'$ with eigenvalue $1$, and $(0, 1, 1)'$ with eigenvalue $2$. The corresponding determinant is $8$. How can I calculate the $3\times3$ symmetric matrix $A$ and $AP$?...
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1answer
37 views

Can you tell how many eigenvectors a matrix has from just the characteristic equation?

If the equation has a repeated root, can you tell without evaluating in the matrix if that repeated root corresponds to more than one eigenvector?
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2answers
48 views

Is $A \in \mathcal{M}_n(\mathbb{C})$ diagonalizable?

$A \in \mathcal{M}_n(\mathbb{C})$ such that $A^2$ has got $n$ distinct non zero eigenvalues. Show that A is diagonalizable. Attempt : As $A^2$ has got $n$ distinct non zero eigenvalues. The ...
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1answer
32 views

Diagonalizing a real normal matrix

Given the matrix $A = \begin{pmatrix} 1 & 2 & 3 \\ 2 & 3 & 4 \\ 3 & 4 & 5 \end{pmatrix}$, how would I find a real orthogonal matrix $P$ such that $PAP^t$ is a diagonal matrix? ...
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0answers
10 views

Routines for diagonalization of banded hermitian matrices?

I have a problem in which I need to diagonalize matrices with many thousands of complex elements three times. I know that the matrices are hermitian and sparse. Specifically, they consist of 9 bands ...
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0answers
17 views

Matrix Numerov Method in three dimensions

Hi can anybody help me to write the following equation in form of matrix by using Numerov's method. $\left(\frac{d^2}{d x_1^2}+\frac{d^2}{d x_2^2}+\frac{d^2}{d x_3^2}+x^2_1+x^2_2+x^2_3+x_1x_2+x_2x_3+...
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2answers
29 views

Reducing the quadratic form

I'm trying to reduce the quadratic form $q(x_1, x_2, x_3, x_4) = x_1x_2 + x_1x_3 + x_1x_4 + x_2x_4$ into a quadratic form of the form $q = λ_1y_1^2 + λ_2y_2^2 + ··· + λ_ry_r^2$ for some real numbers $...
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3answers
64 views

Similarity between a matrix and diagonal matrix

How to prove a matrix and a diagonal matrix are similar? Is there some rule to follow or are there some steps to follow? Like: suppose a matrix $A$ = something, prove that $A$ is similar to a ...
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1answer
22 views

What can I infer about the eigenvalues of the sum of two matrices with known eigendecompositons

I have two matrices and I know the eigenvalues and eigenvectors of both, it also happens to be the case that they share the same set of eigenvalues. I now need to sum the two matrices and take the ...
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1answer
39 views

Show every $X\in SU(2)$ is conjugate to a matrix $\begin{bmatrix} e^{i\theta} & 0 \\ 0 & e^{-i\theta} \end{bmatrix}$

I want to show that any $X\in SU(2)$ is conjugate to a matrix of the \begin{bmatrix} e^{i\theta} & 0 \\ 0 & e^{-i\theta} \end{bmatrix} for $\theta\in \mathbb{R}$. So I guess I want to find $K\...
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0answers
40 views

Diagonalization and the Hadamard product

Let $B \in \mathbb{C}^{n\times n}$ be unitarily diagonalizable such that $B=V\Lambda V^*$. Let $A=B\circ B$ where $\circ$ accounts for the Hadamard product. Then we can say that $A$ is also unitarily ...
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0answers
18 views

For which values ​the matrix is ​diagonalizable

For which values ​​of $a$ matrix $A$ is ​​diagonalizable? $$A = \pmatrix{0&i\\i&a}$$ in the case that it is not diagonalizable determine a base of Jordan Attempt: The minimal polynomial ...
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2answers
32 views

how prove that a linear transformation is diagonalizable, given an eigenvalue and the dimension of its kernel

A question from an exam : (First year mechanical engineering, first course in linear algebra): Let $V$ be the vector space of $2\times2$ matrices, and let $U$ be the subspace of $V$ containing $2\...
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0answers
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Linear algebra- Diagonalization of a symmetric matrix

A linear transformation $$ T:R^3→R^3$$ is defined as $$ T:(x)=Cx$$ where $C$ is a symmetric matrix. a) State the dimensions of the eigenspaces $\mbox{N(C-αI)}$ and $N(c-βI)$ It is also given that: $$...
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1answer
42 views

Diagonalize a Matrix with just the Eigenvalues

If I have a matrix with distinct eigenvalues, why can't I just write a matrix with those as diagonal elements, but I have to do the procedure $P^{-1}AP$? Also I have found this question too that ...
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1answer
30 views

How to prove that $\langle P,A^2 \rangle \le 0 $ for every positive $P$ and skew-symmetric $A$?

I have stumbled upon the following claim, and I wonder if it has a simple proof: Let $P$ be a real $n \times n$ symmetric positive definite matrix. Then for every real skew-symmetric matrix $A$, $\...
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1answer
46 views

Diagonalize the cyclic shift operator

Diagonalize this nxn matrix $\begin{bmatrix} 0&1&0&&&...&&&0\\ 0&0&1\\ &&0&1\\ &&&0&1\\ &&&&0&.\\ &...
2
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1answer
43 views

Square root of matrix

Let $A^2$ be a symmetric matrice such that the spectral theorem allows us to write $$A^2 = P\operatorname{Diag}(\lambda_1 \dots \lambda_n)P^T$$ Suppose $\forall i \in [\![ 1, n]\!], \lambda_i \geq 0$....
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1answer
29 views

Relation between eigenvectors and powers of a matrix for finding out if a graph is disconnected

I'm looking for a quick way to find out whether a graph is disconnected or not. It is if the sum of powers of it's adjacency matrices is a nonzero matrix. To speed up the process of calculating powers,...
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2answers
29 views

Proving $\dim(E_0) \geq n - k$

I found a question from an old exam which I am not really able to wrap my head around. The questions states: Given $k<n$ and $v_1, v_2, ..., v_k \in \mathbb{R} ^n$ non-zero vectors, orthogonal to ...
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0answers
26 views

Theoretical questions about characteristic polynomials, diagonalizability, rank and similarity

I am new to Linear Algebra and would like some feedback regarding my answer to the following question A and B are two square matrices of order n. Prove or refute (with a counterexample) the following ...
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1answer
22 views

Diagonalizability of matrix over C and R respectively

I am new to linear algebra, and was asked the following question: $A$ is a matrix of order n x n, $n \geq 2$, with characteristic polynomial $p(λ)=λ^n-1$. Is $A$ diagonalizable over R? over C? If $A$ ...
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1answer
19 views

Eigenspace for a linear transformation + diagonalizability

I am new to linear algebra, and was asked the question below. I am just looking for some feedback regarding my proposed answer. $T:R_4[x] \to R_4[x]$ is a linear transformation defined by $T(p(x))=p(...
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2answers
15 views

Eigenvalues and eigenvectors in combined Linear Transformations

I am new to linear algebra, and cannot work out the following question, despite the fact that I have been thinking about it for a long while. Let $V$ be a linear space of n dimensions over R, and let ...
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1answer
20 views

Diagonalizability in relation to characteristic polynomial and row equivalence

I am new to linear algebra, and am unsure re the following question: True or False? Let A and B be matrices of n x n. If A and B are diagonalizable and they have the same characteristic polynomial, ...
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0answers
22 views

Similar matrices if only difference is diagonalizable/non-diagonalizable

I am new to linear algebra, and am just looking for some feedback regarding the following solution: True or false? 1.$$\begin{pmatrix}1&0\\0&-1\end{pmatrix}$$ and $$\begin{pmatrix}2&0\\0&...
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2answers
16 views

Diagonalizability in relation to squaring and transposition

True or False? Let A be a square matrix If $A$ is diagonalizable, then $A^2$ is diagonalizable. If $A$ is diagonalizable, then $A^t$ is diagonalizable. Re 1, my answer is that it is correct, but I ...
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4answers
92 views

Are all matrices almost diagonalizable?

Every real $2$ by $2$ matrix that is not diagonalizable is similar to the $2$ by $2$ jordan canonical form, $$ J_2=\begin{bmatrix}s&1\\0&s\end{bmatrix}, $$ where $s$ is the eigenvalue (with ...
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1answer
35 views

Diagonalizability over $\mathbb{C}$ and $\mathbb{R}$ respectively

I am new to Linear Algebra, and would love some feedback regarding the following question, which I found a bit confusing: $$A = \begin{Bmatrix}0&1&0&0\\0&0&1&0\\0&0&0&...
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1answer
26 views

Diagonalise a sparse (symmetric) matrix with elements only on some diagonals

Is there an analytical way or a good approximation or any other mathematical method to diagonalise a sparse (symmetric) matrix with elements only onsome diagonals? For example $$ \begin{bmatrix} B &...
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2answers
56 views

Diagonalizing matrix with fractions [closed]

I'm revising for an exam in linear algebra, and I've found myself stuck on this one specific exercise. I'm supposed to decide a matrix $P$ and a diagonal matrix $D$ from my matrix $H$ (which I'll ...
1
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1answer
29 views

If $A$ is a real Symmetric Matrix of order $n (\geq 2)$ , then there exists a symmetric Matrix $B$ such that $B^{2k+1} = A$.

If $A$ is a real Symmetric Matrix of order $n (\geq 2)$ , then there exists a symmetric Matrix $B$ such that $B^{2k+1} = A$. Is the statement true? I think the statement is true. My Attempt : I ...
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3answers
53 views

Can a matrix be not a multiple of identity, have repeated eigen values and still be diagonalizable?

The question: Diagonalisability of 2×2 matrices with repeated eigenvalues suggests that if a matrix has all its eigen values distinct, it must be diagonalizable. However, any multiple of the ...
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2answers
36 views

How to find eigenvector when the roots are complex

Let $A=\begin{pmatrix}1&-4\\1&1\end{pmatrix}$ then I want to diagonalize this matrix. Doing it's characteristic polynomial I find out that $\lambda_{1,2}=1\pm2i$. Then it's diagonal matrix $$D=...
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0answers
21 views

prove/disprove if each two in $n$ operators can be diagonalizable simultaneously then all can be diagonalizable simultaneously

I have an idea that for $n$ diagonalizable operators $A_1, A_2, ..., A_n \in \ell(V)$. if each $A_i, A_j$ can be diagonalizable simultaneously then all of them can be diagonalizable simultaneously. ...
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3answers
35 views

$A$ square matrix, $\mathrm{rank}(A)=3,$ characteristic polynomial of A is $x^2(x-1)(x-2) \Rightarrow A$ is diagonalizable

I've been trying to prove or disprove the following statement: Let $A$ be a square matrix such that $\mathrm{rank}(A)=3$. Prove or disprove that if the characteristic polynomial of $A$ is $x^2(x-1)(...
1
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1answer
41 views

$A,B\in \mathbb R^{n\times n}$ share $n$ common linearly-independent eigenvectors $\Rightarrow AB=BA$.

I've been trying to prove the following statement: Let $A,B\in \mathbb R^{n\times n}$ be square matrices such that they share $n$ common linearly-independent eigenvectors. Then $AB=BA$. Everything ...
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1answer
19 views

Diagonal Matrix Problem

Could someone check if the solution of the problem is right? Problem: Let $A, B \in \mathbb{C}^{n\times n}$ be selfadjoint ,such that $[A,B] := AB − BA = 0$ Show that there is a unitary matrix ...
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1answer
49 views

Analytic functions and diagonalisation of matrices.

If I have an analytic function $f$ of a square matrix A (like sin(A)), then I know that if the matrix diagnosable then it is possible to find a matrix $$D = P^{-1}AP \tag{1}$$. Then for a function $f(...
1
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1answer
57 views

Unitary Matrices Proof

Problem: Let $A, B \in \mathbb{C}^{n\times n}$ be selfadjoint ,such that $[A,B] := AB − BA = 0$ Show that there is a unitary matrix $U \in \mathbb{C}^{n\times n}$ such that $U^*$A$U$ and $U^*$$...
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2answers
62 views

Proof of Jordan-Chevalley decomposition

Let $A$ be a square matrix over $\mathbb{C}$. Prove there are matrices $D$ and $N$ such that $A = D + N$ such that $D$ is diagonalizable, $N$ is nilpotent and $DN = ND$. I can see that any nilpotent ...