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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8
votes
1answer
61 views

Prove that matrix $A$ diagonalizable if $A^2=I$ using characteristic polynomial

Prove that the matrix $A$ is diagonalizable if $A^2=I$ using characteristic polynomial I saw an answer that used the minimal polynomial of $A$. Can that be proven without using minimal polynomial? ...
2
votes
2answers
52 views

Given a symmetric matrix $A$, find $P$ such that $P^T A P$ is a diagonal matrix

Given $$A =\begin{pmatrix} 0 & 3 & 0 \\ 3 & 0 & 4 \\ 0 & 4 & 0\end{pmatrix}$$ find a matrix $P$ such that $P^T A P$ orthogonally diagonalizes $A$. Verify that $P^TAP$ is ...
2
votes
1answer
45 views

Diagonalizability of a given matrix

I must find out under which conditions the matrix $$A= \left[\begin{array}{ccc|cc}& & & c_0 &\\ & & &c_1&\ddots\\ & & &c_2 &\ddots& c_0\\ &...
0
votes
1answer
41 views

Diagonalizing the matrix (if possible)

Diagonalize the matrix $\begin{bmatrix}0&-4&-6\\-1&0&-3\\1&2&5\end{bmatrix}$ if possible So I know that I can check to see if this is diagonalizable by doing $A = PDP^{-1}$ ...
-1
votes
2answers
27 views

Powers of a matrix using eigen-vectors

Represent a vector as linear combination of eigen-vectors: $$u_0=c_1x_1+....c_nx_n$$ Now, $$Au_0=c_1\lambda_1x_1+....c_n\lambda_nx_n$$ where $\lambda$ is eigen-value. $Ax_i=\lambda_ix_i$ since $\...
4
votes
1answer
78 views
+50

How linear map transform the unit ball?

Let $f:\mathbb{R}^n \to \mathbb{R^n}$ be a linear application, we suppose that $f$ is symmetric ($\langle f(x),y\rangle=\langle x, f(y)\rangle$), without using spectral theorem how we can see that $f$ ...
11
votes
2answers
97 views

Let $S$ be a diagonalizable matrix and $S+5T=I$. Then prove that $T$ is also diagonalizable.

My solution: Since $S$ is diagonalizable, so we can write $S=P^{-1}DP$, where $P$ is an invertible matrix and $D$ is a diagonal matrix. Now $5T=I-S=P^{-1}P-P^{-1}DP=P^{-1}(I-D)P$. So $T=P^{-1}...
0
votes
1answer
40 views

Calculating $\|A\|_2$ in terms of eigenvalues of $A^\ast A$

Let $A$ be a real matrix. I'm supposed to calculate $\|A\|_2$ in terms of the eigenvalues of $A^t A$. I thought to just diagonalize $A^t A$ as $UD^2U^t$ but then I have $\|Ax\|_2 =x^tUD^2U^tx$ instead ...
8
votes
0answers
181 views

When a matrix has same eigenvalues of its column-swapped version?

What are the properties needed for a matrix $A$ to have $\mbox{Spec}(A)= \mbox{Spec}(A \cdot P)$, where \begin{equation} P = \begin{pmatrix} 0 & \cdots & 0 & 1 \\ \vdots & \...
0
votes
2answers
29 views

About matrix diagonalization in C from the characteristic polynomial.

Ok the excercise is: You have one characteristic polynomial, it's: $\lambda^4 + \lambda^2$ Find two matrixes with this polynomial, one of them diagolalizable in C and the other one not. so the ...
0
votes
2answers
27 views

Differences of matrix exponentials

Let $T:V\rightarrow W$ be a linear map of inner product spaces with $T^\ast$ the dual map. I am to calculate $f(\lambda)=\operatorname{tr}e^{-\lambda T^\ast T}-\operatorname{tr}e^{-\lambda TT^\ast }$. ...
0
votes
1answer
40 views

Signatures of symmetric $2 \times 2$ matrices.

I've never seen the term signature of a matrix before this exercise I'm given, and understand it simply means the number of positive eigenvalues. Anyway: Let $A_1, A_2$ be real invertible symmetric $...
2
votes
0answers
30 views

Seemingly strange exercise on unitary Hermitian operator

Let $V$ be a finite dimensional inner product space. Let $T$ be a Hermitian unitary operator. Prove there's a subspace $W$ such that for each $v\in V$ we have $Tv=w-w^\prime$ where $w\in W,w^\...
2
votes
2answers
28 views

Isn't it problematic to cite the Gödel sentence as a proposition asserting 'This sentence is unprovable' since it isn't really on point?

In the proof of Gödel's incompleteness theorem the Diagonalization Lemma is applied to the negated provability predicate $¬Prov_F(x)$: this gives a sentence $G_F$ such that $F ⊢ G_F ↔ ¬Prov_F(⌈G_F⌉) $...
1
vote
2answers
24 views

Terminology: matrix diagonalizable as a bilinear form

If a matrix $P$ is such $P^{-1}MP$ is diagonal, we say that $P$ diagonalizes $M$ (implicitly, as the matrix of an endomorphism). Now, if $P^\top M P$ is diagonal, is it correct to say that $P$ ...
0
votes
0answers
25 views

Representing matrix of orthogonal projection

In an exercise I was given an 3d inner product space and a basis for a subspace and had to orthogonalize it and complete it into an orthogonal basis for the whole space. Then, I was told to find the ...
1
vote
2answers
36 views

“If for every eigenvalue of matrix A - the algebraic multiplicity equals 1 so A is diagonalizable” True/False?

I can't find the answer. I know that A is diagonalizable if and only if its minimal polynomial is a product of distinct linear factors , but I can't determine if its true according to the given ...
0
votes
2answers
47 views

If $A^k=I$ for $k\geq 1$ then $A=I$?

I came across a question: Prove or disprove: $A$ is a square matrix of size $n$ over $\mathbb R$, $P_{(\lambda)}=\lambda^k-1,P_{(A)}=0, k\geq 1 \implies A$ is diagonalizable over $\mathbb R$ ...
1
vote
4answers
81 views

Prove: $A^2\in M_n(\mathbb F)$ is diagonalizable $\implies A$ is diagonalizable (A is invertible)

I do not understand intuitively why this is true but I have a feeling this should be proved using the Cayley-Hamilton theorem. I know that A matrix or linear map is diagonalizable over the field ...
2
votes
5answers
100 views

Factorize a third degree polynomial

It's my first time posting here so I'm not used to describing my problem in mathematics. I'm currently trying to solve a problem which asks if a 3x3 matrix is diagonalizable, I know the method but ...
-1
votes
2answers
49 views

If $A$ is diagonizable then $p(A)$ is diagonalizable

Show that if a matrix $A$ of size $n \times n$ is diagonalizable, then $p(A)$ is diagonalizable for each polynomial $p$.
3
votes
3answers
70 views

Prove that for any diagonalizable matrix $A$, $A^n$ is diagonalizable and also $aA^m+bA^n$

Suppose that A is a diagonalizable matrix. 1) Prove that $A^n$ is diagonalizable 2) Prove that $aA^n + b A^m$ is diagnalizable, for every $a,b\in\mathbb{K}$ I thank you any help or hint you can ...
2
votes
3answers
57 views

Define $L(A) = A^T,$ for $A \in M_n(\mathbb{C}).$ Prove $L$ is diagonalizable and find eigenvalues

Let $L:M_n(\mathbb{C}) \to M_n(\mathbb{C})$ be defined by $L(A) = A^T,$ where $A^T$ is the transpose of $A$ and $M_n(\mathbb{C})$ is the space of all $n \times n$ matrices with complex entries. Prove ...
-2
votes
0answers
31 views

diagonal matrix and differential equations

I need help , be A= 1 0 0 1 0 2 0 0 0 0 1 0 1 0 0 1 real eigenvalues are : ${ 0 ; 1 ; 2 ; 2 }$ Eigenvectors: eigenvalue 0: $[ -1 ; 0;...
1
vote
1answer
42 views

Quadratic Forms Using Derivatives

This link says we can diagonalize a quadratic form $$ f(\vec{x}) = \sum_{i,j=1}^n a_{ij}x_i x_j, $$ $$a_{ij} = a_{ji}, a_{ii} \neq 0$$ using derivatives (?!!!) in a formula like $$f(\vec{x}) = \...
0
votes
1answer
31 views

Power method and convergence

I am working on some practice problems for the convergence of power method for some given recursion relationship and I am trying to generalize/reflect on the question after having been stuck on the ...
0
votes
1answer
30 views

Recursion relationship and linear algebra

I wanted to confirm my intuition about a problem that I got wrong relating to an application of the power method to recursion relations. The question is as follows: For the context of the question, ...
1
vote
1answer
32 views

Minimal polynomial and diagonalizable matrix: property

Quick question We know that if a matrix/linear transformation in a space has dimension n and its minimal polynomial has k different roots with algebraic multiplicity 1, that the matrix/linear ...
1
vote
1answer
35 views

low rank approximations and diagonalization

I would like to discuss or hear an opinion about the following. Given is the (hermitian) $n\times n$ matrix $A = D+M V M^{\dagger}$ with D diagonal. I would like to calculate the eigenvalues (and ...
2
votes
2answers
54 views

Finding an invertible matrix.

I want to find an invertible matrix $P$ where $P^tAP$ is a diagonal matrix. $$A=\begin{pmatrix} 1 & 2 & 1 \\ 2 & 0 & 2 \\ 1 & 2 & 1 \end{pmatrix} $$ I have calculated ...
1
vote
2answers
54 views

If $U^*DU=D=V^*DV$ for diagonal $D$, is $U^*DV$ diagonal too?

All the matrices mentioned are complex $n\times n$ matrices. Let $U, V$ be unitary matrices such that $U^*DU=V^*DV=D$ for a diagonal matrix $D$ with nonnegative diagonal entries. Does this imply that $...
0
votes
2answers
44 views

Two Diagonalizable Matrices Have Common Diagonalisation when $AB=BA$

The following is an exercise: Thm.: $A_{n \times n}$ is diagonalizable iff its minimal polynomial $m_A(t)$ splits into distinct linear factors. Use this theorem to prove that if $A, B \in \...
0
votes
3answers
51 views

Given diagonal matrix, is it possible to find the invertible matrix P?

A is not explicitly given, but A satisfies the following. $A\begin{bmatrix}1\\1\\0\end{bmatrix}=\begin{bmatrix}1\\1\\0\end{bmatrix},A\begin{bmatrix}1\\0\\1\end{bmatrix}=-\begin{bmatrix}1\\0\\1\end{...
0
votes
1answer
26 views

A fact about symmetric matrices and square roots

Is it true that if $A$ is symmetric then any square root is symmetric? I can't prove this using basic symbolic computation, so what if we insist that $A$ is diagonalizable, or even positive definite?
0
votes
1answer
28 views

Finding a polynomial to satisfy a matrix equation

Is there a canonical way of finding a polynomial $p$ such that $$ p\left(\begin{bmatrix} 1& 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 ...
0
votes
0answers
12 views

matrix function diagonalization

Like diagonalization of a constant matrix is it possible to diagonalize a matrix function $\phi(t)$ if $t\in(0,T)$ i.e., if there exist $ P(t)$ suchthat $P^{-1}(t)\phi(t)P(t)=D(t)$ in all cases? or is ...
3
votes
1answer
33 views

Showing a map is nilpotent

Let $\mathbf{A},\mathbf{B}\in\mathrm{M}_n(\mathbb{R})$ such that $\mathbf{A}$ invertible and diagonalisable, and $\mathbf{AB}=\lambda \mathbf{BA}$ for some $\lambda >1$. I want to show that $\...
0
votes
0answers
11 views

Floquet Boundedness for Floquet multiplier $|\lambda_i|=1$

The statement: Consider the system $x'=A(t)x$, where $A(t)$ is a periodic matrix with period T. If $|\lambda_i|=1$ then the corresponding Jordan block to $e^{TR}$ is diagonal. The constant matrix R ...
2
votes
1answer
21 views

Diagonalizability and elementary divisors

How to prove that an $n \times n$ matrix $A$ over a field $\mathbb F$ is diagonzalizable if and only if every elementary divisor of $A$ has degree $1$? I kind of know why this is true but I am not ...
1
vote
2answers
44 views

Finding if a linear transformation is diagonalisable

Hi i am having some trouble tackling this question for my exam revision. Let $M_{(2,2)}(\mathbb{R})$ denote the vector spce of 2x2 matrices over the real numbers. Also, let A denote the matrix $$\...
1
vote
1answer
22 views

Find new bases with respect to which the matrix of the linear map is diagonal

Let $U=\mathbb{R}^3$ and $V=\mathbb{R}^3$. Let $T$ be the linear map $U\rightarrow V$ defined by the matrix $$A= \left[ \begin{matrix} 6 &3&9\\ 3&4&2\\ 3&6&0 \end{matrix} \...
1
vote
2answers
227 views

Prove that an $n \times n$ matrix $A$ over $\mathbb{Z}_2$ is diagonalisable and invertible if and only if $A=I_n$

Through some facts, when $A$ is invertible, I found out that the eigenvalue can't be $0$, since if the eigenvalue is $0$, then $\det(A)=0$, which means that is is not invertible. Since it is over $\...
2
votes
2answers
60 views

Diagonalisation proof

Suppose the nth pass through a manufacturing process is modelled by the linear equations $x_n=A^nx_0$, where $x_0$ is the initial state of the system and $$A=\frac{1}{5} \begin{bmatrix} 3 & 2 ...
3
votes
2answers
81 views

How to diagonalize a matrix with several eigenvalues of zero?

I'm looking to diagonalize a matrix A seen below. (Find a $P$ and a $D$ such that $AP = PD$). $$ \begin{bmatrix} 1 & 2 & 3 & 4 & 5 & 6 \\ 1 & 2 & 3 ...
0
votes
1answer
23 views

Simultaneous Diagonalization of A and B via $\Sigma = A^{-1}B$

I am reading the paper "A Generalized Subspace Approach for Enhancing Speech Corrupted by Colored Noise" by Yi Hu and Philipos C. Loizou. In the paper, they claim that given two matrices $R_{n}$ and $...
1
vote
1answer
32 views

Given two diagonalizable matrices that commute (AB = BA),is AB necessarily diagonalizable?

Prove or disprove: Given two diagonalizable matrices A, B that commute (AB = BA),is AB necessarily diagonalizable?
1
vote
3answers
41 views

Finding Matrix A from Eigenvalues and Eigenvectors (Diagonalization)

Question: Let $A$ be a $3 \times 3$ Matrix such that $[-3,4,1]$ is the eigenvector corresponding to eigenvalue $3$, and $[6,-3,2]$ is an eigenvector corresponding to the eigenvalue $2$. If $v$ = ...
2
votes
1answer
71 views

Rank of square matrix $A$ with $a_{ij}=\lambda_j^{p_i}$, where $p_i$ is an increasing sequence

Let $$ A = \begin{bmatrix} \lambda_1^{p_1} & \lambda_2^{p_1} & \cdots & \lambda_n^{p_1} \\ \lambda_1^{p_2} & \lambda_2^{p_2} & \cdots & \lambda_n^{p_2} \\ \lambda_1^{p_3}...
1
vote
2answers
57 views

Is the following matrix diagonalizable?

Determine if this matrix is diagonalizable. $$ C= \begin{bmatrix} \frac{\sqrt2}2 & 0 & -\frac{\sqrt2}2 \\ 0 & 1 & 0 \\ \frac{\sqrt2}2 & 0 & \frac{\sqrt2}2 \end{bmatrix} $$ I ...
1
vote
3answers
61 views

Is a complex symmetric matrix with positive definite real part diagonalizable?

Let $M \in \mathbb{C}^{n \times n}$ be a complex-symmetric $n \times n$ matrix. That is, $M$ is equal to its own transpose (without conjugation). If the real part of $M$ is positive-definite, then is $...