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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3
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3answers
61 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
50 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
28 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
39 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
28 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
30 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
29 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
51 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
42 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
24 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
27 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
10 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
32 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
20 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
226 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
78 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
31 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
39 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
64 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
56 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
54 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 $...
4
votes
1answer
120 views

Prove there is an orthogonal matrix

Exercise 6.4.15 in Shifrin and Adams' Linear Algebra: a Geometric Approach says Suppose $A$ and $B$ are symmetric and $AB=BA$. Prove there is an orthogonal matrix $Q$ so that both $Q^{-1}AQ$ and $...
0
votes
1answer
37 views

I'm having a difficulty solving this linear algebra problem. I would appreciate any help!

So we are having a matrix $A$: $$\begin{bmatrix} 2&0&0\\ 0&1&1\\ 0&1&1\\ \end{bmatrix}$$ I already found the eigenvalues which are $\lambda_{1}=2$ (of multiplicity $2$) and $\...
2
votes
2answers
38 views

How does diagonalizing a matrix help with eigenvalue calculation?

So in linear algebra class we learned that a matrix $A$ is diagonalizable if it can be written in the form: $$A=PDP^{-1}$$ The useful part was that $A^k$ can be easily computed with $PD^kP^{-1}$. The ...
5
votes
2answers
287 views

Geometric interpretation for eigenvalues and eigenvectors of the cross product's representation as a linear map

Fix ${\bf x} = (x_1,x_2,x_3) \in \Bbb R^3\setminus\{{\bf 0}\}$. We can look at the cross product as a linear map ${\bf x}\times: \Bbb R^3 \to \Bbb R^3$ which is represented in the standard basis by $$\...
1
vote
2answers
33 views

If A has complete distinct eigenvalues and A commutes with M and N, do M and N commute?

I've been thinking about the way that eigenvalues appear on the diagonal of a diagonalized matrix, and found a nice question on it in my textbook: Prove that if a matrix $A \in C_n$ with $n$ distinct ...
1
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0answers
57 views

Unitary Complex Matrix

Let $M$ be a complex matrix $$M:=\begin{bmatrix} 0 & i & -i \\ i & 0 & 1 \\ -i & 1 & 0 \\ \end{bmatrix}$$ 1) Find a unitary complex matrix $Q$ ...
0
votes
1answer
29 views

Basis which makes TWO linear transformation diagonalised at once

Find a basis $\gamma$ with respect to which both of the following lienar transformations on $\mathbb{R^3}$ become diagionalised (the matrices below are the matrices with respect to the standard basis):...
1
vote
1answer
19 views

Spectral Theorem for a Complex Vector Space and corollary

When it says orthogonal projections it must mean they are orthogonal with each other otherwise if it meant they were orthogonal lin transformations then they would be invertible- the only invertible ...
0
votes
1answer
33 views

If A is invertible and orthogonally diagonalizable, is $A^{-1}$ orthogonally diagonalizable as well?

I know that the answer is yes. Are the reciprocal of the eigenvalues of A the eigenvalues for $A^{-1}$? If the eigenvalues for A are $3$ and $2$, would the eigenvalues for $A^{-1}$ be $1/3$ and $1/2$? ...
0
votes
1answer
23 views

When diagonalizing a matrix, in what order should you arrange the the eigenvectors to form the invertible matrix $P$?

I was following this example online to diagonalize a matrix. It lists the eigenvectors as $\lambda =3,2,4$ (note the order). It then arranges each eigenvalue's corresponding eigenvector (3 column ...
0
votes
2answers
58 views

Diagonalizing, Eigenvalues and Eigenspaces

Prove that the matrix $A= \begin{pmatrix} 2 & 0 & -2 \\ 0 & 3 & 0 \\ 0 & 0 & 3 \\ \end{pmatrix} $$ $ is diagonalizable and thus find the ...
0
votes
2answers
40 views

Diagonalizing a 3x3 matrix

Prove that matrix $A$ is diagonalizable, find the bases for the eigenspaces, the diagonalizing matrix $P$, and compute $P^{-1} A P$ where $A= \left(\begin{array}{ccc} 2 & 0 & 3 \\ 0 &...
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0answers
20 views

Restriction of diagonalizable endomorphism to an invariant subspace is diagonalizable - another approach

There are some questions discussing the diagonalizability of a restriction of a diagonalizable endomorphism to an invariant subspace, however, I have a question regarding a certain approach, which ...
1
vote
2answers
27 views

If matrix $A$ is similar to matrix $D$ and $B$ is similar to $E$, than: $AB$ is similar to $DE$?

More specifically: if $A$ & $B$ are diagonalizeable, than is it correct to say that $AB$ is diagonalizeable? (Hints would be more appreciated)
0
votes
1answer
52 views

eigenvalues and eigenvectors of 2x2 block matrix

My question is a really straightforward one: Is there an easier way to find the eigenvalues and/or eigenvectors of a 2x2 block diagonal matrix other than direct diagonalization of the whole matrix? $ ...
0
votes
1answer
35 views

Eigenvalues and Eigenvectors relating to orthogonal basis and diagonal matrices

Find the eigenvalues and eigenvectors of the matrix. $$A = \begin{bmatrix} 1 & 1 & 0 \\ 1 & 0 & -1\\ 0 & -1 & 1 \end{bmatrix}$$ As we have seen in the lectures,...
0
votes
1answer
66 views

Is $A = \left( \begin{matrix} \lambda & 0 \\ 0 & 0 \end{matrix}\right)$ diagonalizable if $\lambda$ is the only eigenvector of $A$?

My book states the following lemma: Suppose that $\lambda$ is the only eigenvalue of $A \in M_{2\times 2}(\mathbb{F})$. Then, $A$ is diagonalizable if and only if $A = \left( \begin{matrix} \lambda &...
-1
votes
1answer
43 views

How to know if the matrix is diagonalizable?

I put the matrix into a row echelon form but I got the matrix without $a$-s. What should I do ? M= $\pmatrix{1 & 0 & 1 \\ 0 & 1 & 0 \\ 0 & 0 & 0}$ It is not OK. I need to ...
1
vote
4answers
41 views

Find the limit a matrix raised to $n$ when $n$ goes to infinity

Let $ A $ be a $ 3\times3 $ matrix such that $$A \left( \begin{array}{ccc} 1 \\ 2 \\ 1 \end{array} \right)=\left( \begin{array}{ccc} 1 \\ 2 \\ 1 \end{array} \right),~~~A \left( \begin{array}{ccc} ...
2
votes
2answers
37 views

For which complex parameters the following matrix is diagonalizable

For all possible complex values of the parameter $\lambda$, determine if the matrix $A$ is diagonalizable and if so find an invertible matrix $C$ and a diagonal matrix $D$ so that $C^{-1}$$DC=A$ $A$ =...
2
votes
1answer
20 views

Does this inner product manipulation make sense?

Suppose A is a normal matrix over $M_n(\mathbb{C})$, with diagonalization $A = PDP^*$. Consider the inner product $<A\mathbf{v}, \mathbf{v}>$ $<A\mathbf{v}, \mathbf{v}> = <P(DP^*\...