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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3answers
44 views

Diagonalization with the given eigenvalue and its vector

Let $-3$ be an eigenvalue of a $3\times3$ singular matrix $P$ and $$P\begin{bmatrix} 5\\ 3\\ -2 \end{bmatrix}=\begin{bmatrix} -20\\ -12\\ 8 \end{bmatrix}.$$ Then find whether $P$ is ...
2
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2answers
30 views

diagonalization of a matrix over finite fields

I'm having a problem with determine whether a matrix is diagonalizable over $\mathbb F_{2}$, over $\mathbb F_{3}$, etc. for example, for the following matrix: $$ \begin{bmatrix} 1 ...
3
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0answers
62 views

Simultaneous diagonalization of commuting matrix

I have 3 diagonalizable matrices $A,B,C$. They commute with each other $[A,B]=[B,C]=[A,C]=0$ [edit] The matrix $A$ is Hermitian but $B$ and $C$ have no properties. [/edit] I can get the eigenvalues ...
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0answers
14 views

Fast way to find exponential of a matrix dot product where one of them is diagonal

Suppose $Q$ is a dot product of diagonal matrix A and matrix B: $$ Q=A\cdot B= \left( \begin{matrix} a_1 & 0 & \cdots & 0 \\ 0 & a_2 & \cdots & 0 \\ ...
0
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1answer
23 views

orthogonal matrix

I have to show the following claim: Let $A\in Mat(n,\mathbb{R})$ be positive definite and symmetric. Show that there exists a Matrix $T\in Mat(n,\mathbb{R})$ such that $T^tAT$ is a diagonal matrix. My ...
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2answers
108 views

$A^2$ is diagonalizable leads to $A$ diagonalizable?

If $A^2$ is diagonalizable, is it necessary true that $A$ is diagonalizable? Also, the opposite: If $A$ is diagonalizable, is it necessary true that $A^2$ is diagonalizable? I'm not sure yet, tried ...
0
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3answers
87 views

for which a, the matrix A is diagonalizable?

A = $ \begin{pmatrix} 2a+3 & 0 & 0 \\ -a-3 & a & a+3 \\ a & a & a+3 \\ \end{pmatrix} $ Characteristic polynomial: $ ...
0
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1answer
29 views

Square matrix A, and Is $A = S^2$?

Given A is a square matrix. A is diagonalizable and has eigenvalues which are real (and bigger than zero). Is it necessary true that $A = S^2$? I believe it is, anyone have any ideas how to solve ...
5
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2answers
72 views

$A$ is diagonalizable and $A^3 = A^2$

If $A$ is diagonalizable and $A^3 = A^2$. Is it necessary true that $A^2 = A$?
1
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1answer
60 views

Why is this matrix diagonalizable?

Given the matrix $$A=\left( \begin{array}{ccc} 0 & -1 & -2 \\ -1 & 0 & -2 \\ -2 & -2 & -3 \\ \end{array} \right)$$ It has the following characteristic polynomial: ...
0
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1answer
24 views

Differentiation involving determinant

This question has arisen by following the proof in the appendix of Louis Liporace's paper on maximum-likelihood estimation, where the paper concerns classes of probabilistic functions (elliptically ...
1
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2answers
38 views

$T$ is diagonalizable with vector space of finite dimension

Let $F$ a field , $V$ a vector space ove $F$ with finite dimension and $T$ a linear operator on $V$. If $T$ is diagonalizable and $c_1,c_2,\ldots,c_n$ are distinct eigenvalues of $T$ and $\{id_V, T, ...
1
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3answers
48 views

Diagonalizable operator of a finite vector space

Let $V$ a vector space of finite dimension, $dim (V) = r$, and $T: V \rightarrow V$ a diagonalizable operator with $ \lambda _1,\lambda_ 2,...,\lambda _r$ distincts eigenvalues of $T$ then $ (T- ...
1
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3answers
41 views

$A$ is not similar to a diagonal matrix over the reals

Let $A = \begin{bmatrix} 6 & -3 & -2 \\ 4 & -1 & -2 \\ 10 & -5 & -3 \end{bmatrix} $ then $A$ is not similar to a diagonal matrix over the reals and it is not similar to a ...
1
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1answer
32 views

Show that if $A$ is a symmetric matrix with all eigenvalues greater than $0$, then it is positive definite.

Prove that if $A$ is a symmetric matrix with all eigenvalues greater than $0$, then it is positive definite. If $A$ is symmetric then there exists an orthogonal matrix $S$, such that $S^TAS$ is a ...
2
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2answers
62 views

Is there an algebraic characterization of when a 2 x 2 matrix is diagonalizable?

All matrices are over the complex numbers. There is, of course, an algebraic characterization of when a 2x2 matrix $$ \left( \begin{array}{cc} a & b\\ c & d\\ \end{array} \right) $$ is not ...
1
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1answer
36 views

diagonalisation unitary matrix

Let $A \in U(n) \subset \mathbb{C}^{n \times n}$ a unitary matrix. Show that: $\exists ~ S\in U(n)$ so that $\bar{S^t}AS=D:=\begin{pmatrix}\lambda_1&&0\\&\ddots & ...
3
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3answers
46 views

$M$ matrix, $\mathrm{rank}\ M=1$. Prove that $det(e^M)=1$ iff $M$ is not diagonalizable

M is a $n\times n$ matrix over $\mathbb R$. with $\mathrm{rank}\ M=1$. Prove that $det(e^M)=1$ if and only if $M$ is not diagonalizable. I really don't know how to start thinking about this.. :/ I'd ...
4
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1answer
63 views

Eigenvalues of a $4\times 4$ parameters matrix

Let $a,b,c,d\in\Bbb{C}$ and $B =\begin{bmatrix} a & b & c & d\\ d & a & b & c\\ c & d & a & b\\ b & c & d & a\\ \end{bmatrix}$ I ...
3
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3answers
81 views

Prove that $A$ is diagonalizable iff $\mbox{tr} A\neq 0$

Prove that $A$ is diagonalizable if and only if $\mbox{tr} A\neq 0$. $A$ is an $n\times n$ matrix over $\mathbb{C}$, and $\mbox{rk} A=1$. If $p(t)$ is the characteristic polynomial of $A$, I know ...
2
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4answers
76 views

Prove $A$ is diagonalizable while $A^{2}=kA$

Let $A$ be an $n$-order real matrix, with: $$A^{2}=kA$$ Prove that $A$ is diagonalizable, i.e., there exists an invertible matrix $P$ such that $P^{−1}AP$ is a diagonal matrix. My thoughts: ...
0
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1answer
39 views

prove that if T is invertible transformation there is polynomial $p$ such that $T^{-1} = p(T) $

I know how to prove this using Hamilton.C but something doesn't make sense to me. if I assume that there is such polynomial p(x), so p(T)T = I . then looking at these polynomials I get: p(x)x = 1 so ...
0
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0answers
32 views

L={(M,W) | M is a Turing Machine that stops on input W } is not R. E.

I've been thinking about how to show this but I'm stuck. on Computability, Complexity, and Languages, Second Edition: Fundamentals of Theoretical Computer Science (Computer Science and Scientific ...
1
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2answers
24 views

Given a symmetric matrix A, find an orthogonal matrix S such that $S^TAS$ is a diagonal matrix

Given the symmetric matrix: $$A = \left( \begin{array}{ccc} 1 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 1 \\ \end{array} \right)$$ find an orthogonal matrix $S$ such that $S^TAS$ is a ...
0
votes
1answer
39 views

real matrices $2\times 2$ and $3\times 3$ that are not similars to a diagonal matrix

Example of real matrices $2\times 2$ and $3\times 3$ that are not similars to a diagonal matrix. I find that $A =\begin{bmatrix} 2 & 1 \\ 0 & 1 \end{bmatrix} $ then i suppose that its ...
4
votes
4answers
539 views

Is every self-inverse matrix diagonalizable?

If $A=A^{-1}$, is there always a matrix C such that $C^{-1}AC$ is a diagonal matrix (containing only -1 and 1 in the main diagonal) ? How can I check with PARI/GP, if a given matrix is ...
3
votes
1answer
45 views

Determine if a particular matrix is diagonalizable

my teacher gave me this excercise: Determine if this matrix is diagonalizable $ \begin{pmatrix} 1 & 1&1&1\\ 1&2&3&4\\ 1&-1&2&-2\\ 0&0&1&-2 ...
0
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0answers
26 views

What is the largest (dense, real, symmetric) random matrix I can diagonalize on a computer?

I have read that 10.000x10.000 is no problem for LAPACK or similar routines. I would like to know if N=20.000 or 40.000 is possible. EDIT: I don't know if it is relevant, but the matrix is positive ...
1
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0answers
13 views

Prove that every symmetric matrix can be diagonalized using similarity transformation even if it has repeated eigenvalue

Prove that every symmetric matrix can be diagonalized using similarity transformation even if it has repeated eigenvalue by showing that the Jordan form of a symmetric matrix has no Jordan block of ...
3
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1answer
22 views

Matrix $A$ with characteristic polynomial

Given: Matrix $A$ with characteristic polynomial $p(x) = (x+3)^2(x-1)(x-5)$ Also given: $\rho(A+2I) + \rho(A+3I) + \rho(A-5I) = 9$ (btw $\rho$ means rank of the matrix) Prove: $A$ is ...
0
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2answers
27 views

Minimal polynomial of diagonalizable matrix

It's a if and only if sentence (have to prove both directions) If a matrix $A$ (over $\mathbb{C}$) is diagonalizable then its minimal polynomial's roots are all of algebraic multiplicity 1. Any idea ...
1
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0answers
34 views

Diagonalizing and finding the eigenvalues of matrix of type $T$.

I have seen some of the solutions type within the math.stackexchange but didn't able to get the clear idea. Consider here n to be $\ge$ 5. $$ T = \begin{bmatrix} \alpha_1 & \beta & & & ...
0
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0answers
18 views

Block Diagonalization related to Direct Sum and Single Eigenvalue?

I'm just a beginner in Linear Algebra, and I've proved myself the following: A matrix $A^{n \times n}$ is block diagonalizable if and only if the base field $F^n$ can be divided into at least two ...
1
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3answers
38 views

examining if a matrix is diagonizable

I was practicing some linear algebra problems and I stopped at this one: Without calculating the eigenvectors, show that the following matrix is diagonalizable and find the diagonal matrix to which ...
0
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2answers
42 views

Orthogonal matrix and eigenvalues

How can I find an orthogonal matrix that can diagonalize the next matrix: $$M = \begin{pmatrix} \ a & b \\\ b & a \end{pmatrix}, b\ne 0.$$ Another question is how can I find the eigenvalues ...
0
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1answer
34 views

Find unitary matrix so that $ P^{-1}BP$ is diagonal.

given is the matrix $ B = \begin{pmatrix} 1 & i & -i \\ -i & 2 & 0 \\ i & 0 & 2 \end{pmatrix} $. I have to find a matrix $P \in U(3)$ (in unitary group, meaning that $P^{-1}$ ...
0
votes
1answer
32 views

invert or transpose

Is this correct: When finding the diagonalization of a matrix $A$ of the form $QDQ^{-1}$ then if you normalize your eigenvectors instead of having to invert $Q$, you could just take $Q^t$. Just ...
0
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1answer
62 views

Show that $f$ is diagonalizable

Given an endomorphism $f$ on the vector space on $\mathbb{R}$ of dimension $n$ such that $f(f(x))=3f(x)-2x$. Let $E_1=\ker(f-Id)$ and $E_2=\ker(f-2Id)$. Show that: 1.$E_1$ and $E_2$ form a direct ...
1
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1answer
30 views

Hermitian Matrix Unitarily Diagonalizable

I am having trouble proving that Hermitian Matrices ($A = A^{*}$) are unitarily diagonalizable ($A = Q^{*}DQ$, where Q is a unitary matrix, $QQ^{*} = I$ and D is a diagonal matrix). I also know that ...
0
votes
1answer
47 views

Diagonalizable A, computing fast.

I have $A =$ $ \begin{pmatrix} a & 0 & 0 \\ b & 0 & 0 \\ 1 & 2 & 1 \\ \end{pmatrix} $ I know that this matrix $A$ is diagonalizable when ...
1
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0answers
54 views

Special expression for 3-linear symmetric map $T(X,Y,Z) = \langle -Jh(X,Y),Z \rangle$ [Ejiri]

For a project in Riemannian Geometry, I have been working out the details of a paper by Ejiri (http://www.ams.org/journals/proc/1982-084-02/S0002-9939-1982-0637177-8/S0002-9939-1982-0637177-8.pdf) ...
2
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2answers
37 views

If HP=PH+P for H,P n×n complex matrices, must H be diagonalizable?

If $F$ is a field of characteristic zero, $H,P$ are $n\times n$ matrices over $F$, $0 \neq \alpha \in F$, and $HP=PH+\alpha P$, then must the minimal polynomial of $H$ be square-free and must $P$ ...
0
votes
2answers
20 views

Diagonalisability with $\lambda = 2,x$

I have a $2 \times 2$ matrix, $Y(x)$, with $2$ eigenvalues: $\lambda = 2,x$ where $x\in \mathbb{R}$ Now $Y(x)$ is only diagonalisable if $\lambda_1,\lambda_2$ are distinct. Are $2,x$ distinct? I ...
3
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1answer
34 views

Minimal polynomial and diagonalization of a block matrix.

Let $A \in \mathbb C^{m\times m}$ and $B \in \mathbb C^{n\times n}$, and let $C=\begin{pmatrix} A & 0 \\ 0 & B\\ \end{pmatrix} \in \mathbb C^{(m+n)\times (m+n)}$. 1) Calculate the minimal ...
2
votes
2answers
55 views

if A diagonalizable then show that $a=0$

Let $A= \begin{pmatrix} 2 & 0 & 0\\ a & 2 & 0 \\ b & c & -1\\ \end{pmatrix}$ if A diagonalizable then show $a=0$. $P_A(x)=|xI-A|=(x-2)^2(x+1)=x^3-3x^2+4$ ...
2
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3answers
49 views

Linear Algebra - Diagonalizable matrix

It's a new topic we learn during the linear algebra class and I need a bit help understanding. Lets say, for example, that I have this matrix: \begin{pmatrix}2&1\\x&8\end{pmatrix} and x ∈ R ...
0
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1answer
28 views

How to find no of possible diagonal adjacent elements(no of adjacent may vary) in a 2 dimensional matrices in any size

How to find no of possible diagonal adjacent elements(no of adjacent may vary) in a 2 dimensional matrices in any size. Eg: i have a 5X5 matrices and i need to collect all possible diagonal 3 adjacent ...
0
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2answers
30 views

Find an orthonormal basis of a particular bilinear form

Let $V=\mathbb{R^3}$. Find an orthonormal basis in which the bilinear form with matrix $A$: $\begin{pmatrix} 2 & -2 & 0 \\ -2 & 1 & -2 \\ 0 & -2 & 0\end{pmatrix}$ has a ...
2
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1answer
27 views

Eigenvectors of this matrix - what's the relation to rotation operator?

I have found the eigenvalues to be 0, 1 and 2. The corresponding eigenvectors are: $\frac{1}{\sqrt 2} (1 , -1, 0)$ and $(0, 0, 1)$ and $\frac{1}{\sqrt 2}(1, 1, 0)$. I found that when $x^2 + 2xy + ...
1
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1answer
40 views

Homework help: What is a diagonal matrix?

As a foreword, this is part of my homework so I have left all the calculations out: I just need an explanation. I was trying to show that D = U†AU is diagonal, where D is the diagonalized matrix, A ...