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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0answers
15 views

Diagonalization of sparse block matrix

I have a real symmetric matrix, \begin{equation} \left( \begin{array}{ccc} 0 & M & M' \\ M ^T & 0 & 0 \\ M ^{ \prime T} & 0 & 0 \end{array} \right) \end{equation} ...
2
votes
1answer
43 views

Complex matrix and diagonalizablity

Let $A\in\mathcal{M}_4(\mathbb C)$ such that $\operatorname{rank}(A)=2$ and $A^{3}=A^2$ $\neq0$. Suppose that $A$ is not diagonalizable. Then 1. One of the Jordan blocks of the Jordan cannonical form ...
2
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3answers
23 views

Rank of a diagonalizable matrix?

What can be said about the rank of a diagonalizable matrix?
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2answers
35 views

Diagonalizing a matrix. Which formulae is correct?

In my coursebook on linear algebra on some page I see that a diagonal matrix $D$ for a matrix $A$ that can be diagonalized ca be found as follows: $$\tag{1}D=T^TAT$$ But reading further I see that my ...
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1answer
40 views

Trouble understanding the diagonal matrix theorem.

The Diagonal Matrix Representation Theorem states: Suppose $A=PDP^{-1}$, where $D$ is a diagonal $nxn$ matrix. If $B$ is the basis for $R^n$ formed from the columns of $P$, then $D$ is the $B$-matrix ...
3
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3answers
65 views

Prove that $T^n$ is diagonalizable.

Prove or give a counterexample: If $V$ is a complex vector space and $\text{dim V} = n$ and $T \in L(V)$, then $T^n$ is diagonalizable. In order to show that $T$ is diagonalizable I need to show ...
1
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3answers
30 views

non-symmetric matrix with orthogonal eigenvectors

Given that a symmetric matrix with real entries has orthogonal eigenvectors, is the converse true? That is, if a matrix has orthogonal eigenvectors, does it have to be symmetrical and real?
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1answer
35 views

Eigen vectors of the matrix whose columns are eigen vectors of the original matrix

Consider a matrix $A$ of dimension $n$X$n$ whose eigen vectors are $y_1,y_2,y_3,...,y_n$ and are linearly independent. What are the properties of the eigen vectors of the matrix $P$ whose columns are ...
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0answers
37 views
+50

Block diagonalizing two matrices simultaneously

There are two matrices $A$ and $B$ which can not be diagonalized simultaneously. Is it possible to block diagonalize them? What if the matrices have an special pattern? Physics of the problem is ...
3
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1answer
52 views

If diagonalizable matrices commute does it neccesarily mean that they can be simultaneously diagonalized?

If matrices $M_1$ and $M_2$ can be simultaneously diagonalized, than they commute, which can be easily shown: \begin{align} M_1M_2&=P^{-1}D_1PP^{-1}D_2P \\ &=P^{-1}D_1D_2P \\ ...
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3answers
50 views

Diagonalizability of a certain $4\times4$ matrix

Question $\bf 3.$ Determine if the following matrix is diagonalizable. (explain your answer) $$A=\pmatrix{ 1 & 4 & -2 & 3 \\ 3 & -3 & 0 & 4 \\ 1 & 1 & 1 ...
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0answers
36 views

Show that isotropic function S(A) and A have same eigenvectors

Given $\boldsymbol{A}$ is a positive definite, symmetric second order tensor and $\boldsymbol{Q}\boldsymbol{S}(\boldsymbol{A})\boldsymbol{Q}^T = \boldsymbol{S}(\boldsymbol{QAQ}^T)$ $\forall ...
1
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3answers
47 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
33 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
70 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 ...
1
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0answers
18 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
votes
1answer
27 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 ...
1
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2answers
110 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
votes
1answer
30 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
77 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
61 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
votes
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
39 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
51 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
36 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
votes
2answers
64 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
votes
3answers
47 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 ...
5
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1answer
67 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 ...
4
votes
3answers
89 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
77 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
votes
1answer
41 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 ...
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
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1answer
40 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
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4answers
568 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
2answers
96 views

Determine if a particular matrix is diagonalizable

my teacher gave me this exercise: Determine if this matrix is diagonalizable $ \begin{pmatrix} 1 & 1&1&1\\ 1&2&3&4\\ 1&-1&2&-2\\ 0&0&1&-2 ...
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0answers
27 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
14 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 ...
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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 & & & ...
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0answers
23 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
40 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
44 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
35 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
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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
34 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 ...