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.

learn more… | top users | synonyms

3
votes
2answers
50 views

Testing the diagonalizability of matrix $B= \left(\begin{array}(\lambda_1 & a & b \\ 0 & \lambda_1 & c\\ 0 & 0 & \lambda_2\end{array}\right)$

How to show that the matrix $$B= \left(\begin{array}(\lambda_1 & a & b \\ 0 & \lambda_1 & c\\ 0 & 0 & \lambda_2\end{array}\right)$$ is diagonalizable when $a\neq0$, when ...
2
votes
2answers
34 views

$A$ is diagonalizable, if $A,B$ have then same eigenvalues, then $B$ is also diagonalizable

Given $A_{n\times n},B_{n\times n} \in \mathbb R$ such that $A$ is diagonalizable then: if $A,B$ have then same eigenvalues, then $B$ is also diagonalizable over $ \mathbb R$. if $A,B$ ...
2
votes
2answers
30 views

Let $A$ be a real symmetric matrix with rank $1$ , then can all the diagonal entries of $A$ be $0$ ?

Let $A$ be a square real symmetric matrix with rank $1$ , then can all the diagonal entries of $A$ be $0$ ? I know that real symmetric matrices are diagonalizable . Also if all the diagonal entries be ...
1
vote
2answers
41 views

Showing that if $A$ is diagonalizable then $A^2-4A+8I$ is diagonalizable

Let $A_{n\times n}$ be a real matrix then: if $A^4 = 8A$ then $A$ is not invertible. if $A$ is diagonalizable over $\mathbb R$ then $A^2-4A+8I$ is diagonalizable over $\mathbb R$. ...
0
votes
1answer
46 views

True/false questions about minimal and characteristic polynomials of a matrix

We have the matrix $A= \begin{pmatrix} 0 &2 &2 \\ 2& 0 &2 \\ 2& 2 & 0 \end{pmatrix}$, then one of the following is true: $f_A(x)=m_A(x) $ The matrix ...
4
votes
1answer
49 views

Finding a matrix given eigenvalues and eigenvectors.

I am asked to construct a $4 \times 4$ symmetric matrix, with given eigenvalues and eigenvectors. I understand how to actually get $A$ as a product of $P^T, D$ and $P$, when $D$ is the diagonal ...
4
votes
3answers
258 views

Eigenvector and eigenvalue for exponential matrix

$X$ is a matrix. Let $v$ be an eigenvector of $e^{X}$ with corresponding eigenvalue $a$. Show that $v$ is also an eigenvector of $e^{X}$ with eigenvalue $e^{a}$ If $X$ is diagonalizable, then we can ...
1
vote
2answers
25 views

If $D$ is a $3\times 3$-matrix of order $6$, then the geometric multiplicity of eigenvalue $-1$ in $D^3$ is two

Consider the subgroup $H \le \operatorname{GL}(3,3)$ generated by the two matrices $$ A = \begin{pmatrix} 0 & 0 & 1 \\ 0 & -1 & 0 \\ 1 & 0 & 0 \end{pmatrix} \quad\mbox{ and ...
10
votes
4answers
163 views

Suppose $e^A = A$, prove that $A$ is diagonalizable

Suppose $e^A = A$, prove that $A$ is diagonalizable, where A is a matrix. What I have tried to do is write $A= D + N$, where $D$ is diagonalizable, $N$ is nilpotent and $DN = ND$. Since $N$ is ...
2
votes
0answers
16 views

What can we say about the eigenvalues and diagonalization of this $2N\times2N$ matrix $A$?

There is a $2N\times2N$ matrix $A$, which is of the form: $A=\left(\begin{array}{cc} B & C\\ -C^{*} & -B^{*} \end{array}\right),$ where $B$ is a hermitian matrix, and $C$ is a symmetric ...
0
votes
1answer
36 views

Eigenvalues and Eigenvectors

For each linear operator $T$ on $V$, find the eigenvalues of $T$ and an ordered basis $\beta$ for $V$ such that $[T]_\beta$ is a diagonal matrix. Where $V = M_{2\times 2}(\mathbb R)$ and $T ...
0
votes
1answer
25 views

Hoffman Theorem of Diagonalizable operators

Lemma: Let $V$ be a finite-dimensional vector space over the field $F$. Let $T$ be linear operator on $V$ such that the minimal polynomial for $T$ is a product of linear factors $$p = ...
0
votes
0answers
19 views

Set of matrices related to simultaneous diagonalization?

I want to understand the properties of the following set of matrices. Given a matrix $R \in \mathbb{R}^{p \times q}$, the set is: $\qquad \{Q : Q^T \operatorname{diag}(d) Q = R^T ...
0
votes
0answers
19 views

Invariant sum with diagonalization of a matrix

Let $A$ be the $4 \times 4$ matrix on $\mathbb{R}$ $A = \left[\begin{array}{cccc} 0 & 1 & 0 & 1\\ 1 & 0 & 1 & 0\\ 0 & 1 & 0 & 1\\ 1 & 0 & 1 & 0\\ ...
1
vote
1answer
42 views

Every matrix $A\in M_2(\mathbb{C})$ is similar to one of two forms [duplicate]

I'm in trouble at the following exercise: (Ex. 6, page 143 - Um Curso de Álgebra Linear; Coelho, Flávio Ulhoa) Show that if $A\in M_2(\mathbb{C})$, then $A$ is similar (or conjugated) to a ...
1
vote
1answer
21 views

Diagonalizable matrix is symmetric with respect to a certain inner product

Let $A\in \mathbb{R}^{n\times n}$ be diagonalizable, then there is and symmetric possitive definite matrix $C\in \mathbb{R}^{n\times n}$ such that A is symmetric w.r.t $<\cdot,\cdot>_C$, ...
2
votes
1answer
38 views

Is there a simple proof that the matrix is diagonalizable?

Let $A^n=D$, where $D$ is a diagonal invertible matrix, and $n\ne0$ is a number. Is there a simple proof (without involving Jordan canonical form) that the matrix $A$ is diagonalizable?
0
votes
2answers
48 views

Find a basis in which matrix will be diagonal

Let $F= \begin{bmatrix} -1 & -2 & 2 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{bmatrix}$ is a matrix of a linear transformation ...
1
vote
1answer
24 views

Minimal polynomial of a $4\times4$ matrix [closed]

I just need to see an example of a non-diagonalizable $4\times4$ matrix over $\mathbb{R}$ whose minimal polynomial is the same as its characteristic polynomial. I saw the question elsewhere and ...
1
vote
4answers
52 views

$A,B \in GL(n,\mathbb C)$ be two diagonalizable matrices such that $AB=BA$ ; then $\exists p(x) \in P_n(\mathbb C)$ , such that $p(A)=B$ or $p(B)=A$?

Let $A,B \in GL(n,\mathbb C)$ be two diagonalizable matrices such that $AB=BA$ ; then does there exist a polynomial $p(x)$ of degree at most $n$ with complex-coefficients , such that $p(A)=B$ ? ...
1
vote
2answers
35 views

If all the eigenvalues of $A$ are distinct, then $B$ can be expressed uniquely as a polynomial in $A$ with degree no more than $n − 1$.

Let $A$ and $B$ be $n×n$ matrices such that $AB = BA$. If all the eigenvalues of $A$ are distinct, then $B$ can be expressed uniquely as a polynomial in $A$ with degree no more than $n − 1$. My Try: ...
1
vote
0answers
58 views

If $A^2$ is diagonalizable then A is also…

I tried to dis/prove the following: A is a 2x2 complex matrix 1) If $A^2$ is diag. then $A$ is also. I mostly tried using $P^{-1} A^2 P = D$ and found out that: $D=P^{-1} A PP^{-1} A P = ...
0
votes
3answers
27 views

Find a matrix from its eigenvalues and corresponding vectors

Suppose $A$ is a $3 \times 3$ matrix with eigenvalues $\lambda_1=-1$ $\lambda_2=0$ and $\lambda_3=1$ and with the corresponding eigenvectors $\vec{v_1}=<1,0,2>$ $\vec{v_2}=<-1,1,0>$ ...
0
votes
0answers
28 views

Matrix diagonalization depending on the eigenvalues and eigenvectors

I'm trying to do an app that can give the eigenvalues and vectors for a given square matrix. I have already found how to compute and display the eigenvalues and vectors, but I don't know in which case ...
1
vote
3answers
29 views

Why does it seem that eigendecomposition requires that the decomposed matrix be diagonal?

The eigen-decomposition of positive semi-definite matrices always exists. Given such a matrix $\mathbf{A}$, then, we have $$\mathbf{Av}=\lambda\mathbf{v}$$ for a given eigen value $\lambda$ and ...
2
votes
2answers
19 views

Diagonalize a complex matrix

We have a matrix $A = \left( \begin{array}{ccc} 0 & 1 \\ -1 & 0 \\ \end{array} \right)$ I have found the eigenvalues to be simple and equal to $$\lambda_1 = i$$ $$\lambda_2 = ...
1
vote
0answers
27 views

Simultaneous diagonalization of two symmetric matrices vs. diagonalization of one nonsymmetric matrix

In physics, when considering the motion of a system with $N$ degrees of freedom described by vector $x$, the linearized equations of motion take the form $$M \ddot{x} = - K x.$$ Here, $M$ is a ...
1
vote
1answer
31 views

Does the constraint $\mathbf{w'1}=1$ make $\mathbf{w}$ an eigenvector of any square symmetric matrix?

Given a positive definite matrix $\mathbf{M}$ such that $\mathbf{w'Mw}=s$ where $s>=0$, and a constraint $\mathbf{w'1}=1$, where the bold faced $\mathbf{1}$ indicates a vector of $1$s, does it not ...
1
vote
1answer
22 views

Two matrices $A, B$ satisfying in characteristic polynomial of $B$ and $A$, respectively.

Let matrices $A,B\in M_n(\mathbb{C})$ such that $A$ satisfy in characteristic polynomial of $B$, and $B$ satisfy in characteristic polynomial of $A$. Can we say that: $A$ is diagonalizable if and ...
1
vote
1answer
32 views

Determinant of a modal matrix used in diagonalization

If A $\in \mathbb{R}^{n\times n}$ is diagonalized as $P^{-1}AP=\Lambda$ with P the modal matrix composed of the eigenvectors of $A$, is there a general way to determine the determinant of $P$, except ...
2
votes
2answers
89 views

Matrix A satisfy in $A^n=2A$, Prove it is diagonalizable

Let $A$ is a $n\times n$ matrix for $n>1$ with complex entries that satisfy in $A^n=2A$. Prove that $A$ is a diagonalizable matrix.
3
votes
0answers
72 views

If $A\in\Bbb \{\pm1,0\}^{n\times n}$ is symmetric of rank $<n$, does $A-I$ have rank $n$?

Supposing we have a symmetric matrix $A\in\Bbb \{\pm1,0\}^{n\times n}$ of rank $m<n$ with all $+1$ or all $-1$ or all $0$ as diagonals and no $0$ on non-diagonals, when do we have $A-I$ to be full ...
0
votes
1answer
31 views

Similar matrix intuition clarification

I've been studying similar matrices (diagonalisation and JCF) and I want to check if my intuition is sound. So take transformation $T: V \rightarrow V$ and take two bases $E$ and $F$ of $V$. Let $A$ ...
2
votes
2answers
28 views

Simultaneous diagonalization

Given two symmetric matrices $A,B\in\Bbb R^n$ how can we find if they are simultaneously diagonalizable? If they have such property how can we find $U$ such that $UAU'$ and $UBU'$ are simultaneously ...
2
votes
1answer
26 views

Diagonalizable transmit to submatrix

If $$\begin{pmatrix} A & B\\ \Large 0 & C \end{pmatrix}$$ is similar to a diagonal matrix, are $A$ and $C$ also similar to diagonal matrices?
1
vote
1answer
43 views

Power of a diagonal matrix

Let's say I have a linear transformation L: $\Bbb R [x] _{\le 2}$ to $\Bbb R [x] _{\le 2}$, a base B (the canonical for simplicity) and that I have the matrix $\Bbb[L]_{\ B}$ . I know how to calculate ...
0
votes
1answer
28 views

On multiplying symmetric matrices by diagonal matrices with roots of unity

Given two symmetric non-zero and non-identity matrices $A,B\in\Bbb C^{n\times n}$ of same rank supposing there exists a non-identity diagonal matrix $D\in\Bbb C^{n\times n}$ containing only roots of ...
0
votes
3answers
29 views

On multiplying by diagonal matrices

Given two matrices $A,B\in\Bbb C^{n\times n}$ supposing there exists a diagonal matrix $D\in\Bbb C^{n\times n}$ such $$AD=DB$$ does that mean $A,B$ are diagonals and hence equal?
1
vote
1answer
36 views

Does invertible diagonalizable matrix have its inverted matrix diagonalizable?

From invertibility, $(PAP^{-1})^{-1}=PA^{-1}P^{-1}=D^{-1}$, the thing is how do you make sure the $P$ doesn't change?
3
votes
1answer
35 views

How to find $[T]^m _m$ for $T: \mathbb R^3 \rightarrow \mathbb R^3$ s.t $T(x) = Ax$ when only $A$ and $m$ are provided$?$

I am presented with the following problem: Let $T: \mathbb R^3 \rightarrow \mathbb R^3$ be linear and define it as such: $T(x) = Ax$ for some $3\times 3$ matrix $A$. Let $m_1, m_2, m_3 \in \mathbb ...
0
votes
0answers
33 views

Diagonalization of a matrix with complex eigenvalues

I'm dealing with the $3 \times 3$ matrix: $$ \begin{pmatrix} 3 & 4 & 0 \\ -4 & 3 & 0 \\ 0 & 0 & 1 \end{pmatrix}. $$ I have the characteristic equation, which is $(1 - L)(L^2 ...
0
votes
1answer
32 views

Why Must A Matrix be Symmetric for Orthogonal Diagonalization

So far, all we are doing in class is determine if the matrix A is symmetric, find the basis for the eigenspace P, and apply Gram Schmidt for it to be orthogonal. My question is; why must A be ...
2
votes
2answers
34 views

Proof about Diagonalization of A

The question asks WHY is it true that $$A^{n} = PD^{n}P^{-1}$$ I can never do proper proving in algebra; what I almost know for sure is that a proof by induction is the way to go here. But how do you ...
0
votes
1answer
26 views

How are the conditions for “diagonalizability” and “upper-triangularizability” of a linear operator different?

My understanding is that a linear operator is basically diagonalizable if it has as many eigenvectors as its dimension. But when can a linear operator be turned into a upper triangular matrix?
1
vote
1answer
22 views

Finding a diagonalizing matrix associated with Jordan

Find the Jordan normal form $J$ of the upper triangular matrix $A = \begin{pmatrix}2 & 0 & 1 & 2 \\ 0 & 2 & 2 & 1 \\ 0 & 0 & 2 & 1\\ 0 &0 & 0& 3 ...
1
vote
2answers
23 views

Unitary transformation

I have a matrix in following form $$A=\begin{bmatrix} 1&0&0&0&0&0&0&0\\ 0&-1&0&0&0&0&0&0\\ 0&0&1&0&0&0&0&0\\ ...
0
votes
2answers
25 views

a and b such that A is similar to B

Let $$A = \begin{bmatrix} -3 & 6 & 0 \\ -2 & 4 & 0 \\ 0 & 0 & -1 \\ \end{bmatrix} $$ I know the Eigenvalues of the matrix are $-1,0$ ...
1
vote
4answers
55 views

Is any square matrix over $\Bbb{C}$ diagonalisable?

I know the answer to the question is false (or so my professor said) but I am not sure why. I read http://www.imsc.res.in/~kapil/papers/matrix/ but did not quite understand it enough to answer my ...
0
votes
0answers
15 views

Inverse problem of covariance matrix – diagonalization of Hermitian operator

I have understood the two things respectively: 1. Use a set of observations to construct a covariance matrix, and then compute the eigenvectors of the matrix. 2. The diagonalization the Hermitian ...
0
votes
2answers
40 views

All values of $k$ for which matrix is diagonisable

Find all values of k for which matrix is diagonalizable: $$A= \begin{bmatrix} 1 & 1 & k\\ 1 & 1 & k\\ 1 & 1 & k \end{bmatrix} $$ The question contains multiple matrices ...