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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2
votes
1answer
23 views

Kernel and diagonalizability of endomorphism $f:\mathbb{R_2[x] \to R_2[x]}$ such that $f(p)=p(1)x^2-p(k),$ for $k \in \mathbb{R}$.

Problem: Let $f:\mathbb{R_2[x] \to R_2[x]}$ be the endomorphism on the space of polynomials of degree less or equal than two such that $$f(p)=p(1)x^2-p(k),$$ for $k \in \mathbb{R}$. I have to ...
1
vote
1answer
22 views

invariant subspace under some conditions

This came out at my linear algebra exam and I was not able to solve it. Let $f\colon \mathbb{R^{3}} \rightarrow \mathbb{R^{3}}$ be a linear transformation such that $\langle f(u),f(v) \rangle = ...
-1
votes
1answer
14 views

Construcut a matrix knowing its mnimal polynomial [on hold]

In Hoffman's Linear Algebra there's the exercise: Find a matriz $3\times 3$ wich the minimal polynomial is $x^2$(section 6.2, page 198 exercise 6) Thank you very much for the help.
1
vote
2answers
53 views

Can a matrix have eigenvalue with infinite multiplicity?

Suppose we have matrix of the form $$ A= \begin{bmatrix} a & -1 \\ 0 & a \\ \end{bmatrix} $$ and we would like to analyze its diagonalizability. By taking the ...
1
vote
2answers
27 views

Show that if $v\in (V_c)^{\perp}$ then $(Av)\in (V_c)^{\perp}$ for a normal matrix $A$ with an eigenvalue $c$

Suppose $A \in M_{n\times n}(\mathbb C)$ is a normal matrix and $c$ is an eigenvalue of $A$. I'm trying to show that if $v\in (V_c)^{\perp}$ then $(Av)\in (V_c)^{\perp}$. I know that if we were ...
0
votes
1answer
27 views

Orthogonally diagonalizing a matrix

Can anybody explain how to orthogonally diagonalize the following matrix: $$ \begin{pmatrix} 9 & \sqrt10 \\ \sqrt10 & 0 \\ \end{pmatrix} $$ Am I correct in ...
1
vote
2answers
30 views

Similarity in two 2x2 Matrices and finding the S in A=SBS-1

I am doing something wrong here and I am not sure what. The object of the exercise is to find the S for similar matrices $A$ and $B$. $A=SBS^{-1}$ with $B=\begin{pmatrix}4& 1\\1& ...
1
vote
1answer
31 views

Eigenvalues of the matrix $AA^*$

Suppose $A \in M_{n\times n}(\mathbb C)$ and let $B=A A^*$. Show that all the eigenvalues of $B$ are non-negative real. Can you please give me an hint how to start the proof? All I know is that ...
-3
votes
1answer
36 views

Let A be an $n \times n$ square matrix. Is $A^t$ $A$ diagonalisable? [closed]

Let $A$ be an $n \times n$ square matrix. Is $A^t$$A$ diagonalisable? True or false
3
votes
2answers
85 views

For which values of $a$ the matrix is diagonalizable

Given the following matrix: $$B=\begin{bmatrix} 1 & 0 & 0 \\ 1 & 0 & a^2 \\ 1 & 1 & 0 \end{bmatrix}$$ I tried to find for which values of $a$, the matrix $B$ is diagonalizable. ...
3
votes
3answers
47 views

If $f_A(x) \ne m_A(x)$ and $A^3=I$ then $A=I$?

Suppose $A \in M_{3\times3}(\mathbb R)$ and $f_A(x) \ne m_A(x)$ where $f_A(x)$ is the characteristic polynomial of $A$ and $m_A(x)$ is the minimal polynomial of $A$. If we were to assume that ...
1
vote
1answer
28 views

Find diagonalizable and nilpotent parts of matrix

Let B = $\begin{bmatrix}1 & -2\\\ 2& -3\\\end{bmatrix}$ I found the jordan form BQ = QJ, where Q = $\begin{bmatrix}2 & 1\\\ 2& 0\\\end{bmatrix}$ and J = $\begin{bmatrix}-1 & 1\\\ ...
1
vote
2answers
31 views

A is singular. Show that A is diagonalizable over $\Bbb{R}$

Let $A\in M_{10}(\Bbb{R})$ be singular matrix such that $\mathrm{rank}(I-A)=4 \ \mathrm{and} \ \mathrm{rank}(3I-A)=7$. Show that $A$ is diagonalizable over $\Bbb{R}$. Which diagonal matrix $A$ is ...
1
vote
1answer
48 views

Prove these quadractic forms are equivalent over $\mathbb{Z_5}$

Consider the following quadractic forms, defined in the field $\mathbb{Z_5}$, $$q(x, y, z, t) = 2y^2 + z^2 + 2t^2 + 4xy + 2xt + 4yt$$ $$q_0(x, y, z, t) = x^2 + y^2 + z^2 + dt^2$$ Prove they are ...
0
votes
1answer
46 views

Find the basis of this bilinear form

Let $A$ be the matrix of a bilinear form in a certain basis $\{v_1,v_2,v_3\}$, where $A$ is: $$ A= \left[ \matrix { 1 & 0 & 2 \\ 0 & -1 & 1 \\ 2 & 1 & 1 } \right] $$ I've ...
3
votes
4answers
67 views

Show that matrix $A$ is NOT diagonalizable.

Let $A$ be a square matrix $A^2=0$ and $A\neq0$ and show that it is not diagonalizable. I decided to use the sample matrix of $$A = \begin{bmatrix}0 & 1\\0 & 0 \end{bmatrix}$$ which satisfies ...
0
votes
1answer
30 views

Is there an explicit formula for this recursive series of matrices

I want to get an efficient way of computing $P_k$ from $P_0$ that satisfies the following recursion: $P_k = FP_{k-1}F^T+Q$ Where $P_k$, $F$ and $Q$ are matrices ($Q$ is diagonal, if it changes ...
0
votes
0answers
19 views

Why is unitary diagonalization works?

I've been told that if you take an Hermitian matrix, find it's eigenvectors, normalize them and write them as columns of a matrix, $P$ then: $$P^{-1}AP = D$$ Where (Magically) $D = ...
3
votes
1answer
38 views

What can we conclude from the equality $m_A(x) = m_B(x)$? [closed]

Suppose $A,B \in M_{n\times n}(\mathbb C)$ and $m_A(x) = m_B(x)$. Is one of the following propositions is true? (1) $f_A(x) = f_B(x)$ (2) A is invertible if and only if B is invertible. I think ...
0
votes
2answers
27 views

What can we say about $(A+I)^3$ in case $f_A(x) = m_A(x)$?

Suppose $A \in M_{3\times3}(\mathbb C)$ and $f_A(x) = m_A(x)$ where $f_A(x)$ is the characteristic polynomial of $A$ and $m_A(x)$ is the minimal polynomial of $A$. If we were to assume $(A+I)^3=0$, ...
0
votes
1answer
21 views

how to form a change of basis matrix with eigenvectors

So I have found $3$ eigenvectors: $E(1): (2,-1,1), E(2): (1,0,0), (0,0,1)$ Where $E(i)$ is just the eigenvalue. So how do I determine my change of basis matrix? In my textbook they just say the ...
2
votes
1answer
41 views

How to say if Eigenvectors of A are orthogonal or not? without computing eigenvectors

I am give matrix : $$A=\begin{bmatrix} 0&-1 & 2 \\ -1 & -1 & 1 \\ 2 & 1 &0 \end{bmatrix} $$ 1. Without finding the eigenvalues and eigenvectors, determine whether the ...
0
votes
0answers
34 views

Is a family of commuting self adjoint operators simultaneously diagonalizable?

Let $V$ be a finite-dimensional inner product space over $\mathbb{R}$. Let $\mathscr{A}$ be a family of self-adjoint operators on $V$ such that $ST=TS$ for all $S,T\in \mathscr{A}$. Then, ...
1
vote
1answer
37 views

What can we say about $m_A(x)$ with respect to $m_{A^2}(x)$ when $A$ is diagonalizable?

Suppose $A$ is a real $n\times n$ matrix and diagonalizable over $R$. Is one of the following propositions is true? (1) $m_{A^2}(x)$ divides $m_A(x)$ (2) $m_A(x)$ divides $m_{A^2}(x)$ I think ...
2
votes
2answers
36 views

Normal, Real, and Square matrix which is diagonalize over C but not over R

I'm trying to find out a normal, real and $\boldsymbol n\times \boldsymbol n$ ($n\ge3$) matrix $A$ which is diagonalize over $C$ but isn't diagonalize over $R$. I know that the following matrix ...
0
votes
0answers
27 views

Explicit diagonalisation transformation for this complex matrix

I need to diagonalise a complex $2\times2$ matrix, $M$. I need the explicit diagonalisation transformation, i.e. a matrix, $S$ s.t. $S^{-1}MS= \text{diag}(m_1,m_2)$. I am talking about a matrix of the ...
7
votes
0answers
69 views

Is a normal matrix satisfying $A^TA=…$ circulant?

Let $A=\{a_{ij}\}$ be a normal matrix such that $a_{ij}\geq 0$ with equality iff $i=j$. Suppose that $$ A^TA=\begin{pmatrix} a & b & \cdots & b\\ b & a & \ddots & \vdots\\ ...
1
vote
1answer
24 views

a question about normal matrices

Let $A$ be a normal matrix and $\lambda$ a scalar. Show that $A-\lambda I$ is also a normal matrix. $A$ is a normal matrix then there is a unitary diagonalization of $A$ over $\mathbb{C}$. ...
1
vote
1answer
56 views

Conditions for diagonalizability of $n\times n$ anti-diagonal matrices

Let $A$ be an $n\times n$ anti-diagonal matrix: $a_{i,j}=0$ unless $i+j=n+1$. A) When is $A$ diagonalizable (what are the conditions on the $a_{i,n+1−i}$)? B) Find the eigenvalues and eigenvectors ...
1
vote
1answer
53 views

Must eigenvectors of unitary matrix be orthogonal?

Suppose $A$ is a square complex unitary matrix. Assume that $u$ and $v$ (in $\mathbb C$) are both different eigenvectors of $A$. Is it possible that the dot product of these two vectors isn't zero? ...
3
votes
5answers
200 views

If $A^4=I$ then $A$ must be diagonalizable?

Suppose we have a real matrix $A$ which satisfies $A^4=I$, can we determine if $A$ is diagonalizable? I believe the answer is that we can't because all we know about the matrix $A$ is that it is ...
2
votes
1answer
55 views

Nilpotent Matrices properties

Let $N \in M_n(\mathbb{F})$ be nilpotent. Prove that for any $1 \leq k \in \mathbb{N}$ a matrix $B \in M_n(\mathbb{F})$ exists such that $B^k=I+N$ I have no idea how to het started here. We've just ...
1
vote
2answers
86 views

Simultaneously diagonalization of two matrices.

Let $A$ be a real symmetric matrix and $B$ a real positive-definite matrix. Is it possible to simultaneously diagonalize of $A$ and $B$? Thank you very much.
0
votes
0answers
59 views

Proving matrix similarity for given matrices

Let $\mathbb{F}$ be a field, and let $A=(a_{ij})_{i,j=1}^n$ and $B=(b_{ij})_{i,j=1}^n$ be matrices in $M_n(\mathbb{F})$ such that: a. $ \forall 1 \leq i,j \leq n: b_{ij}=0 \iff a_{ij}=0$ b. ...
1
vote
2answers
40 views

Two square matrices with the same minimial polynomial are similar for $n=5$ or $n=6$ [duplicate]

Let $\mathbb{F}$ be a field, $\lambda \in \mathbb{F}$ and $A,B \in M_n(\mathbb{F})$ such that $m_A(x)=m_B(x)=(x-\lambda)^k$ and such that the geometric multiplicity of $\lambda$ in $A$ equals to the ...
1
vote
1answer
71 views

About diagonalization [duplicate]

"Let A = $\begin{bmatrix}1 & 1 & 4\\0 & 3 & -4\\0&0&-1\end{bmatrix}$. Is the matrix A diagonalizable? If so find a matrix P that diagonalizes A. Can you write A as a linear ...
0
votes
1answer
29 views

Transformation of coordinate axis to make matrix diagonal

Consider the matrix $$ A= \begin{bmatrix}1/8 & \frac{-5}{8\sqrt{3}} \\ \frac{-5}{8\sqrt{3}} & 11/8 \end{bmatrix} $$ which of the following transformations of the coordinate ...
2
votes
1answer
30 views

$A_i \sim B_i \implies \text{Diag}(A_1 \ldots A_n) \sim \text{Diag}(B_1\ldots B_n) $ [closed]

How do I prove that: $A_i \sim B_i \implies \text{Diag}(A_1 \ldots A_n) \sim \text{Diag}(B_1\ldots B_n) $ Notation: $A\sim B$ meaning is $A$ is similar to $B$. Also, $A_i, B_i$ are square matrices ...
3
votes
1answer
83 views

Two questions about diagonalization

Let A = $\begin{bmatrix}1 & 1 & 4\\0 & 3 & -4\\0&0&-1\end{bmatrix}$. Is the matrix A diagonalizable? If so find a matrix P that diagonalizes A. Can you write A as a linear ...
1
vote
1answer
15 views

Lagrange Method for Presenting Bilinear form as sum of squares

I have the following question in my assignment which I'm having a hard time solving. For the following bilinear form, present find a digonal form (diagonal matrix form): What I thought to do at ...
1
vote
3answers
68 views

Symmetric Matrix Transformation

Here's the question, Let $T$ be the transformation of 2 by 2 real symmetric matrices defined by: \begin{bmatrix}a&b\\b&c\end{bmatrix}>>>>\begin{bmatrix}c&-b\\-b&a\end{bmatrix} ...
0
votes
0answers
6 views

Extremal singular values of $P\Phi D$

Let $A=P\Phi D$ be a matrix where $P$ is a projection matrix such that $R(P)\subset R(\Phi)$ and $D$ is a non-singular diagonal matrix. Is there any relation between $\sigma_{min}(A)$ and ...
1
vote
2answers
48 views

What is the linear space of Eigenvectors associated with a certain Eigenvalue?

The following matrix $A$ has $\lambda=2$ and $\lambda=8$ as its eigenvalues $$ A = \begin{bmatrix} 4 & 2 & 2 \\ 2 & 4 & 2 \\ 2 & 2 & 4 \end{bmatrix}$$ let $P$ be the ...
0
votes
2answers
71 views

Under what assumptions is it correct to say “a matrix is diagonalizable if and only if its eigenvalues are real”?

A $2\times 2$ matrix is diagonalizable if and only if its eigenvalues are real. Which statement is most correct: The proposition is true only if the eigenvalues are all greater than zero. The ...
0
votes
1answer
31 views

If $T^k = Id$ for $k\ge 1$ then $T$ is diagonalizable [duplicate]

Let $V$ a finite dimension space over $\mathbb{C}$ and $T:V\to V$, a linear transformation such that $T^k = Id$ for $k\ge 1$. Prove that $T$ is diagonalizable. I'd be glad for an hint. How do I ...
2
votes
2answers
55 views

How to determine if a 3x3 matrix is diagonalizable?

The matrix is given as: $A=\begin{bmatrix} 0 & 1 & 1 \\0 & 0 & 4 \\ 0 & 0 & 3 \end{bmatrix}$ So the matrix has eigenvalues of $0$ ,$0$,and $3$. The matrix has a free ...
3
votes
4answers
124 views

Diagonalization and find matrix that corresponds to the given condition

Diagonalize the matrix $$ A= \begin{pmatrix} 1 & 2\\ 0 & 3 \end{pmatrix} $$ and find $B^3=A$. I derived $A \sim \text{diag}(1,3)$ but I have problem finding any $B$. I tried to solve it by ...
1
vote
0answers
30 views

Eigenvectors and Generalized Eigenvectors

I've wondered whether someone could calrify me what are Generalized Eigenvectors, and why can I use them to find triangular form of a matrix. Say I have a $3\times3$ matrix, and I want to bring it to ...
2
votes
3answers
43 views

Restrictive definition of diagonalizable matrix

There is a theorem that says that every matrix of rank $r$ can be transformed by means of a finite number of elementary row and column operations into the matrix $$D=\begin{pmatrix} I_r & O_1 \\ ...
1
vote
0answers
7 views

How to diagonalize $D_1U^\dagger D_2U$ with unitary $U$ and real diagonal $D_{1,2}$

Is there a trick to diagonalize the following expression $$D_1U^\dagger D_2U$$ such that $$D_1U^\dagger D_2U=V^\dagger D\,V$$ where $U$ is unitary and $D_1$ and $D_2$ are diagonal and real?