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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1answer
36 views

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 ...
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1answer
22 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 ...
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1answer
27 views

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

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 ...
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1answer
49 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 ...
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1answer
12 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 ...
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3answers
56 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} ...
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0answers
5 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 ...
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1answer
42 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 ...
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2answers
65 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 ...
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1answer
28 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 ...
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1answer
38 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 ...
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4answers
117 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 ...
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0answers
22 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 ...
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2answers
31 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 \\ ...
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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?
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0answers
43 views

Compute the Eigenvectors & Show A is diagonalizable

$A = \begin{bmatrix} 1&2&1 \\ 0&1&0 \\ 1&3&1 \\ \end{bmatrix} $ I computed the eigenvalues: $λ_ 1 = 1$ $λ_ 2 = 0$ $λ_ 3 = 2$ The ...
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1answer
21 views

Proof that triagonal matrices with distinct diagonal elements are similar

I'm trying to prove that if $A,B$ are triangular matrices with distinct elements along their main diagonals then the matrices are similar. I have been interpreting this to mean that the elements ...
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1answer
32 views

Diagonalization of Matrix

I have a problem that says to find the values of k for which the matrix $A$ is NOT diagonalizable over $\mathbb{C}$. I know that I need to find the zeros for the characteristic polynomial and check ...
2
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1answer
28 views

Determining diagonalizability of a linear transformation defined by a matrix.

Suppose $A\in M_n(\Bbb C)$ satisfies $A^6-A^3+I=O$. Prove that if a linear transformation $T:M_n(\Bbb C)\rightarrow M_n(\Bbb C)$ is given by $T(B)=AB$, then $T$ is diagonalizable. How to prove it? ...
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1answer
48 views

Schur decomposition of a matrix with distinct eigenvalues is almost unique

Let $M\in \mathbb C^{n,n}$ have $n$ distinct eigenvalues, and let $U_1, U_2$ be two Schur-forms of $M$. Show that if $U_1, U_2$ have equal diagonals, there is a hermitian diagonal matrix $Q$ such ...
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0answers
30 views

linear algebra diagonalization [duplicate]

$$A =\begin{pmatrix}a &b \\ c & d \end{pmatrix}$$ show that: $A$ is diagonalizable if $(a-d)^2 +4bc > 0$ $A$ is not diagonalizable if $(a-d)^2 + 4bc\leq 0$
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1answer
21 views

What kind of a matrix has a unitary diagonalizing matrix?

Suppose $D = P^{-1} A P$. When is $P$ unitary? In other words, what kind of a matrix $A$ should be, such that $D=P^{\dagger}AP$? i.e. what are the conditions a matrix must have to be able to ...
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1answer
18 views

Diagonalization and linear equation

I have $\mathbf{x}(t) = \mathbf{V} \mathbf{u}(t)$. Now I have to show that $\mathbf{x}'(t) = \mathbf{A} \mathbf{x}(t)$ implies that $\mathbf{u}'(t) = \mathbf{D} \mathbf{u}(t)$. How do I do this? I ...
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1answer
35 views

If $D=P^{-1}AP$, then $f(D)=P^{-1}f(A)P$?

Suppose I have a diagonalizable matrix $A$, such that $D = P^{-1}AP$ Can I apply an element-wise function $f$ and expect that $f(D)=P^{-1}f(A)P$, assuming $f$ is not a linear transofrmation? Or in ...
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1answer
42 views

Does a diagonal matrix commute with every other matrix of the same size?

Does a diagonal matrix commute with every other matrix of the same size? I'm stuck on one line of a proof that I am writing, and I would like to switch order between a non-diagonal and a diagonal ...
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2answers
41 views

Diagonalisable or not?

Let $$A = \begin{pmatrix}a & b\\c & d\end{pmatrix}.$$ Show that 1) $A$ is diagonalisable if $(a - d)^2 + 4 bc > 0$ 2) $A$ is not diagonalisable if $(a - d)^2 + 4 bc < 0$
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1answer
15 views

Complex numbers property proof. [duplicate]

I eas given this quesstion in one of my Linear Algebra course with the excercises regarding minimal polynomialsm eigenvalues and diagonalizable matrix: Show that for any two numbers $a,b \in ...
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1answer
49 views

Factorize matrix determinant

When trying to diagonalize a matrix, say : $$\left(\begin{matrix} 0 & 2 & -1 \\ 3 & -2 & 0 \\ -2 & 2 & 1 \end{matrix}\right)$$ to find the eigenvalues, I have to find ...
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0answers
33 views

Simultaneously diagonalizable without distinct eigenvalues

It is a well known result that if $u$ and $v$ are two diagonalizable endomorphisms of a $\mathbb{C}$ finite-dimensional linear space $E$, if $u$ (or $v$) has distinct eigenvalues and if $u$ and $v$ ...
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1answer
34 views

Setting up a matrix from a recurrence relation to find diagonal matrix?

Considering the recurrence $F_n= F_{n−1}+3F_{n−2}−2F_{n−3}$ where $F_0=0$, $F_1=1$ and $F_2=2$, use diagonalization to find a closed form of the expression. If the sequence is continued the numbers ...
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3answers
30 views

Tell if $A$ is diagonalized using it's characteristic and minimal polynomials

$$A= \left( {\matrix{ 2 & 1 \cr 1 & 2 \cr } } \right)$$ I already calculated that $f_A(x) = (x-3)(x-1)$. Also, the minimal polynomial must be also $(x-3)(x-1)$. How can I use ...
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2answers
40 views

Is this diagonal matrix unique? [closed]

How can I determine if the diagonal matrix is unique? \begin{bmatrix} 1 & \frac{1}{2}-\frac{1}{2}i & \frac{1}{2}+\frac{1}{2}i \\ 2 & -i & i \\ 1 & 1 ...
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1answer
30 views

A is normal if and only if every matrix unitarily equivalent to A is normal

I need to prove that if $A$ and $B$ are unitarily equivalent, then $A$ is normal if and only if $B$ is normal. The proof is as follows: Suppose $A$ is normal and $B = U^*AU$, where $U$ is unitary. ...
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3answers
26 views

Diagonalize Matrix A

I am told to diagonalize Matrix A i solved for P and P inverse ...
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2answers
62 views

New proof about normal matrix is diagonalizable.

I try to prove normal matrix is diagonalizable. I found that $A^*A$ is hermitian matrix. I know that hermitian matrix is diagonalizable. I can not go more. I want to prove statement use only this ...
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1answer
23 views

Show matrix is element in eigenspace

Let $A$ be an $n\times n$ matrix such that $A^2=A$. a) Let $E_{1}(A)=\{x \in \mathbb{R^n} | Ax=x \}$: let $E_{0}(A)=\{{ x \in \mathbb{R^n} | Ax=0\}}$. Let $x$ be any vector in $\mathbb{R^n}$. Show ...
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1answer
52 views

Orthogonal diagonalization of a symmetric matrix

Find an orthogonal matrix $P$ that diagonalizes $$\begin{pmatrix}-1 &4 &-2\\ -3& 4 &0\\ -3 &1& 3\end{pmatrix}.$$ My eigenvalues are 1 , 2 and 3 but my $P$ while verifying is ...
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1answer
124 views

Solving a recurrence with diagonalization?

Considering the recurrence $F_n=F_{n-1}+3F_{n-2}-3F_{n-3}$ where $F_0=0$, $F_1=1$ and $F_2=2$. Use diagonalization to find a closed form expression for $F_n$. So I first continued the recurrence to ...
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0answers
22 views

Solving a homogeneous linear system of differential equations: no complex eigenvectors?

I have to solve the following equation by diagonalization. $ X' = \begin{bmatrix}1 & 1\\1 & -1\end{bmatrix} X$ I was able to determine the complex eigenvalue roots: $det(A-\lambda I)=0$ ...
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1answer
39 views

Diagonalization of a Matrix in terms of other matrices and eigenvalue

Task: Let A be a symmetric matrix having only one eigenvalue λ and C be a matrix that diagonalizes A by a similarity transformation. Find a simplified expression for A in terms of λ, C, and I, the ...
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2answers
32 views

Find bases for eigenspaces of A

$$A = \begin{pmatrix} 6 & 4 \\ -3 & -1\end{pmatrix}$$ Find the bases for eigenspaces $E_{\lambda_1}$ and $E_{\lambda_2}$ of $A$. I don't really know where to start on this problem.
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1answer
29 views

Diagonalizing zero matrix

Consider the matrix $A = 0$ that is diagonalized by the matrix $$S = \begin{bmatrix} 5 & 2 \\ 2 & 1 \end{bmatrix}.$$ What is the diagonal matrix? I'm confused because I thought you could ...
0
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1answer
16 views

If T is a linear operator, has dim V distinct eigen values, then T has a diagonal matrix with respect to some basis of V.

My question is "Why did they write T has a diag matrix w.r.t some basis of V". Give T belongs to L(v), dosent that say we already fixed the basis ?
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2answers
35 views

Diagonalisable matrices over different fields

I believe this fits in with my knowledge of Jordan Normal form, however I am not sure how to approach the question itself? I am especially lost with $F_7$
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2answers
133 views

$AB=BA$ with same eigenvector matrix

I read in G. Strang's Linear Algebra and its Applications that, if $A$ and $B$ are diagonalisable matrices of the form such that $AB=BA$, then their eigenvector matrices $S_1$ and $S_2$ (such that ...
0
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1answer
32 views

Diagonalization with complex eigenvalues

Given $A = \begin{bmatrix} 0 & 1 \\ -4 & 0 \end{bmatrix}$ find a matrix C of the form $ C = \begin{bmatrix} a & -b \\ b & a \end{bmatrix}$ and a matrix P such that $A = PCP^{-1}$. ...
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1answer
38 views

Linear Algebra quadratic forms diagonalization

I have a question that reads: Diagonalize the quadratic form $A(x,x) = 2x^2 - 1/2 y^2 -2xy - 4xz$ by completing the squares, and find the change of basis matrix and the new basis in which A will be ...
0
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1answer
34 views

Linear Algebra Quadratic Form Diagonalization

I asked this question the other day but I still didn't understand it. Hopefully someone can get through to me this time. I have a question that reads: Diagonalize the quadratic form A(x,y) = 3x^2 - ...
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2answers
50 views

Diagonalizing Quadratic Forms. Linear Algebra

I have a question that reads: Diagonalize the quadratic form $A(x,y) = 3x^2 -12xy + 7y^2$ by completing the square. What is diagonalization? Is that when I should find the eigenvector matrix, ...
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0answers
40 views

Is there a proof of the theorem on diagonalization of Hermitian matrices without induction?

If $T: V->V$ and $\dim V=n$ and $T$ is hermitian or skew-hermitian, there exist $n$ eigenvectors which form an orthonormal basis for $V$. And, matrix of $T$ relative to this basis is diagonal ...