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
30 views

Does Fermat's Little Theorem apply to matrices?

I'm working on a problem involving applying FLT to matrices, so any information about how to do this or prove this is true would be helpful. I've been doing some research and experimenting a little, ...
1
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1answer
58 views

How to show that T is diagonalizable iff $\dim S_{\lambda_1}+\dim S_{\lambda_2}+\dots+\dim S_{\lambda_k}=\dim V $

Theorem: $V$ is a vector space on field $F$. and $T:V\to V $ is linear transformation. $\lambda_1,\lambda_2,\dots,\lambda_k$ are eigenvalues and $ S_{\lambda_1},S_{\lambda_2},\dots,S_{\lambda_k} ...
2
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3answers
48 views

Finding the matrix exponential

Find the matrix exponential of $$\begin{bmatrix}1& 1\\ 0& 1\end{bmatrix}.$$ Since this matrix is not diagonalizable, you will have to use the definition of the matrix exponential. ...
0
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1answer
28 views

Proof for real symmertic, skew-symmetric matrices are always diagonalizable?

Was reading a lot about certain kinds of matrices being always diagonalizable, like, real symmetric, skew-symmetric, etc, What would be the proof behind it ?
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3answers
37 views

Find the conditions required for the values of a, b, and c that make the following matrix symmetric.

Set up the system: $$A = \begin{bmatrix} 5& a+b+c& a-b \\ 3& -7& 2\\ 1& a+c & 6 \end{bmatrix}$$ I did it like this: ...
1
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1answer
23 views

finding a power of a matrix

When you are given the eigenvectors and eigenvalues of a matrix A and are asked to solve for A^3, for the formula A = PDP^-1 is your diagonal matrix the identity matrix with the eigenvalues swapped ...
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0answers
10 views

Diagonalization of quadratic forms over $\mathbb{Q}$

I'm having difficulties in finding the diagonal forms of some quadratic forms. I am sure it is not supposed to be that difficult but I guess I am lacking some creativity after overdoze of coffee and ...
0
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1answer
33 views

If A is an nxn matrix of complex numbers such that some power of A is identity, then i want to prove that A is diagonalizable over C

If A is an nxn matrix of complex numbers such that some power of A is identity, then i want to prove that A is diagonalizable over C. How the proof will be followed from the jordan canonical forms?
1
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1answer
31 views

Using EigenValues to form a diagonal matrix

After going through my Linear algebra note, I know if for any matrix $A$, we find the eigenvalues and eigenvectors , we can construct a matrix P, such that $P^{-1}AP$ is a diagonal matrix. Now for ...
1
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1answer
39 views

Hermitian matrix the only diagonizable

During the last lecture one of my professors claimed that the hermitian matrix is the ONLY complex matrix which was diagonizable. This seems strange to mee (not to say a very very strong claim to ...
4
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2answers
50 views

Characteristic polynomial of a mapping from matrices space to matrices space

Let $T$ be the linear map from $M_n \to M_n$ given by TX=AX, while A is as well a matrix $n \times n$ (a) Write out the characteristic polynomials for $T$ (b) Show that if A is ...
2
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2answers
88 views

Diagonalizing the matrix $A$, when $A^2$ is diagonalizable

If the matrix $A^2$ is diagonalizable and $A$ is invertible, is $A$ diagonalizable? I know it is not true if we leave out the invertibility. For example if $ A= \begin{pmatrix} 0 & 1 ...
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3answers
53 views

Is this a diagonalizable matrix?

question: Suppose $A$ is a $3\times3$ matrix such that $\det(A)=0$, $\det(A+2I)=0$, $\det(A-3I)=0$. Is $A$ diagonalizable? Is $A$ invertible? What is the rank of $A$? So the ...
3
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4answers
69 views

why the matrix is diagonalizable?

show that matrix $$A = \begin{bmatrix} a & b \\ 0 & a \end{bmatrix}$$ is diagonalizable iff $b = 0.$ I do not understand why. Cuz if a, b are reals, I can always find a ...
2
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1answer
49 views

Can't show that these matrices are diagonalizable.

Consider that for each $n \times n$ (possibly complex) matrix, $A_{k}$, $0 \leq k \leq m$, we have that \begin{align} A_{0}A_{k} &= kA_{k}, \qquad 1 \leq k \leq m \end{align} and suppose that ...
1
vote
0answers
47 views

When is a matrix diagonalizable?

The matrix $A$ is diagonalizable with $A=SDS^{-1}$ where $D$ is a diagonal matrix and $S$ is the matrix with eigenvectors as columns iff $A$ has linearly independent eigenvectors. Correct?
2
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2answers
61 views

diagonalize a non-normal matrix , without distinct eigenvalues

I wonder how to diagonalize a matrix that is non-normal, and does not have distinct eigenvalues. Let $\lambda_i$ be the eigenvalue, and $v_i$ be the eigenvector with that eigenvalue. I think the ...
3
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1answer
30 views

Linear Algebra question on diagonlization Please check my work

My first question is that is a basis for each eigenspace the same thing as a corresponding eigenvector for an eigenspace? Could someone tell me if im doing this correctly? I have the matrix ...
1
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2answers
52 views

Proof that if powered matrices are equal to $E$, then they are diagonalizable

I need help with this: Prove this: If $A^k=E$, then $A$ is diagonalizable. Could anyone help me please?
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0answers
31 views

Matrix factorisation/decomposition?

Find matrices A and B such that $$ \begin{bmatrix} 4 & 2 & 1 \\ 1 & -4 & 1 \\ \end{bmatrix} =A \begin{bmatrix} 2& 0 & 1 \\ 0 & 3 & 0 \\ \end{bmatrix} B $$ I've only ...
0
votes
2answers
27 views

Diagonalization of a strange transformation

Let be $V$ a vector space on $\mathbb C$ and $\dim V=4$ and let be $f \in \operatorname{End}(V)$ such that $\operatorname{Im}(f^2+a \cdot \operatorname{id}) \subset \ker(f+id)$ where $a=\det f$, ...
1
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2answers
324 views

Proving that $\dim(\mathrm{span}({I_n,A,A^2,…})) \leq n$

Let $A$ be an $n\times n$ matrix. Prove that $\dim(\mathrm{span}({I_n,A,A^2,...})) ≤ n$ I'm at a total loss here... Can someone help me get started?
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0answers
32 views

Diagonalization of a linear transformation in the polynomial vector space

Let $V = R_3[X]$ be the vector space of polynomials with real coefficients of degree at most 3 and consider the linear transformation $V \rightarrow V$ defined by $f_a(p(x))=p(1-ax)$ for each $p(x) ...
0
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1answer
54 views

How can I find a matrix $\bf B$, with positive eigenvalues, such that its square $\bf B^2$ is another matrix $\bf A$?

I've been given a 2x2 matrix $\mathbf A$, its eigenvalues $\lambda_1$ and $\lambda_2$, its eigenvectors $\mathbf v_1$ and $\mathbf v_2$, and a diagonal matrix $\mathbf D = \text{diag}(\lambda _1, ...
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0answers
63 views

Block Diagonalisation of 4x4 Matrix

I'm attempting to find a 4x4 matrix, P, that will convert my matrices, $A = \begin{bmatrix}1&1&0&0\\0&-1&0&0\\0&0&-1&0\\0&0&0&-1\end{bmatrix}$ and, ...
1
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2answers
52 views

Diagonalization of a complex matrix

I'm trying to prove this: Let $A$ be a complex matrix. If $A^2$ is diagonalizable and $A$ is invertible then $A$ is diagonalizable. So, if $\lambda$ is an eigenvalue of $A$ then $\lambda²$ is an ...
0
votes
1answer
28 views

Show that the set of all subsets of an infinite enumerable set is not enumerable

I know this problem involves using Cantor's theorem, but I'm not sure how to show that there are more subsets of an infinite enumerable set than there are positive integers. It seems like a lot of ...
0
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1answer
38 views

Transition matrix homework

Suppose that firms in a particular industry fall into one of three size categories: large, medium and small. If a firm is large one year, the probabilities that it will remain large, fall into the ...
6
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0answers
121 views

Matrix diagonalization theorems and counterexamples: reference-request.

I'm looking for exhaustive list of diagonalization theorems and counterexamples in linear algebra. In this question I understand the question of matrix diagonalization very broadly: suppose we have ...
2
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0answers
50 views

The minimal polynomial of A is dividing $x^{2013} -1$, prove A is diagonalizable over the complex field

$A $ is $nxn$ real matrix. The minimal polynomial of A is dividing $x^{2013} -1$. I need to prove that: (1). A is diagonalizable over the complex field. (2). If A is diagonalizable over the reals, ...
0
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1answer
54 views

How to compute the diagonal matrix for this problem?

I did find the basis but I have no clue in solving the diagonal matrix part of the problem. Could someone please help me?
1
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1answer
39 views

How can be this quadratic diagonalizable?

I have a problem with this quadratic form: $$q(x,y,z) = x^2+ay^2+3z^2+2xy+2xz+2ayz$$ What I have to do is to clasify it depending on the value of $a$ I get stuck in this step: $$A = ...
0
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0answers
15 views

Is any method which allows segmentation of long diagonalizing procedures?

This is a question for a smarter way of numerical computation. When I diagonalize a certain type of Vandermonde-matrices in Pari/GP ("mateigen(M)"), for instance of size 16x16 then this can be ...
0
votes
1answer
56 views

How to diagonalize a matrix?

I have given the matrix \begin{equation} A = \left(\begin{array}{cc} 4 & 2 \\ 4 & 6 \end{array}\right) \end{equation} And I have to diagonalize it, but I have no idea from where to start. ...
0
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3answers
259 views

Diagonalizable properties of triangular matrix

How to show that an upper triangular matrix with identical diagonal entries is diagonalizable iff it is already diagonal?
0
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1answer
60 views

Positive semidefinite but non diagonalizable matrix -proof of non-negative eigenvalues

I have a question about positive semidefinite matrices that are non diagonalizable. Example: \begin{equation} A= \left(\begin{array}{cc} 2 & 1\\ 0 & 2\\ \end{array}\right) \end{equation} ...
0
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2answers
103 views

Is the interval $[0,1]$ equinumerous with $\mathbb{R}$ [duplicate]

I recall reading a proof that showed these two sets were equinumerous, but I'm having trouble finding it. Is there any intuitive method to show that they are in fact equinumerous? It seems like since ...
1
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1answer
52 views

Linear Algebra Dynamical System Help

I was just wondering, for a dynamic system does the origin always have to be an attractor, saddle point, or repellor? Also if a matrix isn't diagonalizable then does that mean the origin cannot be a ...
1
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2answers
63 views

Unitary similarity transformation

I have a matrix: $ A= \dfrac{i}{3} \begin{bmatrix} 1&-2&1\\-2&1&1\\1&1&-2\end{bmatrix} $ Could someone explain me how to find a corresponding diagonal matrix for a ...
2
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1answer
21 views

Linear Algebra Linear transformation Help

If $T:\mathbb{R}^n \rightarrow \mathbb{R}^n$ is a linear transformation, then there exists a basis for $\mathbb{R}^n$ in which $T$ is diagonal. Is this true or false
0
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1answer
38 views

A question about diagonalizable.

For which $x$ is $$M=\begin{pmatrix}4&0&-2\\x&5&4\\0&0&5\end{pmatrix}$$ diagonalizable? I know a matrix which is diagonalizable can be written in the form $A=S\Lambda S^{-1}$ ...
1
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2answers
67 views

A question about diagonalizable matrices

Let $A$ be a square matrix such that $A \ne0$, but $A^k=0$ for some integer $k \gt1$. show that $A$ is not diagonalizable. Could somebody give me some hints?Many thanks
2
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1answer
57 views

The density of diagonalizable matrices of $M_n(\mathbb{C})$ problem.

For any matrix $A = (a_{ij})_{1\leq i,j\leq n} \in M_n(\mathbb{C})$, we pose $||A|| = \max_{1\leq i,j\leq n} |a_{ij}|$. $1.$ Show that $||.||$ define a norm on $M_n(\mathbb{C})$ and that $\forall A, ...
1
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1answer
26 views

Is there any relation between Minimal polynomial of a matrix in M(n,C) and its diagonalizablity?

Is there any relation between Minimal polynomial of a matrix in $M(n,\mathbb C)$ and its diagonalizablity? I want to mean looking at the roots of minimal or characteristic polynomial can we say about ...
1
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1answer
49 views

Which entries could you change to make this 2 by 2 matrix diagonalizable ? [Strang P309 6.2.14]

For all $c \in \mathbb{C}$, the matrix $A = \begin{bmatrix} c & 1 \\ 0 & c \\ \end{bmatrix}$ is not diagonalizable because the rank of $A - cI = 1.$ Change one entry to ...
5
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3answers
52 views

Can non-normal matrices with double eigenvalues never be diagonalized?

Is there a matrix $A$ with $A^TA≠ AA^T$ (non-normality) and double eigenvalue that is still diagonalizable? If $A^TA \neq AA^T$ and $λ_1 =λ_2 = λ$ (double eigenvalue) $\stackrel{?}{⇒}$ not exists ...
1
vote
1answer
31 views

Associated Bilinear Form to Q (Quadratic Form)

I need to diagonalize the quadratic form $Q(x) = {x_{1}}^{2} + 2x_{1}x_{2} + 2{x_{2}}^{2} + 2x_{2}x_{3} + {x_{3}}^{2}$ so I know I need to find the associated Bilinear form with $B(x,x) = Q(x)$ - the ...
2
votes
0answers
86 views

Diagonalization of Vandermonde matrix

Is there a method to diagonalize (at least some) $ n \times n $ Vandermonde matrices? For example invertible matrices which has method to invert them with Cramer method for example, but there is some ...
3
votes
2answers
50 views

Show a matrix is normal - check my proof

Short easy question, I just want someone to double check what I did. We are given that $T$ is an invertible, normal matrix. We are asked to show that $T^{-1}$ is also normal, and find it's unitary ...
6
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2answers
98 views

$\lim_{k \to \infty} A^k$ where $A$ is diagonalizable

I'm reviewing diagonalization and am wondering if the following makes sense. Let $A \in \mathcal{M}_{n \times n}(\mathbb{R})$ be a diagonalizable matrix. That is, there exist matrices $D$ and $P$ such ...