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
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3answers
71 views

If square matrix A satisfying $A^2-4A+4I=0$ does it follow that A is diagonizable?

I am given the following statement and asked to determine whether it is true or false: If A is a n x n matrix, and $A^2-4A+4I=0$, then A is diagonizable. Any help is appreciated, thank you.
1
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0answers
43 views

$A \in M_n$ is diagonalizable $\iff$ the minimal polynomial has distinct roots

I have a proof ,written by someone, of : $A\in M_n$ is diagonalizable $\iff$ the minimal polynomial has distinct roots. The proof says: $A$ is diagonalizable $\iff$ A has n linearly independent ...
1
vote
2answers
30 views

Why does similarity with a diagonal matrix imply that the Jordan normal form must also be diagonal?

If a matrix representation of a linear transformation is similar to a diagonal matrix, why does this imply that the Jordan normal form must also be diagonal?
2
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1answer
32 views

How to compute the matrix $S$ in Sylvester's law of inertia

Sylvester's law of inertia states that for any symmetric matrix $A$ there exist an invertible matrix S such that, $S^T A S = D$, where $D$ is a diagonal matrix which has only entries 0, +1 and −1 ...
1
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0answers
27 views

Diagnalization of block matrix with circulat blocks

I have the following Matrix $A = \begin{pmatrix} X \\ Y \end{pmatrix}$ Where X, and Y are circulant Matrices. I want to diaganlize $AA^T$. I tried the following: $AA^T = \begin{pmatrix} ...
0
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1answer
20 views

Are all adjacency matrices (graph theory) diagonalizables?

If $A$ is an adjacency matrix of a graph $G$ and it can be diagonalized to get it in the form $A=PDP^{-1}$, with $D$ diagonal, is there any graph-theoretic interpretation to the matrices $P$ and $D$?
-1
votes
1answer
48 views

Why do eigenvalues exclusively form the main diagonal in a diagonalizable matrix?

So, why do eigenvalues exclusively form the main diagonal in a diagonalizable matrix? If we have $n\times n$ matrix ($n$ being a natural number) that is diagonalizable, why is it eigenvalues ...
0
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0answers
30 views

Diagonalizing to solve a linear recurrence with complex eigenvalues

I know how to solve for a closed form of linear recurrences whose matrix form has all real eigenvalues. What is the difference when solving one with complex eigenvalues? I can't seem to get this ...
0
votes
1answer
66 views

Can't orthogonally diagonalise this symmetric matrix.

So I have a symmetric matrix A = $\begin{bmatrix} 2 & 2 & -4 \\ 2 & -1 & -2 \\ -4 & -2 & 2 \end{bmatrix}$ and I want to orthogonally diagonalise it. I know that there are ...
3
votes
2answers
94 views

Linear algebra; orthogonal diagonalization

I am currently studying for my most important exam of the year, but there is something that I keep breaking my head about. Help would be greatly appreciated, because my books don't tell me how to find ...
1
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1answer
31 views

Can performing row (column) operations to a set of dependent eigenvectors produce a basis of vectors that are again eigenvectors?

I am having some trouble finding a basis of eigenvectors that diagonalizes two matrices simultaneously. I have found two bases of eigenvectors for two 3x3 matrices. I can't seem to find an ...
0
votes
1answer
47 views

From diagonalizing bases for matrices A and B, how do I find one basis that diagonalizes both matrices?

The first matrix A has eigenvectors (0,1,0), (2,0,1), (1,0,1) The second matrix B has eigenvectors (4,1,2), (1,1,0), (1,0,1). Both sets form a basis for $R^3$. Now, how do I pick out a basis of ...
2
votes
2answers
49 views

Problem concerning eigenspace and rank of some matrix

Problem: Let $N,P \in \mathbb{R}^{n \times n}$ be matrices, and let $P \neq 0$. Suppose that $P = NP$ and that $P$ is diagonalizable. Prove then that $N$ has an eigenspace with dimension greater than ...
1
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1answer
18 views

How to use inner products in C(n) to prove normal matrix is unitarily diagonalizable after knowing that normal matrix is diagonalizable?

I have showed that normal matrix is diagonalizable, but how can I know that the normal matrix is unitarily diagonalizable using inner products?
4
votes
1answer
69 views

Diagonalize the matrix A or explain why it can't be diagonalized

Diagonalize the matrix or explain why it cant be diagonalized $A=\begin{pmatrix}1 & 2 & 4 \\3 & 5 & 2 \\2 & 6 & 1\end{pmatrix}$ Hint: One eigenvalue is $λ=9$ So, i began the ...
2
votes
1answer
52 views

Diagonalization and Commuting Matrices

Attempt: I have shown part (ii) and I have found the eigenvalues and eigenvectors for A, B respectively and shown they can be diagonalised. I need help with (iii) and (iv), for (iii) I can't show ...
2
votes
2answers
88 views

Finding $P$ such that $P^TAP$ is a diagonal matrix

Let $$A = \left(\begin{array}{cc} 2&3 \\ 3&4 \end{array}\right) \in M_n(\mathbb{C})$$ Find $P$ such that $P^TAP = D$ where $D$ is a diagonal matrix. So here's the solution: $$A = ...
0
votes
1answer
23 views

Is it possible to determine if a matrix is not diagonalizable via row operations?

Suppose a matrix can be row reduced to the identity matrix, is this enough to say that it is not diagonalizable? If so, what theorem(s) or logic figures this out?
3
votes
1answer
36 views

Prove $T|_{V_\lambda}$ is diagonalizable

Let $V$, an $n$-dimensional vector space and let $T, S:V\to V$, two diagonalizable linear operators. Show that if $TS=ST$ then every $V_\lambda$ of $S$ is $T$-invariant and the restriction, ...
1
vote
1answer
41 views

Find a Basis $B$ of $R^2$ so that $B$ matrix of $T$ is diagonal

$T([1,1]^t) = [3,7]^t$ $T([1,-1]^t) = [1,1]^t$ Here's what I get: $T= \left(\begin{array}{cc}3 & 1 \\7 & 1\end{array}\right) $ The eigenvectors of $T$ is $E = \left(\begin{array}{cc} .4798 ...
1
vote
3answers
67 views

find the Jordan form and $P$ such that $P^{-1}AP = J$.

Consider the matrix $$A = \left(\begin{array}{cccc} -11&0&-9\\32&1&24\\16&0&13 \end{array}\right)$$ I want to find the Jordan form of $A$, with $1$-s at the bottom and the ...
3
votes
3answers
88 views

Diagonalize a symmetric matrix

let $$A = \left(\begin{array}{cccc} 1&2&3\\2&3&4\\3&4&5 \end{array}\right)$$ I need to find an invertible matrix $P$ such that $P^tAP$ is a diagonal matrix and it's main ...
1
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1answer
42 views

Finding the Jordan Form and basis

$$A= \begin{pmatrix} 2&1&2\\ -1&0&2 \\ 0&0&1 \end{pmatrix}$$ I found that $$f_A(x)=m_A(x) = (x-1)^3.$$ So the Jordan form must be: $$J= \begin{pmatrix} 1&0&0\\ ...
0
votes
3answers
62 views

Find $P$ such that $P^{-1}AP = J$

Let $$A = \begin{bmatrix} 1 & 1 & 0 & -1 \\[0.3em] 0 & -1 & 1 & 2 \\[0.3em] -1 & -1 & 0 & 0 \\ 0 & 0 & 0 & 0 ...
3
votes
1answer
32 views

Application of diagonalization of matrix - Markov chains

Problem: Suppose the employment situation in a country evolves in the following manner: from all the people that are unemployed in some year, $1/16$ of them finds a job next year. Furthermore, from ...
0
votes
1answer
90 views

Prove that $T$ is not diagonizable

I'm having difficulties with this exercise, can anyone give me a hand? Let $T:R^3 \rightarrow R^3$ be a linear transformation. It's know that $(1,1,0), (1,1,1)$ are eigenvectors of $T$ and: ...
1
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2answers
35 views

Prove that $A$ is diagonalizable and find similar matrices

Let $A$ be a matrix $(3x3)$ such that: $A(1,1,1)^t=(2,2,2)^t$ and $rank(2I + A) \lt rank(2I-A)$ I need to prove that $A$ is an diagonalizable matrix and find all the matrices that are similar to it. ...
1
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0answers
18 views

Finding point closest to origin on a hyperboloid

(1) Let A be 3x3 real symmetric matrix. The eigenvalues of $A$ are $\lambda_1 = -6, \lambda_2 = 1, \lambda_3=4$ $q(x_1,x_2,x_3) = -x_1^2 + x_2^2 -x^2_3 + 10x_1x_3 = 1$. $A$ is the matrix of $q$. I ...
1
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2answers
71 views

Diagonalizing a matrix. Is it necessary to use $P^{-1}AP=D$?

The matrix $D$ comes from $P^{-1}AP$, and has the form: $$ \begin{bmatrix} \gamma_1 & & \\ & \ddots & \\ & & \gamma_n \end{bmatrix} $$ When asked to diagonalize, can I just ...
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0answers
34 views

How to do the diagonal decomposition of this matrix?

Given a matrix G as a N*N symmetric matrix, in which $$G_{ij} = \begin{cases} \sum_{j=1}^{N} \frac{1}{R_{ij}}, &\quad \mbox{if j = i} \\ -\frac{1}{R_{ij}}, ...
0
votes
2answers
82 views

2014 IMC first problem first day (eigenvalues of a product of symmetric matrices).

This was the first problem of the IMC 2014. Let $A$ and $B$ be two $n\times n$ symmetric matrices with real entries which have all their eigenvalues strictly larger than $1$. Prove all the ...
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1answer
36 views

Connection between $S$ and $A$ when $S^{-1} A S$ is a diagonal matrix

In diagonalizing a matrix $A$, we use a matrix $S$, which consists of eigenvectors of $A$. To compute $S$, we simply take each eigenvector of $A$ and write it as a linear combination of the standard ...
0
votes
1answer
53 views

Is the converse of the Spectral Theorem true?

In the book by Friedberg, Insel and Spence, symmetric matrices are orthogonally diagonalizable, and over the complex number field, normal matrices are orthogonally diagonalizable -- this is all from ...
2
votes
4answers
283 views

If a matrix has positive, real eigenvalues, is it always symmetric?

We know that symmetric matrices are orthogonally diagonalizable and have real eigenvalues. Is the converse true? Does a matrix with real eigenvalues have to be symmetric? A class of symmetric ...
0
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1answer
15 views

Eigenvectors of a Symmetric Endomorphism

Prove that there isn't any symmetric endomorphism $f$ of $\mathbb R^3$ such that $e_1=(1,0,1)$ and $e_2=(1,1,1)$ are eigenvectors of $f$. I don't know how to do it, any hint?
2
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2answers
33 views

Enough evidence to conclude that a linear operator is diagonalizable

I was going over the following problem : (a) Let $T$ be a linear operator on a finite dimensional vector space $V$, such that $T^2=I$. Prove that for any $v \in V$, $v-Tv$ is either an eigenvector ...
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1answer
41 views

Making a matrix diagonal with its eigenvectors

I'm trying to make my matrix diagonal. this is my matrix (for matlab and octave) ...
0
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0answers
34 views

Help understanding a theorem about diagonalizable matrices

So while studying for my Linear Algebra test, I'm required to study some theorems and their proofs, and I have trouble understanding a particular part of the proof for the following (I'm translating ...
1
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1answer
75 views

If $A$ is a $12 \times 12$ real matrix such that $A^{17}=I$ , is $A$ diagonalizable ? Are all eigenvalues of $A$ real ?

If $A$ is a $12 \times 12$ real matrix such that $A^{17}=I$ , is $A$ diagonalizable ? Are all eigenvalues of $A$ real ?
1
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1answer
28 views

Solving the equation $A^{-1}=\alpha A+\beta I$ for diagonalizable matrix $A$.

Suppose that $A$ is an invertible $5 \times 5$ matrix with characteristic polynomial $(\lambda-2)^3(\lambda+2)^2$. If $A$ is diagonalize find $\alpha$ and $\beta$ such that. $$A^{-1}=\alpha A+\beta ...
0
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1answer
20 views

What is this notation supposed to mean? $diag\{ A_1, A_2, \cdots, A_N \}$

A paper has the following equation which I do not understand how to calculate the $diag$ function: $J = diag\{ A_1 \otimes A_1, A_2 \otimes A_2, \cdots, A_N \otimes A_1N \} \dot{}(Q^T \otimes ...
3
votes
2answers
39 views

Diagonliazing matrix

Suppose I have a linear operator $T : V \to V : v \mapsto A v$ I want to find the diagonalized version of $A$. Why do people don't just calculate the eigenvalues of $A$ and put them on a diagonal? ...
2
votes
4answers
78 views

What is the difference between using $PAP^{-1}$ and $PAP^{T}$ to diagonalize a matrix?

What is the difference between using $PAP^{-1}$ and $PAP^{T}$ to diagonalize a matrix? Can both methods be used to diagonalize a diagonalizable matrix $A$? Also does $A$ been symmetric or not ...
1
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0answers
21 views

Conjugate-diagonalizable matrix

I saw recently this weird definition in an exam: A matrix $A\in\mathcal M_n(\Bbb C)$ is said to be co-diagonalizable if there exists an invertible matrix $P$ and a diagonal matrix $D$ such that ...
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0answers
15 views

Reference for this notion: Conjugate-diagonalizable matrix

I saw recently this weird definition in an exam: A matrix $A\in\mathcal M_n(\Bbb C)$ is said to be co-diagonalizable if there exists an invertible matrix $P$ and a diagonal matrix $D$ such that ...
0
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0answers
32 views

Nonlinear Lie group from Fulton & Harris

On page 138 of my copy of the celebrated Representation Theory by Fulton & Harris, a proof is outlined to show that the real group of $3\times 3$ upper-triangular unipotent matrices modulo a ...
0
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2answers
31 views

What does it mean to find a basis that “diagonalizes” a transformation?

I'm having a hugely hard time wrapping my head around this statement. I am trying to figure it out on my own but I just don't get it. The terminology is weird to me and I can't really picture what it ...
1
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1answer
24 views

Taking limits of norms of a matrix raised to the nth power:

Given a matrix $$ A = \begin{bmatrix} 0 & 3 \\ -2 & 5 \\ \end{bmatrix} $$ and a vector $x = \begin{bmatrix}1&0\end{bmatrix}$, compute ...
0
votes
2answers
116 views

Find the value of $k$ for which matrix is diagonalizable

Consider the matrix $$A = \begin{bmatrix} 1 & 0 & 1 \\ 0 & 2 & k \\ 0 & 0 & 2 \\ \end{bmatrix}$$ where $k$ is a real number. The ...
0
votes
3answers
39 views

Diagonizable matrix

Got this matrix: \begin{bmatrix} 1 & 2 \\ -2 & 5 \end{bmatrix} I should determine if the matrix is diagonalizable or not. I found the eigenvalues ( only one) = 3. My eigenvector is then ...