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
34 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 ...
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3answers
59 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 ...
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3answers
71 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
41 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\\ ...
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3answers
60 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 ...
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1answer
27 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 ...
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1answer
87 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: ...
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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. ...
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0answers
16 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 ...
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2answers
70 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
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2answers
71 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
30 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
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1answer
52 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 ...
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4answers
271 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
32 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
40 views

Making a matrix diagonal with its eigenvectors

I'm trying to make my matrix diagonal. this is my matrix (for matlab and octave) ...
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0answers
30 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 ...
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1answer
74 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 ?
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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
19 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 ...
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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? ...
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4answers
70 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 ...
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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
30 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 ...
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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 ...
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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 ...
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2answers
90 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
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3answers
37 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 ...
3
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2answers
60 views

Proof: are restriction to operators diagnolizable if the operator is?

"Let $V$ be a vector space of finite dimension $n\ge1\ \ T:V\Rightarrow V$ a linear operator and $S$ a T-invariant space of $V$. Consider the restriction $$T|_S:S\Rightarrow S$$ and $$(T|_S)(w):=T(w) ...
0
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1answer
38 views

Diagonally dominant matrix

Assume $A$ is a positive definite matrix, and $B$ is a matrix with zero row sum. Does matrix $A$ exist such that $AB$ is strictly diagonally dominant?
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0answers
20 views

How to find a real valued similarity transform for block diagonalization

I have a real-valued square matrix $A$ with $n$ eigenvalues with zero real part and $m$ eigenvalues with non-zero real part. How do I find a real-valued similarity transform $T$ such that $A = ...
0
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1answer
31 views

Eigenvectors times diagonal matrix, still eigenvectors?

Suppose we have a $n\times n$ real symmetric positive definite matrix $\Sigma$, and $V=(v_1,...,v_n)$ whose columns are the eigenvectors corresponding to the $n$ eigenvalues $\lambda_1\geq \lambda_2 ...
0
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1answer
53 views
1
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1answer
83 views

For which $\beta \in \mathbb{C}$ is the matrix $A=\bigl(\begin{smallmatrix} 0&1\\1&\beta \end{smallmatrix}\bigr)$ diagonalisable?

I have got a question refering to the following problem. Let $K=\mathbb{C}$. For which $\beta \in \mathbb{C}$ is this matrix diagonalisable? $$A=\pmatrix{0&1\\1&\beta}$$ I think that it is ...
0
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1answer
19 views

Prove that $B^n$ is diagonalisable for all $n=2,3,\dots$ and that every eigenvalue of $B^2$ is the square of some eigenvalue of $B$.

I would like to ask you for some help in the following problem: Suppose that a matrix $B$ is diagonalisable over $\mathbb{C}$. Prove that $B^n$ is diagonalisable for all $n=2,3,\dots$ and that ...
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1answer
38 views

prove that matrix is diagonal by matrix rank and eigenvalue rank

$A$ is matrix $9\times9$ with rank of $5$, there is rank$(A-3I)=5$, the matrix has another eigenvalue of 5. I need to prove that $A$ is diagonal and find the similar diagonal matrix of $A$. I'm stuck, ...
1
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1answer
34 views

W is T-invariant. Define $\bar T: V/W \to V/W, \bar T(v+W)=T(v)+W$.Prove if $T_W$ and $\bar T$ are diagonalizable without common eigenvalue, then is T

$T$ is a linear operator on a finite dimensional vector space $V$, and $W$ be a $T$-invariant subspace of $V$. Define $\bar T: V/W\to V/W$ by $\bar T(v+W)=T(v)+W$. It can be proved that $\bar T$ is ...
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2answers
59 views

Compute power of a matrix $A$ as $n\rightarrow \infty$

We are given $A^p=A ...A$(p times) And we are given matrix A: $A=\begin{vmatrix}0.6&-0.4&0\\-0.4&0.6&0\\0&0&0.5\end{vmatrix}$ I need to compute $A^p$ as p approach Infinity. ...
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2answers
18 views

Special method of solution for $A\vec x=\vec b$ where $A$ is a square matrix such that $A^tA$ is diagonal and has full rank?

Is there any special shorter method of solution other than cramer's rule for solving a system of $n$ linear equations in $n$ unknowns $A\vec x=\vec b$ where the square matrix $A$ has the property that ...
1
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1answer
30 views

Roots of Matrices and Diagonalization

Question: For which of the following matrices $A_i$ is there A complex matrix $B$ such that $B^2 = A_i$; A self-adjoint complex matrix $B$ such that $B^2 = A_i$; A real matrix $B$ such that $B^2 = ...
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1answer
31 views

Prove that $A$ diagonalizable.

Let $A$ be an $n \times n$ matrix, and let $v_1,...,v_n$ be a basis of $R^n$ so that each $v_i$ is an eigenvector of $A$. Prove that $A$ diagonalizable. Does the diagonalization of $A = QDQ^{-1}$ ...
4
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3answers
264 views

What's so useful about diagonalizing a matrix?

I'm told the the purpose of diagonalisation is to bring the matrix in a 'nice' form that allows one to quickly compute with it. However in writing the matrix in this nice diagonal form you have to ...
2
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2answers
58 views

For what values of k is this singular matrix diagonalizable?

So the matrix is the following: \begin{bmatrix} 1 &1 &k \\ 1&1 &k \\ 1&1 &k \end{bmatrix} I've found the eigan values which are $0$ with an algebraic multiplicity of $2$ ...
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0answers
36 views

Interpreting a diagonalized matrix

I'm doing a practice question before a test: The stress in a solid at a point P can be described by a matrix T called the stress matrix (or stress tensor). If n is a normal vector to a plane cutting ...
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2answers
70 views

Matrix diagonalisable in R, but not in C.

I know is quite easy to find a matrix $A\in\mathbb{R}^{2,2}$ that is diagonalisable if the base field is $\mathbb{C}$, but not diagonalisable if the base field is $\mathbb{R}$. The easiest example can ...
0
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1answer
21 views

Finding eigenvalues/vectors of a matrix and proving it is not diagonalisable.

I have got the following matrix. $$\begin{pmatrix} -7 &4 \\ -9 &5 \end{pmatrix}$$ I need to find the eigenvalues, eigenvectors and $\textbf{prove}$ that it is not diagonalisable. I have ...
5
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3answers
222 views

Prove that $ND = DN$ where $D$ is a diagonalizable and $N$ is a nilpotent matrix.

Let $A$ be an $n \times n$ complex matrix. Prove that there exist a diagonalizable matrix $D$ and a nilpotent matrix $N$ such that a. A = D + N b. DN = ND and show that these matrices are uniquely ...