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
22 views

Transition matrix question,

In diagonalizing a matrix A, we use a matrix S, which consists of eigenvectors of A. To compute S, we simply take each eigenvector and write it as a linear combination of the standard basis. So if ...
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0answers
28 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
262 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 ...
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1answer
14 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?
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2answers
30 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
38 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 ...
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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
69 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
20 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 ...
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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
23 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
83 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 ...
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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
35 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
19 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 = ...
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1answer
30 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 ...
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1answer
49 views
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1answer
82 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 ...
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1answer
18 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
31 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, ...
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2answers
26 views

How to determine diagonalization [closed]

Let $p(t) = t^2(t^2 + 4t + 4)(t - 5)^7$ be the characteristic polynomial of a matrix $M$. Suppose $\dim Null(M) = 2$ and $\dim Null (M - 5I) = 7$. Indicate which of the following statements is true: ...
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
58 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
29 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 $A = QDQ^{-1}$ ...
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3answers
253 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
55 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
35 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
65 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 ...
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3answers
215 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 ...
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1answer
97 views

Diagonalize tri-diagonal symmetric matrix

How to diagonalize the following matrix? \begin{pmatrix} 2 & -1 & 0 & 0 & 0 & \cdots \\ -1 & 2 & -1 & 0 & 0 & \cdots \\ 0 & -1 & 2 & -1 & 0 ...
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1answer
23 views

Symmetric matrix with with all positive/zero elements. How to ensure it is PSD?

I have the following matrix A: symmetric all positive and/or zero values the main diagonal is all the same variable, x. To ensure that the matrix A, is PSD, must I only ensure that x>=0? It seems ...
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1answer
27 views

What is this Toeplitz like matrix called and how do I represent it as a convolution?

I have a matrix that is used to represent the Green's function in a popular class of fast E & M solvers (CG-FFT). The matrix represents distances, that are later filled in using the appropriate ...
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0answers
12 views

Compute the derivatives of an equation

I have an equation which is equal to: $(-c/2)ln(x) + (-c/2)tr(diag(B^TSB)x^{-1})$ Where $c$ is a constant, $tr$ represents the trace, $diag$ represents the diagonal. $B$, $S$ and $x$ are three ...
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4answers
81 views

Understanding the Jordan form via this example

The Question: I do not understand why the Jordan Form of the matrix: $A:=\begin{pmatrix} 1 & 1 \\ -1 & 3 \end{pmatrix}$ is: $J:=\begin{pmatrix} 2 & 1 \\ 0 & 2 \end{pmatrix}$? Here is ...
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1answer
20 views

Proof if $A$ is normal then it is nondefective

What is the proof that if $A$ ($m\times m$ Matrix) is normal i.e $(AA^{\ast} = A^{\ast}A)$ then $A$ is non defective i.e (for each eigenvalue of $A$, its algebraic multiplicity is equal to the ...
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2answers
48 views

Diagonalize the $n \times n$ matrix with ones along both diagonals.

I'm having some trouble diagonalizing this nxn matrix with ones along both diagonals: $\begin{bmatrix} 1&0&0&\cdots&0&0&1\\0&1&0&\cdots&0&1&0\\ ...
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2answers
27 views

Find a and b such that the matrix is diagonalizable

Find a and b such that the matrix $$ \left( \begin{array}{ccc} 1 & a \\ 0 & b \\ \end{array} \right) $$ is diagonalizable. I know that $$ D = S^{-1} A S $$ where S is a matrix made of the ...
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1answer
31 views

Linear Algebra 3x3 matrix diagonalization Row operation before inversing

Hello I am diagonalizing the matrix $$\begin{bmatrix} -1 & 2 & 2 \\ 2 & 2 & -1 \\ 2 & -1 & 2 \end{bmatrix}.$$ The eigenvalues I found are $-3$ and $3$. The eigenvectors are ...
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2answers
58 views

Prove that the eigenvectors of this matrix are a basis in $\mathbb{R}^n$

Let $A \in \mathbb{R}^{n \times n}$ and $w \in \mathbb{R}^n$. Suppose that, $w_i>0$ and $a_{i,j} = w_i / w_j$ for all $i,j=1,\dots,n$. Note that from the construction comes that ...
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1answer
25 views

Prove that the columns of the similarity matrix of a diagonalization are the eigenvectors

I'm interested in eigendecomposition of a matrix. It is clear for me, that you can eigendecompose a matrix if and only if it is diagonalizable. I'm looking for a short proof for that statement, that ...