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Questions tagged [diagonalization]

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.

2
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2answers
33 views

Jordan normal as transformation with respect to the basis of eigenvectors

I have the follwing matrix A: \begin{equation} A = \begin{pmatrix} 2 & 0 & 1 & -3 \\ 0 & 2 & 10 & 4 \\ 0 & 0 & 2 & 0 \\ 0 & 0 & 0 & 3 ...
0
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1answer
32 views

Prove that all normal matrices are semi-simple using Schur's decomposition

Is there any elegant proof that shows that all normal matrices are semi-simple that comes from Schur's decomposition or its corrolaries? There is a proof that normal matrices are unitary diagonizable ...
0
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1answer
39 views

Diagonizability of $n \times n$ matrix with two distinct eigenvalues and eigenspace dimension $n-1$

Let $\mathbf{A}$ be an $n \times n$ matrix with two distinct eigenvalues $\lambda_1$ and $\lambda_2$. If the dimension of the eigenspace $E(\lambda_1)$ is $n-1$, then $\mathbf{A}$ is diagonizable. ...
1
vote
1answer
19 views

Simultaneous Diagonalization With Non-Similar Eigenvectors

So I've been given two diagonal matrices with non matching eigenvectors, A:$$ \begin{matrix} 2 & 0 & 0 \\ 0 & 2 & 0 \\ -1 & 0 & 1 \\ \end{matrix} $$ and B: $$ ...
12
votes
3answers
2k views

Are these square matrices always diagonalisable?

When trying to solve a physics problem on decoupling a system of ODEs, I found myself needing to address the following problem: Let $A_n\in M_n(\mathbb R)$ be the matrix with all $1$s above its ...
0
votes
1answer
27 views

$A,B$ are simultaneously diagonalizable. $A = A^*$. Is $B = B^*$? [closed]

Title says it all really. $A = PD_AP^{-1}$, $B = PD_{B}P^{-1}$. We know $A = A^*$. Can we say for sure that $B = B^*$?
1
vote
1answer
3k views

Principal axis of a matrix

I try to find the definition of the main axis of a matrix. I saw this phrase in some exercise: Let $A$ be a positive matrix, $f:G\longrightarrow \mathbb{R}$ a smooth function, $G$ an open set in $\...
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0answers
25 views

Derivation of eigenvalues of a symmetric matrix

Given a diagonal matrix $D$, we symmetrize matrix $W$ by the following transformation: $S \equiv D^{1/2} W D^{-1/2} $ Now, $S$ is a symmetric matrix which is diagonalizable as $S = X \Lambda X^T$ ...
1
vote
1answer
36 views

Find the Matrix by given information [closed]

Can we obtain a matrix if we have its eigenvalues, algebraic multiplicity and geometric multiplicity of each eigenvalue ?
1
vote
1answer
83 views

Is every diagonalizable matrix a matrix exponential?

I know it is true in $\mbox{SL}_2(\Bbb C)$ and I think it is true in $M_n(\Bbb C)$ because if $M=PDP^{-1}$, we might be able to write D as $\exp(E)$ for some $E\in M_n(\Bbb C)$ as the exponential is ...
1
vote
3answers
106 views

If given $A^n = \alpha I$ prove $A$ is diagonizable

Given matrix $ A \in{M_n(\mathbb{C})}, 0\ne\alpha\in{\mathbb{C}}$ such that $A^n = \alpha I$ prove that $A$ is diagonizable. I've tried proving that the minimal polynomial($m_A$) splits into linear ...
0
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0answers
20 views

Diagonalisable and similar Matrix

So for a similar matrix being there exist invertible matrix P such that $A=P^{-1}BP$. The main question is : i) what intuition exist for a similar matrix (I’m aware linear transformation is the ...
2
votes
1answer
57 views

If $B$ satisfies $A B = B A^{-1}$, show that $B^2$ is diagonalisable

Let $A$ be a $n \times n$ non-singular matrix having distinct eigenvalues. If $B$ is a matrix satisfying $A B = B A^{-1}$, show that $B^2$ is diagonalisable. Answer: Let $\lambda_i, \ i=1,2,3, \...
1
vote
1answer
31 views

Matrix similar to permutation matrix

Given an invertible matrix $A$, when is it possible to find a matrix $S$ such that $$A = SPS^{-1},$$ where $P$ is a permutation matrix? When this is possible, is there a practical method (say, similar ...
0
votes
1answer
32 views

What value in d makes this matrix diagonalisable over the field R?

Question: What value in $d$ makes the matrix $$ \begin{bmatrix} 0 & d & 0 \\ 1 & 0 & d \\ d & 1 & 0 \\ \end{bmatrix} $$ diagonalisable over the field ...
2
votes
0answers
49 views

Proof on diagonalizable

Suppose $T\in\mathcal{L} (\mathbb{R^5})$ is defined by $$T(x_1,\dots,x_5) = (x_1+\dots+x_5,\dots,x_1+\dots+x_5)$$ Proof if T is diagonalizable. Proof: $$T(x_1,\dots,x_5) = (x_1+\dots+x_5,\dots,x_1+\...
0
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0answers
11 views

Why not different diagonal elements in orthogonal reduction of quadratic form?

Suppose we have quadratic form $(3x)^2 -(2y)^2-z^2-(4xy)+(8xz)+(12yz)$ and we are asked to reduce it to cannonical form. The steps would be: Find eigen values (3,6,-9 in this case) of the matrix of ...
1
vote
1answer
112 views

Knowing $M^2 + M = \begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix}$, find the eigenvalues of $M$

Let $M \in M_2(\mathbb R)$ and $$A = \begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix}$$ such that $$M^2 + M = A$$ Find the eigenvalues of $M$. Here is what I did: Let $\lambda$ be an eigenvalue ...
0
votes
1answer
41 views

Generalised Eigenvectors Issue

Ok so i have been doing a few questions on 'Diagonalising' defective matrices, the method I've been using to find generalized Eigenvectors is to make the previous Eigenvector the subject. However i ...
0
votes
4answers
70 views

If $A^2 + b A + c I = 0$ why does $A$ have to be diagonalizable?

If $A \in \mathbb C^{n \times n}$, $b, c$ are scalars such that $c \ne \frac 14 b^2$ and $$A^2 + b A + c I = 0$$ why does $A$ have to be diagonalizable? I am getting to a point where I think $A$ has ...
0
votes
1answer
35 views

Show that a matrix is diagonalizable for all values of a

I have to show that the matrix A is Diagonalizable for all values of a. Given the matrix A. \begin{pmatrix} a+3 & 4 \\ 5 & 5 \end{pmatrix} I first found the characteristic polynomial for A:...
2
votes
2answers
65 views

why any matrix of $SU(2)$ is diagonalizable?

"Any matrix $A\in SU(2)$ can be written as $\begin{pmatrix}z & -\overline w \\ w & \overline z\end{pmatrix}$ for some $w,z\in\mathbb C$ with $|w|^2+|z|^2=1$. Moreover, every matrix of this ...
0
votes
3answers
56 views

Let $A$ be a $3\times 3$ matrix with characteristic polynomial $x^3-3x+a$, for what values of $a$ given matrix must be diagonalizable.

Let $A$ be a $3\times 3$ matrix with characteristic polynomial $x^3-3x+a$. For what values of $a$ given matrix must be diagonalizable. I am talking about diagonalizability over reals. Efforts: If a ...
1
vote
1answer
49 views

What value of b make the following matrix diagonalisable

$$ \begin{bmatrix} 1 & -1 \\ 1 & b \\ \end{bmatrix} $$ It seems like there's no two distinct eigenvalues to make it diagonalisable. Can anyone confirm that is true?
1
vote
1answer
23 views

Proving that any unitary matrix can be diagonalised by a similar matrix

I'm having struggles with understanding important facts about spectral theorem in finite dimensional spaces. For hermitian matrices, I saw in classes that the similarity matrix that diagonalises any ...
1
vote
2answers
53 views

Existence of diagonalizable matrix

Let $A \in \mathbb{C}^{4 \times4}$ and $A^5 = I$. Is $A$ always diagonalizable? How to show $|\mbox{tr} (A)| \leq 4$? If $\mbox{tr} (A) = 4$, can we determine matrix $A$? Some ...
0
votes
2answers
51 views

How can I block diagonalise this matrix?

I have this matrix: $$A = \left( \begin{array}{cccc} 0 & 0 & 0 & -1 \\ 1 & 0 & 0 & -1 \\ 0 & 1 & 0 & -1 \\ 0 & 0 & 1 & -1 \\ \end{array} \right)$$ ...
3
votes
2answers
67 views

Representations of $\mathbb{G}_m$

I know that the multiplicative affine group scheme $\mathbb{G}_m$ is diagonalizable, since the algebra that represents it is $k[X,X^{-1}]$, which is isomorphic to the group algebra $k[\mathbb{Z}]$. ...
0
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0answers
45 views

How come a matrix not diagonalizable if the geometric multiplicity is less than the algebraic multiplicity?

I understand that the algebraic multiplicity is the number of times an eigenvalue appears as a root of the characteristic polynomial. I also understand that the geometric multiplicity is the dimension ...
4
votes
0answers
55 views

If zero is an eigenvalue are dimensions lost?

This is likely a silly question so sorry in advance. However, I am wondering if I am right in thinking that if zero is an eigenvalue, then some dimension must be lost. My understanding is that ...
0
votes
3answers
69 views

Similarity between a matrix and diagonal matrix

How to prove a matrix and a diagonal matrix are similar? Is there some rule to follow or are there some steps to follow? Like: suppose a matrix $A$ = something, prove that $A$ is similar to a ...
0
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0answers
52 views

Properties of $n \times n$ complex matrix with $A^m = I$

If $A^m = I_n$ then what can we say about the eigenvalues and diagonalizablity of $A$? The equation given above is an annihilating polynomial of $A$ and therefore minimal polynomial divides it. Since ...
1
vote
4answers
105 views

Are all matrices almost diagonalizable?

Every real $2$ by $2$ matrix that is not diagonalizable is similar to the $2$ by $2$ jordan canonical form, $$ J_2=\begin{bmatrix}s&1\\0&s\end{bmatrix}, $$ where $s$ is the eigenvalue (with ...
3
votes
1answer
94 views

(Unitary) diagonalization of $A = I-xy^*$

I'm continuing to prepare for a Linear Algebra exam and found another problem that puzzles me. Let $A = I+xy^*$, where $x,y \in \mathbb{C}^m (\neq 0)$. (a) Determine a necessary and ...
0
votes
1answer
22 views

Alternative definition of diagonalisable transformation

Supposedly a transformation $T: V \to V$ is diagonalisable iff there exists a basis of $V$ consisting only of eigenvectors of $T$. Can someone show me why this is true? I don't really know where to ...
2
votes
1answer
70 views

Eigenvalues, diagonalization and convergence of matrices

I am trying to wrap my head around some basic results in Linear Algebra. I am trying to avoid more abstract concepts like Rank-Nullity, and stay in simple properties at an introductory level. (I've ...
3
votes
2answers
3k views

Block-diagonalizing an antisymmetric matrix

I was wondering how to block-diagonalize a $10 \times 10$ antisymmetric matrix into block matrices along the diagonal. Can I just diagonalize each non-diagonal block? Thanks!
4
votes
1answer
558 views

Is a Jordan block not further block diagonalizable?

We can always find a Jordan canonical form of a matrix A. It is a block diagonal matrix. Is it true that each block cannot be reduced to a matrix with more blocks in diagonal? In other words, for a ...
9
votes
1answer
123 views

When can two matrices have zero diagonal in the same basis?

It's common to ask if two Hermitian matrices $A$ and $B$ can be diagonalized in the same basis. Is there an efficient way to check if they can be made hollow in the same basis? By hollow, I mean that ...
0
votes
1answer
44 views
1
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0answers
23 views

show that $C(C[u])=K[u]$

Let $E$ be a $K$-vector space of dimension $n$, and $u$ be a diagonalizable linear map from $E$ to $E$. Let $C[u]$ be the set of linear maps from $E$ to $E$ which commute with $u$. In other words,...
0
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0answers
10 views

What would be the restrictions to diagonalization of this type transformation of Laplacian matrix?

This question is a little specific. I have read about graph theory and saw in one place the following transformation of an Laplacian matrix $L$: \begin{eqnarray} ULW=\begin{bmatrix} -(l_{11}-l_{22})&...
0
votes
3answers
37 views

Find 3x3 matrix by determinant and 2 eigenvalues/-vectors

I have two eigenvectors: $(2, 1, -1)'$ with eigenvalue $1$, and $(0, 1, 1)'$ with eigenvalue $2$. The corresponding determinant is $8$. How can I calculate the $3\times3$ symmetric matrix $A$ and $AP$?...
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2answers
52 views

Is $A \in \mathcal{M}_n(\mathbb{C})$ diagonalizable?

$A \in \mathcal{M}_n(\mathbb{C})$ such that $A^2$ has got $n$ distinct non zero eigenvalues. Show that A is diagonalizable. Attempt : As $A^2$ has got $n$ distinct non zero eigenvalues. The ...
0
votes
2answers
997 views

Showing that two matrices are similar by showing they are similar to the same diagonal matrix

The exercise gives two matrices $A$ and $B$ and asks you to show they are similar by showing that they are similar to the same diagonal matrix, and then after that find and invertible matrix $P$ such ...
0
votes
1answer
37 views

Can you tell how many eigenvectors a matrix has from just the characteristic equation?

If the equation has a repeated root, can you tell without evaluating in the matrix if that repeated root corresponds to more than one eigenvector?
0
votes
1answer
767 views

When diagonalizing a matrix, in what order should you arrange the the eigenvectors to form the invertible matrix $P$?

I was following this example online to diagonalize a matrix. It lists the eigenvectors as $\lambda =3,2,4$ (note the order). It then arranges each eigenvalue's corresponding eigenvector (3 column ...
0
votes
1answer
35 views

Diagonalizing a real normal matrix

Given the matrix $A = \begin{pmatrix} 1 & 2 & 3 \\ 2 & 3 & 4 \\ 3 & 4 & 5 \end{pmatrix}$, how would I find a real orthogonal matrix $P$ such that $PAP^t$ is a diagonal matrix? ...
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0answers
10 views

Routines for diagonalization of banded hermitian matrices?

I have a problem in which I need to diagonalize matrices with many thousands of complex elements three times. I know that the matrices are hermitian and sparse. Specifically, they consist of 9 bands ...
11
votes
1answer
14k views

Hermitian Matrices are Diagonalizable

I am trying to prove that Hermitian Matrices are diagonalizable. I have already proven that Hermitian Matrices have real roots and any two eigenvectors associated with two distinct eigen values are ...