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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.

12
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465 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 ...
8
votes
0answers
235 views

When a matrix has same eigenvalues of its column-swapped version?

What are the properties needed for a matrix $A$ to have $\mbox{Spec}(A)= \mbox{Spec}(A \cdot P)$, where \begin{equation} P = \begin{pmatrix} 0 & \cdots & 0 & 1 \\ \vdots & \...
6
votes
0answers
86 views

Why is it useful to know when a linear operator on a vector space is diagonalizable?

I'm currently taking a conceptual course in linear algebra, and I'm trying to understand why it would be theoretically useful or illuminating to know when a linear operator (or its matrix ...
5
votes
0answers
1k views

Understanding part of the proof of Spectral Theorem for symmetric matrices

I'm reading a textbook where the Spectral Theorem for symmetric matrices is proven. I understand almost everything about the proof except for one thing. The theorem is stated as follows: Theorem: Let ...
5
votes
0answers
125 views

Diagonalization of a big scary matrix

I would need to diagonalize this tridiagonal block matrix $M$: $$M = \begin{bmatrix} A & B & & \\ B^T & A & B & \\ & B^T & A & B \\ & & \ddots & \...
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 ...
4
votes
0answers
32 views

How to 'diagonalise' this special $(N+1)\times(N)$ matrix (described in the text)?

Here is a special $(N+1)\times N$ matrix: $$A=\begin{pmatrix}a_1&a_2&a_3&\ldots&a_N\\b_1&0&0&\ldots&0\\0&b_2&0&\ldots&0\\0&0&b_3&\ldots&...
4
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0answers
66 views

What can we say about the eigenvalues and diagonalization of this $2N\times2N$ matrix $A$?

There is a $2N\times2N$ matrix $A$, where $N$ is a positive integer, which is of the form: $A=\left(\begin{array}{cc} B & C\\ -C^{*} & -B^{*} \end{array}\right),$ where $B$ is a hermitian ...
4
votes
0answers
172 views

Diagonalization of matrices of the form BAB

If $A$ and $B$ are real symmetric matrices, the matrix $BAB$ is also a real symmetric matrix. My question is : knowing the orthogonal diagonalization of $A$ and $B$, can we obtain the orthonormal ...
3
votes
0answers
60 views

Diagonalizability condition involving direct sum of nullspace and range

I'm working on the following problem from Axler's Linear Algebra Done Right (Exercise 5.C.5): Suppose $V$ is a finite-dimensional complex vector space and $T\in \cal{L}(V)$. Prove that $T$ is ...
3
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0answers
47 views

Matrix functions which are not scalar coefficient power series expansions.

Inspired by this question, I started to wondering (firstly) if there exist any (useful / famous) matrix functions which are not defined as power series expansions with scalar coefficients, but rather ...
3
votes
0answers
94 views

Simultaneous orthogonal diagonalization

Statement Let $A, B \in M_{n\times{}n}(\mathbb{R})$ be symmetric and commutative such that $AB=BA$. Then $A$ and $B$ are simultaneously orthonormally diagonalizable. i.e. there is $Q \in O(n)$ such ...
3
votes
0answers
90 views

Is a principal submatrix of a diagonalizable matrix diagonalizable?

Let $A$ be diagonalizable, i.e., $A=X \Lambda X^{-1}$ for some diagonal matrix $\Lambda$. Consider $B$ which is a principal submatrix of $A$. Does there exist an invertible matrix $Y$ and a diagonal ...
3
votes
0answers
143 views

Measure of diagonal dominance

Given any (singular or non-singular) matrix, is there a standard measure for how close the matrix is close to diagonal dominance? For example this matrix: \begin{pmatrix} 5 &2 &2 \\ 5 &1 &...
3
votes
0answers
49 views

Proof of diagonalizing

Let $k<n$ and $v_{1},v_{2},...,v_{k} \in \mathbb{R}^{n}$ be non-zero vectors, orthogonal with respect to the standard inner product in $\mathbb{R}^{n}$, where the vectors of $\mathbb{R}^{n}$ ...
3
votes
0answers
105 views

Interpretation of a projection on one eigenspace (from symmetric matrices) to another

I have a reference $N \times N$ symmetric matrix -- with distinct eigenvalues -- decomposed using SVD as: $$ R_{ref} = V_{ref} D_{ref} V^{-1}_{ref} $$ If i get a matrix $S$ -- with distinct ...
3
votes
0answers
83 views

If $A\in\Bbb \{\pm1,0\}^{n\times n}$ is symmetric of rank $<n$, does $A-I$ have rank $n$?

Supposing we have a symmetric matrix $A\in\Bbb \{\pm1,0\}^{n\times n}$ of rank $m<n$ with all $+1$ or all $-1$ or all $0$ as diagonals and no $0$ on non-diagonals, when do we have $A-I$ to be full ...
3
votes
0answers
430 views

Simultaneous diagonalization of commuting matrix

I have 3 diagonalizable matrices $A,B,C$. They commute with each other $[A,B]=[B,C]=[A,C]=0$ [edit] The matrix $A$ is Hermitian but $B$ and $C$ have no properties. [/edit] I can get the eigenvalues ...
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+\...
2
votes
0answers
20 views

For which values ​the matrix is ​diagonalizable

For which values ​​of $a$ matrix $A$ is ​​diagonalizable? $$A = \pmatrix{0&i\\i&a}$$ in the case that it is not diagonalizable determine a base of Jordan Attempt: The minimal polynomial ...
2
votes
0answers
37 views

Spectral decomposition of an operator

Given the operator: $$\begin{pmatrix} i & 0 & - 4\\ 0 & - 3i & 0\\ 2 & 0 & - i \end{pmatrix}$$ Now $det(\lambda1-A)=(z+3i)(z^2+9)=(z+3i)^2(z-3i)$, I know a theorem that says ...
2
votes
0answers
90 views

Is it possible for a $3 \times 3$ matrix to have rank 1 but not be diagonalizable?

Is it possible for a $3 \times 3$ matrix to have rank $1$ but not be diagonalizable? If the matrix only has the top left entry, then obviously it is already diagonal. But what about other two entries?...
2
votes
0answers
36 views

Does this fact about the minimal polynomial give an efficient diagonalizability criterion?

In a very nice paper "When Is a Linear Operator Diagonalizable?" by Marco Abate (Amer. Math. Monthly 104 (1997), 824-830) I found the following nice description of the minimal polynomial $\mu(T)$ of ...
2
votes
0answers
60 views

Order of eigenvectors sometimes important?

For the matrix $$\left(\begin{array}{cc} 2 & -1 \\ -1 & 2 \\ \end{array}\right)$$ I find $\lambda=1,3$ which gives me $\lambda_{1}=\frac{1}{\sqrt{2}}\left(\begin{array}{c} 1 \\ ...
2
votes
0answers
56 views

Subspace of $M(3,\Bbb R)$ true or false questions

Let $W \subset M(3,\Bbb R)$ be a vector subspace such that $\dim W=7$ I need to determine if this statements are true of false: $\exists\: A \in W$ such that $A$ is not null and A is diagonalizable ...
2
votes
0answers
40 views

Generalizing matrix diagonalizability to abstract tensors

It is well-known that, if $V$ is (say) a real vector space of finite dimension $n$, then there can be established an isomorphism between the space of endomorphisms $\mathrm{End}(V)$ (or the space of $...
2
votes
0answers
99 views

About Jacobi Method for computing eigenvalues of matrices ( Trefethen and Bau Numerical Linear Algebra Lecture 30 Question 30.1

Here, the problem is to derive the following formula for Jacobi Method for computing the eigenvalues of a matrix A, by diagonalizing sub-matrix of A in which the real $2$$\times$$2$ symmetric matrix ...
2
votes
0answers
76 views

Diagonalization of $2\times 2$ Lorentz-symmetric matrix

$\newcommand\pair[1]{\left\langle #1 \right\rangle}$ Consider Minkowski plane $\Bbb L^2 = (\Bbb R^2, \pair{\cdot,\cdot}_L )$, where$\renewcommand\vec[1]{{\boldsymbol #1}}$ $$\pair{\vec{u},\vec{v}}_L\...
2
votes
0answers
42 views

Smooth family of operators induces a smooth family of projections.

Let $H$ be a Hilbert space, and let $A:U \rightarrow L(H)$ be a smooth$^{1}$ family of unbounded closed operators with common dense domain. Suppose furthermore that $A_{t}$ is normal for each $t \in U$...
2
votes
0answers
339 views

Show the form of $J$ and $P$ for Leslie matrix $A$ when $A = PJP^{-1}$

I'm trying to solve this for a homework assignment. The Jordan Normal form theorem states that every complex $n \times n$ matrix $A$ van be written as $A=PJP^{-1}$, where $J$ is the diagonal matrix ...
2
votes
0answers
75 views

How to calc action of operator on functions without diagonalizing the operator?

I have an hermitian operator $$ h = -\frac{1}{2} \frac{\partial^2}{\partial x^2} + v(x) $$ acting on square integrable functions that live in some subspace $V$ of $R$. If I want a computer to ...
2
votes
0answers
108 views

Projection of a symmetric matrix onto the probability simplex

The definition of probability simplex in $\mathbb{S}^n$: $$\{X\in \mathbb{S}^n: X\geq 0, \text{tr}(X) = 1\},$$ where $\mathbb{S}^n$ is the vector space of symmetric $n\times n$ matrices. A ...
2
votes
0answers
79 views

Seemingly strange exercise on unitary Hermitian operator

Let $V$ be a finite dimensional inner product space. Let $T$ be a Hermitian unitary operator. Prove there's a subspace $W$ such that for each $v\in V$ we have $Tv=w-w^\prime$ where $w\in W,w^\...
2
votes
0answers
1k views

Finding an eigenbasis for a matrix & diagonalization

I'm trying to find an eigenbasis for matrix A = $\begin{bmatrix}1&-1&1\\-1&1&-1\\1&-1&1\end{bmatrix}$ so that I can use the result to diagonalize A. Because the characteristic ...
2
votes
0answers
126 views

Diagonalization of infinite matrix

How would you diagonalize the following infinite symmetric matrix? $$T = \left[ \begin{array}{cccc} M \Omega^{2} + 2 \sigma & -c_{1} & -c_{2} & \cdots \\ -c_{1} & m_{1}w_{1}^{2} + ...
2
votes
0answers
614 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 ...
2
votes
0answers
25 views

Normal form over $\mathbb{Z}$ of matrices of order $2$

Suppose $M \in GL_k(\mathbb{Z})$ is of order $2$. That is, $M^2 = 1$ and $M \ne 1$. Then is it true that upto a change of $\mathbb{Z}$ basis, $M$ has the form $$\begin{pmatrix}J \\ & J \\ & &...
2
votes
0answers
69 views

Compute the Eigenvectors & Show A is diagonalizable

$A = \begin{bmatrix} 1&2&1 \\ 0&1&0 \\ 1&3&1 \\ \end{bmatrix} $ I computed the eigenvalues: $λ_ 1 = 1$ $λ_ 2 = 0$ $λ_ 3 = 2$ The ...
2
votes
0answers
59 views

Eigenvalues with constraints?

Note: This is a short version of https://math.stackexchange.com/questions/979834/ For a $n$-dimensional symmetric matrix A, orthogonal matrix C exists so that the relationship $$AC = CD$$ is ...
2
votes
0answers
49 views

$A$ matrix is diagonalisable if $\exists S : S^{-1}AS $ is a diagonal matrix, how can I find S?

Per definition a matrix $A$ is diagonalisable if there exists a matrix S such that $S^{-1}AS$ is a diagonal matrix. My question is how do I find the matrix $S$? Is it always the combination of the ...
2
votes
0answers
515 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 ...
2
votes
0answers
290 views

Proving that a matrix $A$ is diagonalizable

I have to prove this result: If $A \in M_n (F)$ has $n$ distinct eigenvalues then $A$ is diagonalizable. My attempt at proof : Let $\lambda_1,\lambda_2,\ldots,\lambda_n$ be the distinct eigenvalues ...
2
votes
0answers
265 views

When are two operators simultaneously diagonalizable?

I am reading a paper and they have diagonalised both operators in an equation, on a separable Hilbert space, with respect to the same basis. My question is, when can two operators be simultaneously ...
1
vote
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,...
1
vote
0answers
43 views

Diagonalization and the Hadamard product

Let $B \in \mathbb{C}^{n\times n}$ be unitarily diagonalizable such that $B=V\Lambda V^*$. Let $A=B\circ B$ where $\circ$ accounts for the Hadamard product. Then we can say that $A$ is also unitarily ...
1
vote
0answers
50 views

Linear algebra- Diagonalization of a symmetric matrix

A linear transformation $$ T:R^3→R^3$$ is defined as $$ T:(x)=Cx$$ where $C$ is a symmetric matrix. a) State the dimensions of the eigenspaces $\mbox{N(C-αI)}$ and $N(c-βI)$ It is also given that: $$...
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vote
0answers
28 views

Theoretical questions about characteristic polynomials, diagonalizability, rank and similarity

I am new to Linear Algebra and would like some feedback regarding my answer to the following question A and B are two square matrices of order n. Prove or refute (with a counterexample) the following ...
1
vote
0answers
24 views

Similar matrices if only difference is diagonalizable/non-diagonalizable

I am new to linear algebra, and am just looking for some feedback regarding the following solution: True or false? 1.$$\begin{pmatrix}1&0\\0&-1\end{pmatrix}$$ and $$\begin{pmatrix}2&0\\0&...
1
vote
0answers
47 views

Symmetric matrix with n rows has n linearly independent eigenvectors?

I'm trying to better understand SPD matrices and have started with symmetric matrices. It's apparent to me that symmetric matrices have real eigenvalues, but how to show that the algebraic ...
1
vote
0answers
71 views

If A is anti-hermitian, show that $|\det(1+A)|^2 \geqslant 1$

Given that A is anti-hermitian, i.e. $A^\dagger = -A$, show, by diagonalising $iA$, that $$|\det(1+A)|^2 \geqslant 1$$ $\hspace{3mm}$ So what I thought was that I need to show that $\det(1+A)[\det(1+...