Questions tagged [eigenvalues-eigenvectors]

Eigenvalues are numbers associated to a linear operator from a vector space $V$ to itself: $\lambda$ is an eigenvalue of $T\colon V\to V$ if the map $x\mapsto \lambda x-Tx$ is not injective. An eigenvector corresponding to $\lambda$ is a non-trivial solution to $\lambda x - Tx = 0$.

Filter by
Sorted by
Tagged with
0
votes
2answers
29 views

What are the eigenvalues and eigenvectors of $A^2 - 3A + 4I$, given the eigenvalues and eigenvectors of $A$?

Let eigenvalues of $2 \times 2$ matrix $A$ be $1,-2$ and eigenvectors be $x_1$ & $x_2$ respectively. Then eigenvalues and eigenvectors of $A^2-3A+4I$ would be? We know that eigenvalues can be ...
1
vote
0answers
33 views

Show that $A$ is diagonalizable if and only if $\sum_{\lambda \in \operatorname{spec}_{F}(A)}V_{\lambda} =V$

Let $V_\lambda$ be the eigenspace that corresponds to $\lambda$ and $$\operatorname{spec}_{F}(A) = \{\lambda \in F: V_{\lambda} \neq\{0\}\}$$ I'm asked to show that $A$ is diagonalizable if and only ...
0
votes
0answers
22 views

Orthogonality of Eigen vectors for the summation of matrices

If a matrix, let's say $A$, can be decomposed into $A_1$ and $A_2$, and the eigenvectors of $A_1$ and $A_2$ are orthogonal. Does it guarantees that the Eigenvectors of $A$ would also be orthogonal?
0
votes
1answer
29 views

Eigenvalues and Eigenvectors for Operator

I am trying to understand some of the tools of functional analysis and I came across this exercise: Given $L^2(\mathbb{R})$ and the family of operators $\{T_a\}_{a \in \mathbb{R}}$ such that: $(T_af)...
1
vote
2answers
23 views

The asymptotic behavior of a solution to the matrix differential equation $\frac{du}{dt} = A u(t)$, where $A$ satisfies certain criteria

$\mathbf {The \ Problem \ is}:$ Let, $\operatorname {u(t) = (u_1(t),u_2(t))}$ where $t\gt 0$ be the unique solution of the differential equation $\operatorname {du/dt} = Au(t)$ where $\operatorname {u(...
0
votes
1answer
26 views

Eigenvalues of $BB^T$ where $B$ is the Incidence Matrix

Let $I_n, J_n \in M_n(\mathbb{R})$ where $I_n$ is the identity and $(J_n)_{ij} = 1$ for all $i,j$. (1) Calculate the determinant, eigenvalues and eigenvectors of $I_n$ and $J_n$. (2) ...
2
votes
2answers
38 views

Intuition behind Eigenvalue solution matrix

I've been watching the excellent course by 3Blue1Brown on Linear Algebra which is oriented towards giving students intuition into Linear Algebra concepts. I am trying to find an intuitive way to ...
1
vote
1answer
38 views

$T: \mathbb{R}^n \to \mathbb{R}^n$ linear with $n > 1$. Then there is $M$ with $\dim M = 2$ and $T(M) \subset M$.

Let $T: \mathbb{R}^n \to \mathbb{R}^n$ be a linear map, with $n > 1$. Prove that there is a subspace $M \subset \mathbb{R}^n$, with $\dim M = 2$ such that $T(M) \subset M$. This question is from ...
0
votes
0answers
26 views

Similarity Invariance of Trace for Matrix Multiplication

From wiki, I know that for a square matrix A and any invertible matrix P of the same dimensions, the following would hold: $$tr(P^{-1} A P) = tr(A)$$ In my case, A is a density matrix. The question ...
0
votes
1answer
21 views

How to determine the eigenvalues for this matrix

Take a look at this matrix (Jacobian matrix): $$ A = \frac{\partial f}{\partial x} \Big\vert_{x=x_{e_1}} = \begin{bmatrix} 0 & 1 & 0 \\ \dfrac{2\mu g}{(1+\mu y_0)} & -\dfrac{k}{m} & -...
3
votes
1answer
56 views

An insight into the concept of eigenvectors

I am trying to interpret the concept of eigenvalues and eigenvectors geometrically. In order to ensure that my thinking is headed in the right direction, I am writing this up. I want a verification of ...
0
votes
1answer
31 views

For which values of $a,b,c,d,e\in\mathbb{R}$, is this $4\times 4$ matrix diagonalizable?

For which values of $a,b,c,d,e\in\mathbb{R}$, is the matrix $A$ diagonalizable? Here, $$A=\left( \begin{array}{cc} 3 & 0 & 0 & 0 \\ a & 2 & d & e \\ b & 0 & 1 & ...
3
votes
2answers
49 views

Finding the Eigenvectors

Consider the matrix given below: $\begin{pmatrix} 1 & 1 \\ 1 & 0 \end{pmatrix}.$ The eigenvalues for this matrix are $\dfrac{1+\sqrt 5}{2},\dfrac{1-\sqrt 5}{2}.$ I am facing trouble ...
0
votes
1answer
26 views

Prove that every polynomial $p \in P_n(\mathbb{C})$ is the characteristic polynomial of some linear map.

Prove that every polynomial $p \in P_n(\mathbb{C})$ is the characteristic polynomial of some linear map. First attempt. If $p \in P_n(\mathbb{C})$, then $p(x) = (x - a_1)\cdots(x - a_m)$ where $n \...
0
votes
1answer
34 views

“Generalised eigenvectors” $Ax=\lambda Bx$ Proof of B-orthogonality?

I came across this in a textbook, and it is not the usual definition of generalized eigenvectors I've seen. The generalised eigenvectors of matrices A and B are vectors that satisfy $Ax=\lambda Bx$ ...
1
vote
2answers
34 views

Relations between eigenvalues and determinant of an integer matrix

Let $A \in M_{n}(\mathbb{Z})$. (1) Prove that if $k \in \mathbb{Z}$ is an eigenvalue of $A$, then $k$ divides $\det A$. (2) Let $j \in \mathbb{Z}$ such that the sum of all entries in each ...
1
vote
1answer
20 views

Relating the coefficients of the characteristic polynomial of a symmetric matrix to the determinants of its principal submatrices

I've been thinking about this this problem: Let $M$ be a symmetric matrix. Recall that the eigenvalues of $M$ are the roots of the characteristic polynomial of M: $p(x) := det(xI-M) = \prod\limits_{...
1
vote
1answer
28 views

Question regarding the multiplicity of eigenvalues of a matrix polynomial .

$\mathbf {The \ Problem \ is}:$ If $\operatorname p(x)$ is a polynomial over an algebraically closed field $\mathbb F[x]$ and let $A$ be an $n×n$ square matrix, then it is known that every eigenvalue ...
1
vote
0answers
12 views

In a primitive, stochastic matrix, the 2nd largest eigenvalue depends on which property of the elements of matrix?

We know the largest eigenvalue of a primitive, stochastic matrix is 1. The 2nd largest eigenvalue is strictly smaller than 1. But can we get more information of 2nd largest eigenvalue from the ...
1
vote
1answer
37 views

Conditions for two eigenvectors of $AA$ to be eigenvectors of $A$

Let $V$ be a $d$-dimensional real vector space with inner product $\langle\cdot{,}\cdot\rangle$, and suppose $A$ is a symmetric linear map $V\to V$. Then, by the spectral theorem, $A$ has an ...
5
votes
3answers
198 views

Is every eigenvector of AA an eigenvector of A?

Let $V$ be a (finite-dimensional) vector space and $A \colon V \to V$ a linear map. Is it true that, if $v$ is an eigenvector of $A\circ A$, then $v$ is an eigenvector of $A$? I know the converse ...
1
vote
0answers
29 views

Eigen values of a tensor product of the linear transformations

I want to investigate what kind of eigen values cat tensor product $A \otimes B$ of two linear transformations have, s.t $(A \otimes B)z = \lambda z$. If $z$ is a simple tensor then $\lambda = \alpha ...
0
votes
1answer
41 views

Show that $v^tA^{-1}u\ne -1$ if and only if $A+uv^t$ is invertible

Show that $v^tA^{-1}u\ne -1$ if and only if $A+uv^t$ is invertible, where $A$ square matrix and $u,v \in \mathbb{R}^n$ I feel I want to use eigenvalues. I know, $A+uv^t$ is invertible if and only ...
0
votes
2answers
54 views

How to find eigenvalues of this given matrix

Consider the matrix $M$=\begin{bmatrix}9&1&1&1&1&1&1&1&1&1\\ 1&9&1&1&1&1&1&1&1&1\\ 1&1&9&1&1&1&1&1&...
0
votes
0answers
26 views

Eigenvector of matrices with same diagonals

I have 3 covariance matrices (each of whose diagonals are all equal) and their SVDs are of the form, $$A_1 = UD_1U^T, \quad A_2 = UD_2U^T, \quad A_3 = UD_3U^T$$ Now, I am looking at the matrix $...
2
votes
1answer
50 views

Find determinant of $B$ when $B=2A+A^{-1}-I$

Let $A$ be a $4\times 4$ real matrix with eigenvalues $1,\ -1,\ 2,\ -2$. If $$B=2A+A^{-1}-I$$ then determinant of $B$ is? We are provided with eigenvalues then from Cayley Hamilton theorem, $A$ ...
1
vote
1answer
48 views

Solution of $AX+XA=B$ through eigenvectors of $A$

I see that Matlab uses the spectral decomposition to solve the continuous Lyapunov equation $$AX+XA=B$$ The formula they use for positive definite $A$ with matrix of eigenvectors $U$ and column ...
0
votes
1answer
38 views

Eigenvector of Laplacian of ring graph [duplicate]

I know the eigenvectors of the Laplacian of a ring graph with $n$ vertices are $$x_k(u) = \sin \left( \frac{2 \pi k u}{n} \right)$$ and $$y_k(u) = \cos \left( \frac{2 \pi k u}{n} \right)$$ for $1 \...
2
votes
3answers
29 views

Show that for any invertible $A$, if $P\equiv\sqrt{A^\dagger A}$ then $U\equiv A P^{-1}$ is unitary

Please bare in mind, that I am self-teaching this area at the moment and so additional explanation where possible would be greatly appreciated. Let $A:V \to V$ be an arbitrary invertible operator. ...
1
vote
2answers
37 views

Finding an eigenvector for a matrix such that the entries of each column sum 1

Let $A \in \mathbb{R}^{n \times n}$ a matrix such that each entrie of $A$ (I denoted them by $\left[ A \right] _{ij}$) satisfies that $\left[ A \right] _{ij} \geq 0$ and if you sum the entries of each ...
0
votes
0answers
16 views

Elements of eigenvectors of Laplacian matrix of complete graph

Assume we have a complete graph with $N$ nodes. If we show the Laplacian graph with $L$, we know that one eigenvalue of $L$ is equal to zero and all the others are equal to $N$, so we have $N-1$ ...
0
votes
1answer
18 views

Preservation of negativity of eigenvalues of the Jacobian.matrix under coordinate change.

Let $U$, $V$ be open sets in $\mathbb{R}^m$, and $\phi : U \rightarrow V$ be a diffeomorphism. Denote the inverse map by $\psi$. Let $f: U \rightarrow \mathbb{R}^m$ smooth. Define a map $g:V \...
0
votes
3answers
87 views

Prove that vector is eigenvector.

Let $A$ be a matrix of size $5$ (real values) and real $\lambda_{1}$, $\lambda_{2}$, $\lambda_{3}$, such that: vector $(1,1,1,1,1)$ is eigenvector of $A$ corresponding to eigenvalue $\lambda_{1}$ ...
2
votes
0answers
61 views

Are the minimum of two symmetric matrices closer to each other than the two matrices themselves?

Consider any two real symmetric matrices $A,B\in\mathbb{S}^n$ with spectral decompositions: $$A=U_A D_A U_A',\quad B=U_BD_BU_B'$$ Now, let $$A^-\equiv U_A\min(D_A,0)U_A',\quad B^-=U_B\min(D_B,0)U_B'$$ ...
2
votes
0answers
18 views

Sample from distribution given by sum of eigenvalues of symmetric matrix

I'm interested in (efficiently) sampling from the following density: $$ p(X) \propto \exp\Big(\sum_{i=1}^n \lambda_i(X)\Big),\quad\textrm{on}\ \{X \in \textrm{Sym}(n) : \lVert X \rVert_F \le 1\}, $$ ...
1
vote
1answer
25 views

spectral radius of products of symmetric positive definite matrices

I have matrices $A, B, C, D \in \mathbb R^{n\times n}$ which are all symmetric positive (semi-)definite. If I have that \begin{align} \rho(AB) &< \gamma^2, \\ \rho(CD) &< \gamma^2, \...
0
votes
1answer
26 views

Algebraic formula of the pseudoinverse (Moore-Penrose) of symmetric positive semidefinite matrixes

I was reading the Wikipedia page about the Pseudoinverse or Moore-Penrose inverse, where they say that, given a generic matrix $A \in R^{nxm}$, if the matrix is full rank (i.e. rank=$min\{n,m\}$), ...
1
vote
2answers
33 views

Eigen Decomposition Check

I am following the wiki entry on eigen dicomposition with the following matrix: $$A = \begin{pmatrix} 0 & 1 \\ 1 & 0 \\ \end{pmatrix}$$ I wish to find a diagonalizing matrix T.S. $$T^...
1
vote
1answer
24 views

Eigenvalues of an endomorphism over a polynomial ring

I am currently preparing for a math exam and am stuck on the following question: Let $\Bbb K$ be a field, let $\Bbb K[T]$ be the polynomial ring over the variable $T$ over $\Bbb K$, and let $\...
0
votes
1answer
24 views

Eigenvalue of $AB$ where $A,B$ have prescribed row sums

Let $A$ and $B$ be $n\times n$ matrices such that the sum of elements of each row of $A$ is $1$ and the sum of elements of each row of $B$ is $2$. Prove that one eigenvalue of $AB$ is $2$. My ...
3
votes
1answer
56 views

Let $A$ be a given matrix over an algebraically closed field. If $U(B) := AB−BA$ is diagonalizable, then so is $A$?

After solving a similar problem in Hoffman and Kunze, I'm trying this one for a long time, but I didn't get success. I tried use the fact that the eigenvalues of $U$ are $\lambda_i-\lambda_j$ with $1\...
1
vote
1answer
12 views

Methods for Definiteness of a matrix are giving different results

I've found several different sets of rules to check for definitness. They are: Determinant of all upper-left submatrices, shown here Check the sign of the eigenvalues of A, which is the definition ...
0
votes
0answers
26 views

Is it possible to go from Eigenvectors to the original nxn matrix from which it was decomposed?

I understand how eigenvectors can be decomposed from a matrix. Now let's say, I have a set of eigenvectors in hand, is it possible to compute the original matrix from it? Also, does it make sense to ...
3
votes
3answers
121 views

How to find eigenvalues of the matrix

This is a question from our end-semester exam: How to find the eigenvalues of the given matrix: M=\begin{bmatrix} 5,1,1,1,1,1\\ 1,5,1,1,1,1\\ 1,1,5,1,1,1\\ 1,1,1,5,1,1\\ 1,1,1,1,4,0\\ 1,1,1,1,0,4\\...
0
votes
0answers
10 views

Are there any nice properties we can talk about after using PCA to transform data to a new basis?

I'm not talking about "compressing" data here....assume we kept all the eigenvector basis after performing PCA, and then used them to transform our data into a new basis of the same dimension as the ...
2
votes
1answer
35 views

Eigenvalues of an outer product

Let $w, v \in \mathbb{R}^d$. What is known about the eigenvalues of the outer product matrix $vw^{\top} + wv^{\top}$?
0
votes
1answer
27 views

Minimizing product of a vector with a symmetric matrix

Is the minimum of ${\left\lVert{x^TA}\right\lVert}$, where $x \in \mathbb{R}^n$, and $A^{n \times n}$ is a symmetric real matrix, related to the smallest eigenvalue of $A$? I read about Rayleigh ...
0
votes
4answers
52 views

Matrix diagonalization. Is $A = PDP^{-1} = P^{-1}DP$?

Diagonalization of a square matrix $A$ consists in finding matrices $P$ and $D$ such that $A=PD P^{-1}$ where $P$ is a matrix composed of the eigenvectors of $A$, $D$ is the diagonal matrix ...
2
votes
4answers
62 views

Eigenvalue of $A-aI_3$

Question: Let $A=\begin{pmatrix} a+1 & 1 & 1 \\ 1 & a+1 & 1 \\ 1 & 1 & a+1\end{pmatrix}$. Show that $A-aI_3$ has eigenvalue of 3. Also find eigenvector. My thinking: I ...
5
votes
1answer
45 views

Explanation for Eigenvalues or Characteristic values of Projection Operator

Question: I need to give what are possible eigenvalues/characteristic value of projection operator with an explanation. My Attempt: Let E be any projection on vector space V. Assume R be the range ...