Stack Exchange Network

Stack Exchange network consists of 174 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

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 $\lambda x - Tx = 0$.

0
votes
1answer
12 views

Numerical Linear Algebra invertible

Let u and v be two non-zero vectors in $R^n$. Let A=I+u$v^T$. First we show that u$v^T$ is symmetric iff u=cu. Second question: Assume $v^T$u=-1, Find the spectrum of A=I+u$v^T$ in addition to its ...
0
votes
0answers
16 views

How to solve a system of differential equations with complex numbers?

So I have a general understanding of how to do these types of problems but I do not know how to get rid of the complex numbers. I found that the eigenvalues are 3i and -3i and the eigenvectors are [(-...
2
votes
5answers
36 views

Quick question about eigenvalues of $3\times3$ matrix

I'm looking over solutions from past exams. For this problem, the solution states “By inspection, we see that $2$ is an eigenvalue”. Given arbitrary $3\times3$ matrix below$$\begin{...
-1
votes
2answers
37 views

show that a E is a subspace of V [on hold]

I got a little problem with a math problem here it is if you guys can help me it will be wonderful thanks a lot. 1) Let $V$ be a vector space, $f$ an endomorphism of $V$ 2) Let $\lambda$ be the ...
0
votes
1answer
31 views

Does the power method converge?

I want to check if the power method converges for the matrix $A$ and the vector $\vec{v} $ where $$A=\begin{pmatrix}\lambda & 1 \\ 0 & -\lambda\end{pmatrix} \text{ and } \vec{v} =\begin{...
1
vote
0answers
18 views

Differential Equations: How to categorize graph and clockwise vs. counter-clockwise from eigenvalues?

I'm studying for my Final and having a hard time understanding the criteria for category (Sink, Spiral Sink, Center, etc.) and how to tell whether the direction is clockwise (CW) or counter-clockwise (...
0
votes
0answers
20 views

Non trivial solution of a system?

So i wanted to know if this statement is true: If there is a free variable in a row reduced matrix, does this imply that the system has a non trivial solution ? E.g. (this excercise is done in ...
0
votes
1answer
38 views

Symmetric Boundary Conditions/Eigenvalues (PDEs)

Consider the following eigenvalue problem for the Laplacian $-\Delta u = \lambda u$ in $U$ $u + a \left(\frac{\partial u}{\partial v}\right)$ on $\partial U$ where $v$ is the outward unit normal to ...
5
votes
6answers
270 views

Eigenvalues and Eigenvectors of Sum of Symmetric Matrix

Question: Let A = \begin{bmatrix} 1 & 1 \\ 1 & 1 \\ \end{bmatrix} Find all eigenvalues and eigenvectors of the martrix: $$\sum_{n=1}^{100} A^n = A^{100} +A^{99} +...+A^2+A$$ I know that ...
0
votes
1answer
28 views

Real-valued matrices and their eigenvalues

Why are the eigenvalues of the symmetric matrix $A^TB^TBA$ equal to the eigenvalues of the matrix $B^TBAA^T$ where $A \in \mathbb{R}^{3 \times 3}$ and $B$ is either: $B = \begin{pmatrix} {1}, {0}, {0}...
1
vote
1answer
39 views

Finding a set of matrices based on eigenvalues and eigenvectors with constraints

I'm trying to solve the following problem and hope for some helpful insights on how to approach this: In the 3-dimensional case, for and a given set of eigenvalues and eigenvectors and chosen values $...
1
vote
1answer
23 views

Basis of a transformation matrix for diagonal matrix

So I have this question here which says: Let $T:\Bbb R^3\to\Bbb R^3$ defined by $$T\left(\begin{matrix}x_1 \\ x_2 \\ x_3 \end{matrix}\right)=\left(\begin{matrix}-x_1+7x_2-x_3 \\ x_2 \\ 15x_2-...
4
votes
1answer
30 views

Let $U$ be the set of all $n×n$ matrices with real entries such that all their eigenvalues belong to $\mathbb C \setminus \mathbb R $.

Let $U$ be the set of all $n×n$ matrices with real entries such that all their eigenvalues belong to $\mathbb C \setminus \mathbb R $, and $X = M_n(\mathbb R)$. Is $U$ open? I know that set of ...
0
votes
1answer
31 views

What is the real characteristic equation? [duplicate]

My book says that you form the character equation as $\begin{vmatrix} \lambda I - A \end{vmatrix}$. However I see occasionally on stack exchange and other resources that define the character equation ...
1
vote
1answer
36 views

Sum of Symmetric Positive Definite Matrix and Scalar of Identity

If $A$ is an $n\times n$ symmetric positive definite matrix with the smallest eigenvalue $\lambda$, then for any $\mu>-\lambda$, $A+\mu I$ is positive definite. I am trying to show this, but I am ...
0
votes
0answers
36 views

maximum eigenvalue and eigenvector of an altered adjacency matrix of a directed irreducible graph

The matrix $B$ is a non-negative irreducible block matrix (Adjacency matrix of a directed irreducible graph) with $0$ or $1$ entries as follows: $B= \left[ \begin{array}{c|c|c} 0 &B_{12}&B_{...
0
votes
1answer
22 views

Suggestions for making proof flow better

I am relatively new to writing proofs and I would appreciate some criticism to improve my proof writing skills. Here is proof to prove that if $A^2 = 0$, then 0 is the only eigenvalue of A for my ...
1
vote
0answers
21 views

Different conditions for self-adjoint property

Consider the following differential equation $$f(x)\partial_ty(x,t)+\hat{M}y(x,t)=0\,,$$ where $\hat{M}$ is a differential operator with respect to $x$ and assume $y(x,t)=T(t)v(x)$. Then, $$f(x)T'(t)v(...
1
vote
2answers
39 views

linear combination of some matrices is identity matrix

Assume $T$ is a $n\times n$ matrix over number field $\mathbb{F}$. If $\lambda$ is not an eigenvalue of $T$, we know $T-\lambda E$ is invertible matrix where $E$ is the identity matrix. Now if we have ...
0
votes
1answer
31 views

Are they similar matrix

Do $\begin{bmatrix} 0&i&0\\0&0&1\\0&0&0 \end{bmatrix} $ and $\begin{bmatrix} 0&0&0\\-i&0&0\\0&1&0 \end{bmatrix} $ are similar.Is this True/false ...
3
votes
0answers
13 views

Show the spectral radius of a matrix is smaller than 1

Let $\hat{\bf H}$ be a $p\hat{N}\times p \hat{N}$ sparse matrix consisting of $p\times p$ blocks, where each block is of size $\hat{N}\times\hat{N}$. The values in $\hat{\bf H}$ is illustrated below (...
0
votes
2answers
39 views

Is this $3\times3$ matrix diagonalizable?

After browsing through similar posts, I was wondering if I am understanding the meaning of "$n$ distinct eigenvalues" for the following theorem. If the $n\times n$ matrix $A$ has $n$ distinct ...
2
votes
3answers
21 views

Finding the sec. Eigenvector, when knowing the first Eigenvector and Eigenvalue

So here is my problem, Let A be a symmetric Matrix (2x2) with EV=(-2, 3) and the Eigenvalue being -5. Find the 2. Eigenvector to the second Eigenvalue. The only info i can think of is that the EV ...
0
votes
0answers
9 views

Given that a symmetric Matrix with only two eigen values and eigen space E_{2}, Is it possible to find the entries in the Matrix

I tried to solve this using linear equations, but then I strongly sense that there is something wrong with the question.
0
votes
0answers
24 views

Prove that the solution of $x'(t)=Ax+\frac{\sin t}{t+1} (1,\cdots,1)^T$ is bounded

I am trying to show that the solution $x(t)$ of $$x'(t)=Ax+\frac{\sin t}{t+1} (1,\cdots,1)^T$$ is bounded on the interval $(0,\infty)$, where the eigenvalues of the $n \times n$ matrix $A$ have ...
0
votes
1answer
15 views

Show that $L_A$ acts on by orthogonal transformation and in particular rotation.

Let $A$ be a $3\times 3$ orthogonal matrix with determinant $=1$. Let $v$ be an eigen vector corresponding to $1$ of $A$.Let $W=\text{span}\{v\}$. Show that $L_A$ preserves $W^\perp$ and it acts ...
0
votes
1answer
33 views

Determine the values for a for which the matrix is diagonizable.

"Determine the values for a for which the matrix A is diagonizable. " $$ A = \begin{bmatrix}1&1\\a&1\end{bmatrix}$$ My first attempt solving this problem was to find the characteristic ...
2
votes
1answer
39 views

Distance from eigenspace of matrix

In linear algebra, is there a separate name / concept for the notion of distance between linear vector subspaces? I'm asking this because I'm considering a problem in numerical linear algebra where ...
0
votes
0answers
45 views

Change in eigenvalues due to perturbation to a correlation matrix

Let $A$ be a $m \times n$ matrix defined as $ A = \Big[\frac{a_1}{\|a_1\|} \cdots \frac{a_n}{\|a_n\|}\Big]$ and $a_k \in \mathbb{R}^{m\times 1}$ where $k \in [1,\dots,n]$. Now, we define a ...
1
vote
1answer
37 views

Is there any easy way to find the determinant of a 4x4 matrix?

I have a 4x4 covariance matrix and want to find the eigenvalues. I know part of the process is to find the determinant: $$\tiny{\begin{align}\begin{vmatrix} 3.33−\lambda & −1.00 & 3.33 & ...
1
vote
1answer
50 views

Solving a linear system of differential equations

Given that $v_1 = \begin{bmatrix}1&1\end{bmatrix}$ and $v_2 = \begin{bmatrix}2 &1\end{bmatrix}$ are eigenvectors of the matrix $$ \begin{bmatrix}-1&-2\\1&-4\end{bmatrix} $$ which is a $...
7
votes
1answer
130 views

Find eigenvectors of the $(n+1) \times (n+1)$-matrix

Find eigenvectors of the $(n+1) \times (n+1)$-matrix: $$\left(\begin {array}{cccccccc} 0&0&0&0&0&0&-1&0\\ 0&0&0&0&0&-2&0&n\\ 0&0&0&...
0
votes
1answer
40 views

Questions on symmetric matrices and skew-symmetric matrices

Let $A$ be a $3\times 3\;$ symmetric matrix. Let $U$ be the set of all $3\times 3\;$ skew-symmetric matrices. Let $T : U\to U$ be defined as $T(B)=AB+BA.$ Prove that $T$ is bijective iff the sum of ...
0
votes
0answers
30 views

Prove that $ \mbox{Tr}(AB)\leq \sum_{i=1}^n \lambda_{i}(A) \lambda_i(B)$, where $A, B$ are $n \times n$ Hermitian matrices

Suppose $A, B$ are $n \times n$ Hermitian matrices, i.e., $ A^{T}=\bar{A}$ and $B^{T}=\bar{B} $, prove that $$ \operatorname{Tr}(AB)\leq \sum_{i=1}^n \lambda_{i}(A)\lambda_i(B), $$ where $ \...
0
votes
0answers
43 views

Eigenvalues of random matrix

I am studying random matrix and stuck by a problem. Is there any way that I can calculate or describe eigenvalues of random matrix? My first attempt was as follows: Let $A$ be random matrix s.t. $A=(...
2
votes
0answers
21 views

Find the analytical form of the eigenvalue of a special sparse matrix.

Let $\hat{\bf H}$ be a $p\hat{N}\times p \hat{N}$ sparse matrix consisting of $p\times p$ blocks, where each block is of size $\hat{N}\times\hat{N}$. The values in $\hat{\bf H}$ is illustrated below (...
1
vote
2answers
47 views

Find $2\times 2$ symmetric matrix $A$ given two eigenvalues and one eigenvector

I am having trouble finding the symmetric matrix $A$ given eigenvalues $1$ and $4$ and eigenvector $(1, 1)$ corresponding to eigenvalue $1$. I feel like I'd have to use the equation $A=PD(P^{-1})$, ...
1
vote
4answers
72 views

Prove that if $A^2=0$, then $0$ is the only eigenvalue of $A$. [duplicate]

Does my proof hold up to prove that $0$ is the only eigenvalue of $A$ if $A^2 = 0$? Let $A$ be an $n \times n$ matrix. $A^2 = A*A$ because of matrix multiplication. If $A = k$, where $k \neq 0$, ...
1
vote
1answer
45 views

Prove that $x(t)$ is bounded

In this problem, I tried the following: First I show that $$x(t) = e^{tA}x_0 + \int\limits_0^t e^{(t-s)A}f(s)ds$$ Then I take the norm for both sides $$\|x(t)\|\leq Ke^{-\alpha t}\|x_0\|+\left\|\int\...
-3
votes
3answers
63 views

What are the eigenvalues of $\begin{bmatrix}0 & I\\\alpha I & D\end{bmatrix}$ where $D$ is diagonal?

Is there a trick to calculate the eigenvalues of $\begin{bmatrix}0 & I\\\alpha I & D\end{bmatrix}$ where $D$ is diagonal? I'm looking for some kind of expression in terms of $\alpha$ and $D$ ...
0
votes
0answers
19 views

An upper bound for the largest eigen value of $A^tA$

I would like to compute an upper bound on the largest eigenvalue of $A^tA$, where $A$ is an $n \times p$ real-valued matrix. This bound should be sharper than the Gerschgorin bound. I should also be ...
1
vote
3answers
83 views

Easier way to find eigenvalues of Matrices?

I am trying to find eigenvalues for this matrix, A = $\begin{bmatrix} 3 & 2 & -3 \\ -3 & -4 & 9 \\ -1 & -2 & 5 \\ \end{bmatrix}$ I find the characteristic equation here: $(\...
1
vote
1answer
26 views

Singular and eigen values properties…

Let $A\in\mathcal {M}_n(\mathbb{R})$, we will denote $\lambda_{\max}(A)$ the biggest eigenvalue of $A$ in absolute value, as for $B\in\mathcal M_{m,n}(\mathbb{R})$ we will denote $\sigma_{\max}(B)$ ...
0
votes
1answer
22 views

Spectral radius equal to 1 and convergence

The following theorem is well-known: $$ \lim_k A^k = 0 \text{ if and only if } \rho(A)<1 $$ (see wiki for context and proofs). What if now $\rho(A)=1$ and $\lambda\neq -1$ for all $\lambda \in ...
0
votes
1answer
26 views

$A^t\to 0$ when its row sum is strictly less than one?

$A_{n\times n}$ is a matrix having each row sum $<1$ and its largest eigenvalue is also $<1$. I need to show $A^t\to 0,\text{ i.e } a^t_{ij}\to 0\forall i,j\text{ as } t\to\infty$ given that $0&...
0
votes
0answers
39 views

Notation for eigenvalues when $\lambda$ is not available

Let me start by saying that I have never seen any other notation for an eigenvalue apart from $\lambda$. However, I'm writing a text in which $\lambda$ denotes something else and plays a main role, ...
2
votes
4answers
47 views

If $A^2=I$ and $\lambda \not =-1$, then $A=I$ [duplicate]

Given an $n\times n$ matrix $A$ with $A^2=I$, assume that $-1$ is not an eigenvalue of $A$. Prove $A=I$. Proof attempt: Since $A^2=I,$ we have $A^{-1}=A$. Using that fact that $\det(A)=\prod_{i=1}^n\...
2
votes
2answers
52 views

Find a matrix B such that $B^5 = A$ [duplicate]

I am being asked to find a matrix $B$ where $B^5 = A$ $$A = \begin{bmatrix} 1 & 3 \\ 3 & 1 \end{bmatrix}$$ In the first part of the question I was asked to find the eigenvalues & ...
0
votes
1answer
25 views

How do I find this eigenvector for a symmetric Matrix?

I have a symmetric matrix A, whose eigenvalues are $\lambda_1 = 6,~ \lambda_2 = 3,~ \lambda_3 = 2$ and eigenvectors are $\vec{v_1} = (1, 1, 1),~\vec{v_2} = (1,1,-1)$. How do I find the third ...
0
votes
0answers
22 views

Proof of partial order on the set of symmetric matrices [closed]

I was reading a article, and it says as fallows: Occasionally, we will make use of the following convenient notation: for two symmetric matrices M, N we write M ≼ N iff N − M ≽ 0. It is not ...