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 $\lambda x - Tx = 0$.

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About the eigenvalues of a block Toeplitz (tridiagonal) matrix

I have found the following $n\times n$ squared matrix in one stability analysis problem (i.e. I have to identify the sign of its eigenvalues) $$ A(\theta) = \begin{bmatrix} W(\theta)+W(\theta)^T & ...
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When is the following trace inequality valid?

I have $A = A^T$ (and can have any real eigenvalue) and $B = B^T \succeq 0$ and want to know if the following holds $$ trace(AB) \leq 0 \iff \lambda_{max} (AB) \leq 0 $$ I know that the matrix $AB$ ...
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Which non-negative matrices have negative eigenvalues?

It's easy to proof by counterexample that non-negative matrices can have negative eigenvalues. For example, the following matrix have -1 as an eigenvalue: $$ A = \begin{bmatrix} 0 & 0 & 0 ...
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1answer
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Bases for eigenspace

Consider a d dimensional subspace of $\mathbb{R}^n$ which is in the span of $d<n$ eigenvectors. Then any vector in the subspace can be represented as a linear combination of the d eigenvectors. ...
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33 views

Relationship between eigenvalues of Hermetian matrices

Suppose that we have two $m\times m$ matrices $A$ and $B$ which are Hermetian, with $|B_{ij}|\leq |A_{ij}|$ for $i,j = 1,2 \cdots, m$. Can we say anything about the relationship between the largest ...
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A set of linear algebra questions?

Could you help me with these questions, I figured most of them out on my own, but I'm not completely sure if I'm correct. a) $A=\begin{bmatrix}a^2&ab&ac\\ ab&b^2&bc\\ ac&bc&c^...
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1answer
41 views

Do eigenvalues of a linear transformation over an infinite dimensional vector space appear in conjugate pairs?

While attempting to answer a question here (namely, the finite dimensional case of the title question: Prove that if $\lambda$ is an eigenvalue of $T$, a linear transformation whose matrix ...
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0answers
8 views

When do bounds for eigenvalues become strict?

Let $A$ be a real square matrix with eigenvalues $\lambda_k(A), \, k=1, \dots, n$. Further, let $S = (A+A^T)/2$ denote the symmetric part of $A$. Bendixson (1902) showed that $$ \min_j \lambda_j(S) \...
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1answer
51 views

Prove or disprove: if $\lambda$ is an eigenvalue of square matrix T then so is $\bar\lambda$

I think this is true. I know that complex eigenvalues come out as conjugate pairs, but can't think of a formal proof. any lead?
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6 views

PCA for antisymmetric matrix

PCA (Principle Component Analysis) is often used to convert a symmetric matrix to lower dimension one. My question is whether there is semiliar method for antisymmetric matrix? As we known, the ...
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1answer
39 views

How can these row-equivalent matrices have the same determinant?

I am trying to prove that the geometric multiplicity of an eigenvector is bounded by the algebraic multiplicity. One particular proof of this theorem that I like is contained in the answer by Mariano ...
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2answers
28 views

The lower bound of the smallest eigenvalue of a symmetric positive definite matrix

I encounter a symmetric positive definite matrix whose features are all diagonal entries are $1$. all the other entries are in $[0, 1)$, but the matrix is not diagonally dominant. Now I am ...
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1answer
27 views

Singular Value Decomposition of Commuting Matrices

If two square matrices $M_1$ and $M_2$ commute, does it mean that the $U$ and $V^\dagger$ appearing in their singular value decompositions are the same? Specifically, does it imply that $$M_1 = U \...
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2answers
32 views

Find eigenvalues from characterestic equation

Equation of A : $λ^2 + 4λ - 12 = 0$ Find eigen values of $A$ and $A^3$ Find expression of $A^{-1}$ in terms of A I have no idea how to start solving it
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1answer
47 views

Suppose that A satisfies $A^2 - 3A +2I = 0$. Find the eigenvalues of $A$ and $A^2$

knowing that A satisfies the equation $A^2 - 3A +2I = 0$ . I want to find the eigenvalues of $A$ and $A^2$. I don't know where to start. Can you explain how to solve such type of questions ?
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1answer
33 views

Skew-symmetric matrix and its eigenvalues

I checked some examples and I always received that skew-symmetric matrix of even dimension has only pure imaginary eigenvalues. For example: $\begin{bmatrix} 0 & 2 & 3 & 1 \\ -2 & ...
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1answer
17 views

Calculate Real value Matrix from complex eigenvalues?

What I ask is exactly this Algorithm for real matrix given the complex eigenvalues But in my case, Im looking for 4*4 matrix which gives 4 pairs of complex eigenvalues. To be specific, I have ...
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1answer
88 views

Determine the eigenvalue of a real matrix

I think to this question for two days : Let $A$ be a $3\times3$ real matrix such that $\det(A) = 1$ and $A^{-1}= A^T$. Prove that one of the eigenvalues is equal to $1$. I used the fact that ...
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27 views

Minimum eigenvalue of a sum of symmetric matrices

Let $\{v_i\}$ be some orthonormal basis in $\mathbb{R}^n$, and let $\{w_i\}$ be a set of positive weights such that $\sum_{i=1}^n w_i = 1$. I am interested in bounding the smallest eigenvalue of the ...
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10 views

Quadratic eigenvalue equation for a Bloch problem

The Bloch problem in a tight-binding nearest neighbors form is following: $$ (\mathbf{a} c^{-1} + \mathbf{w} + \mathbf{a}^\dagger c) \phi = 0, ~ \mathbf{w} = \mathbf{w}^\dagger$$ $\mathbf{a}$, $\...
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1answer
61 views

Eigenvalues and eigenvectors of $A^TA$ and $A$

For a square matrix $A$, I was wondering what the condition(s) are for the eigenvalues of $A^TA$ to be the same as the eigenvalues of $A$. Also what are the condition(s) for the eigenvectors of $A^TA$...
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33 views

When does a symmetric matrix have repeated eigenvalues?

I understand that for each of the $N$ eigenvalues (regardless of repeated or not) of an $N\times N$ symmetric matrix, the algebraic multiplicity and geometric multiplicity are equal. This means if an ...
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33 views

For which values of t does a matrix not have eigenvalues

I need help solving this problem "For which values of real parameter t does the matrix: \begin{bmatrix} π^2t^2 & 36\\ -36 & 0 \\ \end{bmatrix} NOT have real eigenvalues. Thank you.
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1answer
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Simple algortihm for evaluating eigenvectors

In our lecture notes, there is a line where one has to find eigenvectors for the previously calculated eigenvalues $\lambda_1$ $=$ $\sqrt{61} \over 2$ - $3 \over 2$, $\lambda_1$ $=$ -$\sqrt{...
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39 views

Eigenvalue perturbation of singular matrix

Consider a Hermitian matrix $\mathbf{A_0} \in \mathbb{C}^{N \times N}$ with one singularity, i.e. its eigenvalues in increasing order are: \begin{equation} 0 < \lambda_2 \leq \lambda_3 \leq \cdots \...
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If $AB = BA$ for $A,B \in \mathcal{L}(V,V)$, then $A$ and $B$ have these properties [duplicate]

There is a base such $A$ and $B$ are both upper triangular on these base, and if $A$ and $B$ are diagonalizable, then $A$ and $B$ are diagonalizable simultaneously. For the first I have no idea. To ...
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1answer
53 views

Given matrix $A$ such that $\forall x : |Ax| > |x|$, and eigenvalue $\lambda$ of $A$. Show $|\lambda |\geq 1$.

Say matrix $A$ has the property that for any non-zero vector $x$, left-multiplication of $x$ by $A$ increases the magnitude. That is, $\forall x$ $$ |x| > 0 \implies |Ax| > |x| $$ Is it true ...
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1answer
53 views

Is there an effect for the eigenvalues on vectors other than the Eigenvectors?

Does having an eigenvalue greater than one mean that the magnitude of any vector multiplied by the matrix will be increased?
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33 views

How to solve this Sturm Liouville problem?

$\dfrac{d^2\phi}{dx^2} + (\lambda - x^4)\phi = 0$ Would really appreciate a solution or a significant hint because I could find anything that's helpful in my textbook. Thanks!
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1answer
23 views

Change eigenvalues of correlation matrix and transform into original basis

I use the Random Matrix Theory to filter out the information from the correlation matrix that is associated with noise - Marcenko Pastur band. That is straight forward. Then I follow Rosenow, Bernd, ...
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2answers
23 views

how to estimate the eigenvalue of a covariance matrix?

if $x_i\in\mathbb R^n$ and $\max_i\|x_i\|_2\le 1$ $$A=\frac{1}{n}\sum_{i=1}^nx_ix_i^T$$ $\lambda$ is a eigenvalue of $A$, how to prove $\lambda\in[0,1]$?
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How can I solve an optimization problem $x^T A x$ with constraint $x^T x = 1$?

Let $A \in \mathbb{R}^{n \times n}$ be a positive definite matrix. \begin{align} &\operatorname*{minimize}_{x \in \mathbb{R}^n} & & x^T A x \\ &\text{subject to} ...
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1answer
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What does a function of matrices do to the eigenvalues of matrices in its domain? Two examples and request for generalization if possible

I think, for example, that if $\lambda$ is an eigenvalue of a matrix $A$, then $\lambda^2$ is an eigenvalue for $A^2$ and that $\frac{1}{\lambda}$ is an eigenvalue for $A^{-1}$ provided $A$ is ...
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21 views

The eigenvalue of Laplace problem on a domain with a line removed

Let $\Omega\subset \mathbb R^2$ be given such that $\Omega$ is open bounded with nice boundary. Let $\Gamma$ be a finite segment in $\Omega$. We also define $\Omega_1:=\Omega\setminus \Gamma$. ...
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1answer
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Linear Algebra Eigenvalues and Eigenvectors [closed]

So I have a 2x2 matrix where equation 1(EQN1) is 1 and 2; equation2(EQN2) 2: 4 and 3 The determinant is det(A-λI)=0 When I first solve the eigenvalues I get λ=5, λ=-1 Now this is where I am lost,...
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1answer
34 views

Finding $S$, $D$, and $S^{-1}$ such that $A = SDS^{-1}$

Let $A = \begin{bmatrix}18&12\\-40&-26\end{bmatrix}$Find $S$, $D$, and $S^{-1}$ such that $A = SDS^{-1}$ So I did $\det(A-\lambda I)$ to get the char. poly. eqn. and got eigenvalues $\...
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1answer
25 views

Suppose $A$ is an invertible $n \times n$ matrix and $v$ is an eigenvector of $A$ with associated eigenvalue $4$. Convince yourself that $v$

Suppose $A$ is an invertible $n \times n$ matrix and $v$ is an eigenvector of $A$ with associated eigenvalue $4$. Convince yourself that $v$ is an eigenvector of the following matrices, and find the ...
2
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2answers
24 views

Expressing $v$ as a linear combination of $v_1, v_2, v_3$ and Finding $Av$

Let $v_1 \begin{bmatrix}0\\-2\\2\end{bmatrix}, v_2 = \begin{bmatrix}1\\2\\0\end{bmatrix}$ and $v_3 = \begin{bmatrix}2\\0\\-1\end{bmatrix}$ be eigenvectors of the matrix $A$ which correspond to the ...
2
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1answer
20 views

How can I compute $A(v_1 + v_2)$ where $v_1$ and $v_2$ are eigenvectors of the matrix A

If $v_1 = \begin{bmatrix}5\\3\end{bmatrix}$ and $v_2 = \begin{bmatrix}3\\1\end{bmatrix}$ are eigenvectors of a matrix $A$ corresponding to the eigenvalues $\lambda_1 = -1$ and $\lambda_2 = 4$ ...
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0answers
9 views

What is the following operator which satisfies:

Question What is the following linear operator as an explicit expression of $s$ given the eigenfunction and the eigenvalue: $$ \hat O a^s = e^a a^s $$ Where $a$ is an arbitrary constant. $$ \hat ...
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1answer
26 views

Solving the quadratic formula to determine stability of a system

I am trying to solve the $2\times 2$ matrix $$\begin{bmatrix} 0 &1 \\ -k &-b \end{bmatrix}$$ for a relationship between the variables $k$ and $b$ to determine when a system is stable. ...
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1answer
33 views

Why is $<T\vec x,\vec y>=<\vec x,T^*\vec y>$ for hermitian matrix T?

In this video in 5:15 there's a proof that every hermitian matrix has real eigenvalues. I don't understand the step: $<\vec x, L\vec y>=<L^*\vec x,\vec y>$. I know that I can pull out ...
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1answer
29 views

Assessing the geometric multiplicity of an eigenvalue.

Suppose $\lambda_1 = 1$ is an eigenvalue of the hypothetical matrix $\mathbf A$ $$ \mathbf A = \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{11} -1 & a_{12} +1 &...
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1answer
42 views

Diagonalizing the matrix (if possible)

Diagonalize the matrix $\begin{bmatrix}0&-4&-6\\-1&0&-3\\1&2&5\end{bmatrix}$ if possible So I know that I can check to see if this is diagonalizable by doing $A = PDP^{-1}$ ...
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1answer
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Proof: $\underset{\|q\|=1}{\max} q^TAq = \lambda_{\max}$ with $q$ the corresponding eigenvector ($A$ symmetric)

This problem is quite old and there should be similar problems. I know the following technique: \begin{equation} \begin{aligned} q^TAq=q^TU\Lambda U^Tq=(U^Tq)^T\Lambda (U^Tq) \end{aligned} \...
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0answers
21 views

Stability theorem in numerical eigenvalue problem

This paper mentions the stability theorem in $ 6.1 $ as following: If $ A_{n \times n} $ and $ E_{n \times n} $ are real and symmetric matrix and $ \hat{A} = A + E. $ Let $ \lambda_{1}, \lambda_{2}, \...
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1answer
47 views

Find the eigenvalues and eigenspaces for the matrix

Find the eigenvalues and eigenspaces for the matrix $\begin{bmatrix}4&2&2\\2&4&2\\2&2&4\end{bmatrix}$ I know there is a trick to this one with out doing a $3 \times 3$ ...
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2answers
34 views

How can i find the dimension of the eigenspace?

The matrix $A = \begin{bmatrix}9&-1\\1&7\end{bmatrix}$ has one eigenvalue of multiplicity 2. Find this eigenvalue and the dimension of the eigenspace. So I found the eigenvalue by doing $A - ...
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1answer
28 views

Transition matrix has independent eigenvectors

If $ A_{n \times n} $ is a transition matrix $ - $ a positive matrix in which the sum of all entries in each row or columns equals $ 1 - $ is it true that $ A $ must always have $ n $ linearly ...
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1answer
28 views

For which value of $k$ does the matrix $A$ have one real eigenvalue of multiplicity $2$?

For which value of $k$ does the matrix $$A = \begin{bmatrix}-6&k\\-1&-2\end{bmatrix}$$ have one real eigenvalue of multiplicity $2$? So I understand that if the discriminant is $0$ such that ...