# Tagged Questions

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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### Find the eigenvalues of a symmetric matrix

Find the eigenvalues of a $3 \times 3$ symmetric matrix with $1$ on the main diagonal and $\frac{1}{\sqrt 3}$ off the main diagonal. Since each row on addition give the same value, one of the three ...
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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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### 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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### 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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### 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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### 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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### 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: 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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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 $\... 1answer 25 views ### Suppose$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 ... 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 ... 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$\lambda_2 = 4$... 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. $$\hat ... 1answer 26 views ### Solving the quadratic formula to determine stability of a system I am trying to solve the 2\times 2 matrix$$\begin{bmatrix} 0 &1 \\ -k &-b \end{bmatrix}$$for a relationship between the variables k and b to determine when a system is stable. ... 1answer 33 views ### Why is <T\vec x,\vec y>=<\vec x,T^*\vec y> for hermitian matrix T? In this video in 5:15 there's a proof that every hermitian matrix has real eigenvalues. I don't understand the step: <\vec x, L\vec y>=<L^*\vec x,\vec y>. I know that I can pull out ... 1answer 29 views ### Assessing the geometric multiplicity of an eigenvalue. Suppose \lambda_1 = 1 is an eigenvalue of the hypothetical matrix \mathbf A$$ \mathbf A = \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{11} -1 & a_{12} +1 &... 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$A = PDP^{-1}$... 1answer 16 views ### Proof:$\underset{\|q\|=1}{\max} q^TAq = \lambda_{\max}$with$q$the corresponding eigenvector ($Asymmetric) This problem is quite old and there should be similar problems. I know the following technique: \begin{aligned} q^TAq=q^TU\Lambda U^Tq=(U^Tq)^T\Lambda (U^Tq) \end{aligned} \... 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}, \...
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 $3 \times 3$ ...