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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How to get an eigenvector of a $3\times 3$ matrix that has first column and a row of zeros

I have the following matrix $$ \begin{bmatrix} 1& 0& 0\\ 0& 1& 1\\ 0& 1& 1 \end{bmatrix} $$ First I got the eigenvalues which are $0$, $1$, $2$. I tried to get the ...
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5 views

Stieltjes transformation of e.d.f. sample eigenvalues

If the eigen-decomposition of sample covariance matrix is $S=PDP'$ where $D$ is a diagonal matrix with eigen value of $S$ and $P$ are eigenvectors. If we define the empirical distribution function of ...
2
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1answer
41 views

How does negating a matrix affect its eigenvalues?

I'm working on the following problem: "If $Ax = \lambda x$, find an eigenvalue and an eigenvector of $e^{At}$ and also of $-e^{-At}$." So far, I have figured that $e^{\lambda t}$ will be an ...
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8answers
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Assume that the square matrix A has an eigenvalue of 0. Is A invertible? Why or why not?

Just wanted some input to see if my proof is satisfactory or if it needs some cleaning up. Here is what I have. Proof:Suppose A is square and invertible and for the sake of contradiction let 0 be an ...
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0answers
13 views

Finding characteristic roots and characteristic vectors

V is a two-dimensional vector space over the field of real numbers, with a basis $v_1, v_2$. Find the characteristic roots and corresponding characteristic vectors for T defined by $v_1(T) = v_1 + ...
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Any transformation in $A(V)$ satisfies a polynomial of degree $2$ over $\mathbb F$. [on hold]

$V$ is a two-dimensional vector space over a field $\mathbb F$; prove that every element in $A(V)$, the ring of linear transformations on $V$, satisfies a polynomial of degree $2$ over $\mathbb F$.
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1answer
12 views

How to prove that the spectral radius of a linear operator is the infimum over all subordinate norms of the corresponding norm of the operator.

I am trying to understand a proof I have seen of the following theorem: $$\rho(A)=\inf_{\|\cdot\|}\|A\|.$$ I understand that to do this, the idea is to show that 1) $\rho(A)\leq\|A\|$ for any norm, ...
2
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1answer
23 views

Eigenvalue of (1-0) matrix

Assume I have 2 matrices, each of size nxn with only 1 and 0 as entries in both. (n>10) The first matrix (call it A) has each row summing up to 2 (ie: on each row, it has two "1" and n-2 "0"). It is ...
2
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0answers
38 views

Question about an eigenvalue problem

I have a question... How can I show that the eigenvalue problem $$y''+λy=0$$ $$y(0)=0,$$ $$ y'(0)=\frac{y'(1)}{2}$$ is NOT a Sturm-Liouville problem?
4
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1answer
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What are eigenvalues of higher order finite differences matrices?

I know (at first empirically, then read somewhere) that for for second order finite differences matrices like $$\begin{pmatrix} -2&1&0&0&0\\ 1&-2&1&0&0\\ ...
2
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0answers
48 views

Second Derivative of Eigenvalue

Consider the potential $V_t=tx$ and the corresponding Schrödinger operator $H_t=- \frac{\partial ^2}{\partial x^2}+V_t$ on $L^2([0,R])$ with Neumann or Dirichlet boundary conditions. Let ...
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eigen values and vectors in this question

I have the following matrix to be solved for landa. here is the matrix with my answer (sorry for the wrong equation writing format): but the correct equation from this matrix should be (when ...
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2answers
19 views

Optimization of Product of Different Objective functions (Ex.: Maximize The Product of projections of a complex vector)

Suppose We have this optimization problem which is convex $\mathbf{x}={\arg}\: \underset{\mathbf{x}}\max f_{i}\left (\mathbf{x} \right )$ But the product of different objective function is ...
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1answer
43 views

Find an $n\times n$ integer matrix with determinant $1$ and $n$ distinct positive eigenvalues

I feel pretty stupid for doing this, but here goes anyway. Earlier today I asked: Find an $n\times n$ integer matrix with determinant 1 and $n$ distinct eigenvalues. As it turns out, for my problem I ...
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0answers
22 views

3-Species Population Model

I am trying to solve a 3-species predator-prey system in matlab. Here is the equation: $$\frac{d}{dt} \begin{bmatrix} N_1 \\ N_2 \\ N_3 \\ \end{bmatrix} = \begin{bmatrix} N_1 & 0 & 0 \\ 0 ...
4
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2answers
63 views

Find an $n\times n$ integer matrix with determinant 1 and $n$ distinct eigenvalues

Pretty much what the title suggests: for any positive integer $n$, I'm looking for an $n$-by-$n$ matrix with integer entries, determinant $1$ and $n$ eigenvalues. In case it is absolutely useless to ...
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1answer
12 views

Calculating eigenvectors where there is only 1 non zero number in matrix

So I am attempting to find the equilibrium points of a nonlinear system and I am getting the following jacobian matrix: $$ \begin{pmatrix} 1 & 0 \\ 0 & 1.5 \\ ...
0
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1answer
42 views

Is there a significance to a matrix having an eigenvector equal to a column vector within a matrix?

Consider matrix $A$: $\begin{bmatrix} 2 & -2\\ 3 & -3\\ \end{bmatrix}$ After little computing, we find the eigenvectors (and their corresponding eigenvalues) to be equal to $E_{\lambda=0}= ...
0
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1answer
15 views

How would you compute eigenvectors from this linear system?

I am stuck on a problem and I do not know how to obtain the eigenvectors: $\frac{dY}{dt}=\bigl(\begin{smallmatrix} -2&0\\ -3&1 \end{smallmatrix} \bigr)Y$ Work: I obtained the eigenvalues ...
3
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4answers
186 views

If all eigenvalues are 1 or -1, is then $A^{12}=I$?

True or false: If all the eigenvalues of A are either $\lambda=1$ or $\lambda = -1$ then $A^{12}$= I If we have a matrix $$\mathbf A = \begin{pmatrix}1&0\\0&-1\end{pmatrix}$$ this has ...
4
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4answers
194 views

Every n × n-matrix A with real entries has at least one real eigenvalue. [duplicate]

I have a true/false question: Every n × n-matrix A with real entries has at least one real eigenvalue. I am thinking that this is true but I would like to hear ...
0
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0answers
5 views

Apply MDS when all eigenvalues are negative

I want to map the vectors from n-dimensional to two-dimensional space. I am using MDS ( Multidimensional scaling) method as described in the paper: An Introduction to MDS. On 10 page it is written ...
0
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1answer
15 views

Complex Eigenvalue Periodic

This is probably a simple question, but I'm having trouble finding a clear answer. Let's say we have a system, with the complex eigenvalue: $$\lambda = \alpha + i\beta$$ I know that $\alpha < 0$ ...
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16 views

Low rank matrix square root

I'm trying to perform canonical correlation analysis (CCA) between matrices $X$ ($n \times p$) and $Y$ ($n \times k$), with covariance matrices $S_{X}=XX^T/(n-1)$ and $S_{Y}=YY^T/(n-1)$ respectively, ...
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+50

Characteristic vector components expressed as polynomial in $A$ and $\lambda$

If $\lambda$ is a simple root of $A$, a characteristic vector $x$ associated with $\lambda$ can always be taken to be a vector whose components are polynomials in $\lambda$ and the elements of $A$. ...
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2answers
33 views

Finding real, distinct eigenvalues for arbitrary constants

Let $A= \begin{bmatrix} 1 & 1 & 0 \\ -4 & -3 & 1 \\ k & 0 & 0 \end{bmatrix}$. Find all values of $k$ such that $A$ has three real distinct eigenvalues. I have obtained the ...
4
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1answer
24 views

Eigenvectors of sums of matrices

Suppose we know all eigenvalues and eigenvectors of the hermitian matrices $A$ and $B$, what does this say about the eigenvectors of $A+\varepsilon B$ for small $\varepsilon$?
2
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1answer
25 views

Proof of a theorem connecting Gerschgorin circles and eigenvalues

How do I prove that, given $A\in\mathbb C^{n\times n}$, if $A$ is irreducible and $\lambda$ is an eigenvalue of $A$ such that $\lambda\in\partial\left(\displaystyle\bigcup_{i=1}^nK_i\right)$, where ...
0
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1answer
17 views

Common eigenvector of a sequence of compact operators

Let $H$ be a separable, infinite-dimensional Hilbert space and suppose we have a sequence of norm-one compact operators $(A_n)$ on $H$ which all have 1 as an eigenvalue. Can we pass to a subsequence ...
0
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1answer
36 views

Problem determining eigenvalues of a Hermitian matrix

Suppose that you've got an $n \times n$ irreducible matrix $A$ with strictly positive real entries and eigenvalues $\lambda_i$, $i=1,...,m$, arranged so that $|\lambda_1| > \cdots > ...
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2answers
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Definition: Eigenvalues of a matrix

1) Can a non-square matrix have eigenvalues? Why? 2) True or false: If the characteristic polynomial of a matrix A is p($\lambda$)=$\lambda$^2+1, then A is invertible. Thank you!
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Odd Eigenvalue Problem

So, I'm trying to solve the following eigenvalue problem for the eigenvalues: $$u''(x)+\lambda^2u(x)=0$$ $$u(0)=u(1)$$ $$u'(0)=u'(1)$$ Of course, the two eigenvectors are cosine and sine, and the ...
0
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1answer
30 views

Matrix multiplication and rank reduction? - What is the minimal polynomial?

given is a matrix A with $\begin{pmatrix} a & 1 & 0 & \cdots & 0 \\ 0 & a & 1 & \cdots & 0 \\ \cdots & \cdots & \cdots & \cdots & \cdots \\ 0 & 0 ...
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2answers
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Transition Matrix eigenvalues constaints

I have a Transition Matrix, i.e. a matrix whose items are bounded between 0 and 1 and either rows or columns sum to one. I would like to know if it is possible that in any such matrices the ...
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3answers
48 views

Besides being symmetric, when will a matrix have ONLY real eigenvalues?

I realize that when a matrix is symmetric, then it must have all real eigenvalues. However, I am doing research on matrices for my own pleasure and I cannot find a mathematical proof or explanation ...
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1answer
16 views

sum of two matrices question given condition

How can it be proved that two matrices being orthogonally diagonalizable indicates that their sum is also?
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1answer
26 views

Why are eigenvectors of an invertible matrix linearly independent?

Question asked in the title! Thanks guys
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120 views

Show that $G$ ( subgroup of $\mathrm{GL}(E)$) is finite.

I came across with, I think, a difficult problem : Let E a Hermitian space with a Hermitian norm $||\ ||$. We provide $\mathcal{L}(E)$ with the norm $|||\ \ |||$ subordinated to $||\ ||$. ...
0
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2answers
26 views

Eigen Value & Eigen Vector Pairwise Relationship

Having same eigen values implies eigen vectors are linearly dependent. But why does it not imply that the eigen vectors are same? Are the eigen value and eigen vector pairs not unique for non-zero ...
0
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0answers
22 views

Find eigenvalues using matrix representation?

Let $V$ denote the space $P_2(t)\subset \Bbb R[t]$ of polynomials with real coefficients of degree at most $2$. Let $L : V \to V$ be the linear operator given by: $$ L(f(t)) = (t-1)\cdot ...
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0answers
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It would be possible to use the covariance matrix $C'=XX^T$ instead of the standard $C=X^TX$ to get the same result on PCA?

It would be possible to use the covariance matrix $C'=XX^T$ instead of the standard $C=X^TX$ to get the same result on PCA? If so, what are the next steps to retrieve the data with reduced ...
2
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1answer
34 views

Find Eigenvalues of multiplied Matrices when the corresponding Eigenvalues are known

I am trying to find the eigenvalues or in particular the largest eigenvalue of a transformation which consists of two matrices: $A = B C$. Assuming I know the EV of both matrices $B$ and $C$, is ...
0
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1answer
31 views

Finding Eigenvalues with Gershgorin-Discs

$A=\begin{pmatrix}-5 & 0 & 0 \\ 2 & 2 & 1 \\ 3 & -5 & 4\end{pmatrix}$ Find the Gerschgorin-discs, where the eigenvalues of $A$ lie. According to the formula if we ...
0
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1answer
38 views

Maximizing the product of projections of a vector on another vectors

I want to get the $N\times1$ complex vector $\mathbf{x}$ which maximizes this real valued function $f=\mathbf{x}^{H}\left (\mathbf{a}_{1} \mathbf{a}_{1}^{H}\mathbf{x}\mathbf{x}^{H}\mathbf{a}_{2} ...
0
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1answer
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How many iterations are generally required when using the power iteration method?

Suppose I have an n x n matrix and I want to find the dominant eigenvalue and its associated eigenvector. Given these dimensions, what is the minimum number of iterations of the power iteration ...
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Is the tensor formed by the tensor product of two positive semi-definite symmetric tensors itself positive semi-definite?

Let $A_{ij}$ and $B_{ij}$ be positive semi-definite and symmetric tensors. My question is then if we form a new rank-4 tensor $C_{ijkl}=A_{ij}B_{kl}$, does $C_{ijkl}$ also inherit the positive ...
3
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1answer
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Determine the dimension of a vector space.

Let $A$ be an $n \times n$ diagonal matrix with characteristic polynomial $(x-a)^p(x-b)^q$,where $a$ and $b$ are distinct real numbers. Let $V$ be the real vector space of all $n \times n$ matrices ...
2
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1answer
59 views

Conditions for “$AA^T=A^TA$ implies $A$ symmetric” to hold.

This claim arose in this question Show that $A$ is symmetric, with $A \in M_n(\mathbb R)$ where it is assumed additionally that $AA^TA$ is symmetric. I'm considering weakening hypotheses. Let $A$ be ...
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1answer
18 views

Prove that adjacency matrix has negative eigenvalue

We are given non-oriented graph without loops. Task is to prove that adjacency matrix of that graph has negative eigenvalue. I put some effort into drawing a proof here , but it seems that I'm ...
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2answers
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Matrix with no real eigenvalues

Given an $n \times n$ matrix $A$ with real entries such that $A^2=-I$. Prove that A has no real eigenvalues. We can easily prove the following additional statements about $A$ by taking determinants ...