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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Link between the two definitions of a “hyperbolic point”

The common definition of a hyperbolic point for a flow of a vector field $f$ is a fixed point in which the eigenvalues of the Jacobian matrix of $f$ all have non-zero real parts ...
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2answers
33 views

eigenvalues ​​and eigenvectors

Compute the eigenvalues ​​and eigenvectors of the following matrix T: $$T =\begin{pmatrix} 1&4\\ 0&1 \end{pmatrix}$$ I have $(T-\lambda I)=0$ and from this I found $(\lambda -1)^2$, so ...
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1answer
15 views

Can the matrix product $PA$ be skew-symmetric with $P=P^T>0$ and $A$ Hurwitz?

Let a real (square) matrix $\mathbf A$ is Hurwitz (i.e., all the eigenvalues of $\mathbf A$ have negative real parts). And let $\mathbf P$ is a real symmetric positive definite matrix. What will be ...
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0answers
17 views

Is the eigenvalue decomposition equal to the singular value decomposition for real symmetric matrices?

Question is as the title states. I've read something similar for hermitian matrices, but am unsure if this is correct as well for real symmetric matrices.
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0answers
41 views

Eigenvalue formula for 4x4 symmetric matrix

Is there a formula/algorithm that is accurate to used in finite precision arithmetic (aka numerical stable ) for small symmetric matrix of size 4x4. Additionally I'm looking if it require similar ...
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2answers
19 views

Matrix representation of a linear operator

As I'm studying for my final, my book keeps skipping alot of steps and I don't know how tthey get from point a to point b - probably because its elementary at that stage in the book, except not to me ...
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0answers
15 views

Is there a bound on largest eigenvalue for covariance matrix of discrete random variable?

I have a random variable $Z=(Z_1,\ldots,Z_p)$. Each component can take values in {-1,0,1}. Is there a way to bound the largest eigenvalue of Cov(Z)? Actually, I have a latent multinormal variable ...
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0answers
16 views

Ia it possible to use the deflation algorithm to compute the eigenvalues of a large sparse matrix

I am trying to compute the eigenvalues of a large sparse matrix (about 10% of the values are nonzero). The matrix is real valued, but since it is accumulated by a stochastic process it is not fully ...
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1answer
40 views

Characterization of a matrix with eigenvalues equal to one

Consider an $m\times m$ non-negative matrix $A$ where elements of $A$ can take many different values e.g. they are functions of a variable z. Suppose $A$ is such that one of its eigenvalues is equal ...
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0answers
18 views

Stability of a block matrix with a stable upper left corner

Given a $n\times n$ matrix $A$ is stable, can it be proven that $G=\left(\begin{array}{cc}A & B\\C & -d\end{array}\right)$, where ...
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0answers
47 views

a proof question regarding to eigenvalues and diagonalization [on hold]

Let the scalar field be $\mathbb{F}$. Let $T: V\rightarrow V$ be a linear operator represented by the $n\times n$ matrix $A= [T]_{\alpha\alpha}$. Suppose that the characteristic polynomial of $A$ ...
3
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1answer
67 views

Is it true that $u + v$ is an eigenvector corresponding to the eigenvalue $\lambda$?

Let $A$ be an $n \times n$ matrix, and $u, v$ be eigenvectors corresponding to an eigenvalue $\lambda$ of $ A$ (that is, $Au = \lambda u$ and $Av = \lambda v$). Is it true that $u + v$ is an ...
0
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2answers
49 views

Characteristic polynomial and eigenvalues of a $3 \times3$ matrix.

Hi so I have to find the characteristic polynomials and the eigenvalues of the matrix: $$A = \begin{bmatrix}1 & 0 & 3\\2 & -2 & 2\\3 & 0 & 1\end{bmatrix}$$ So I know you use ...
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0answers
23 views

Which casses of matrices contain A and which contain B? Linear Algebra

Am pretty confused about classes, I don't know what it means, so so I can't really do part_A and I need your help with it? For part B, I got all eigen = 1 for matrix A, and 0 for matrix B, Is this ...
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1answer
19 views

How do you get nullspace N(A) to be orthogonal to C(A^H)

In the picture below, C(A) is given in number7, but I am doing number_8. Ii did a gauss jordan where by i subtracted R2-iR1 to get 0 belo 1st pivot and 1 as the second pivot in column2, row2. Then I ...
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0answers
32 views

Why does s = z+1?

What exactly is Laplace transform? motivated me to ask why unit function is 1/s by Laplace transform and 1/(1-z) by Z-transform? Both seem to be integrals of delta-pulse and secondary integration ...
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0answers
34 views

Show that isotropic function S(A) and A have same eigenvectors

Given $\boldsymbol{A}$ is a positive definite, symmetric second order tensor and $\boldsymbol{Q}\boldsymbol{S}(\boldsymbol{A})\boldsymbol{Q}^T = \boldsymbol{S}(\boldsymbol{QAQ}^T)$ $\forall ...
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4answers
182 views

If $\lambda$ is an eigenvalue of $A^2$, then either $\sqrt{\lambda}$ or $-\sqrt{\lambda}$ is an eigenvalue of $A$

$A$ is an $n\times n$ matrix of complex numbers. Prove that if $\lambda$ is an eigenvalue of $A^2,$ then $\sqrt{\lambda}$ or $-\sqrt{\lambda}$ is an eigenvalue of $A.$ If $\lambda$ is an eigenvalue ...
0
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2answers
48 views

How are signs on eigen vectors chosen, am confused? Linear Algebra

I have found the eigen vaues, I also know that you can find the eigenvectors through a Gausian Jordan. -- x1, gauss jordan gives me rows(1 -1/3 ,, 0 0 ), so [a, b] = [1,3] For vector x2, GJ gives ...
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1answer
32 views

Relation on the determinant of a matrix and the product of its diagonal entries?

Let $A$ be a $3\times 3$ symmetric matrix, with three real eigenvalues $\lambda_1,\lambda_2,\lambda_3$, and diagonal entries $a_1,a_2,a_3$, is it true that \begin{equation*} \det ...
0
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0answers
15 views

Complexity of eigensolvers for sparse matrices

What is the usual complexity of the common (iterative) eigensolvers (Arnoldi, Lanczos, ...) for a very sparse (tridiagonal) symmetric $n\times n$-matrix? Can one eigenpair be computed in ...
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1answer
64 views

Is the following Eigenvalue inequality holds or not?

Can anyone help me with the following problem? Suppose $u=(u_1,u_2,...u_n)^T$, $e=(1,1,...1)^T$, and we have $u\geq e$. Now for any symmetric matrix $A\in S^n$ with $diag(A)=0$, can we claim the ...
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1answer
43 views

The geometric multiplicity

By given this matrix: \begin{pmatrix}0&a&0\\0&0&1\\0&0&0\end{pmatrix} Why for any a which is not 0 the geometric multiplicity = 1? and why for a = 0 the g.m. = 2? I don't ...
4
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0answers
56 views

Linear Algebra, Eigenvalues and Eigenvectors Exercise

I have a question from an exercise. I am given a vector space over the field $\mathbb{R}^{3}$ with 2 dimensions and I am asked to find a basis of eigenvectors. I found the eigenvalues but I have ...
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3answers
47 views

Diagonalization with the given eigenvalue and its vector

Let $-3$ be an eigenvalue of a $3\times3$ singular matrix $P$ and $$P\begin{bmatrix} 5\\ 3\\ -2 \end{bmatrix}=\begin{bmatrix} -20\\ -12\\ 8 \end{bmatrix}.$$ Then find whether $P$ is ...
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5answers
83 views

If $A^2$ is the zero matrix, show that $A$ is linearly dependent?

The original question was show that $0$ is an eigenvalue for the matrix $A$. This was a straightforward practice of righthand multiplication of $Ax = \lambda x \Rightarrow AAx = A \lambda x ...
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1answer
41 views

Question about proving symmetric matrices are diagonalizable

Definition : If a n by n matrix $A$ is orthogonally congruent to another matrix $B$, then there exist an orthogonal matrix $C$ such that $$A = C^{-1}BC$$ Theorem: If $A$ is symmetric, then $A$ is ...
0
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0answers
50 views

Why should I care about eigenvectors/eigenvalues [duplicate]

I've been studying pattern recognition/machine learning and the theory behind it for some time now and I notice that I find myself seeing the same things over and over again, yet without fully ...
2
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1answer
28 views

Orthogonal projector onto an eigenspace of a self-adjoint operator

Suppose that $A$ is a self-adjoint linear operator on a Euclidean finite-dimensional space $V$. Is it true that any orthogonal projector $P_\lambda$ onto an eigenspace of $A$ can be represented as a ...
2
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3answers
319 views

Suppose A has eigenvalues 1,2, 4.

a) What is the trace of $A^2$ b) What is the determinant of $(A^{-1})^T$ I need someone to check my answers and correct me, am especially not sure about part a), help me me out; for a), I did--- ...
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1answer
59 views

Trace of power of stochastic matrix

I would like to know if this statement is true. Having a stochastic matrix (rows sum up to 1), with a positive (non-negative) diagonal, then it holds that $$\text{trace}({W^2})\leq ...
3
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1answer
48 views

$3 \times 3$ real matrix: relation with determinants

$A$ is a $3 \times 3$ matrix with real entries such that $\operatorname{det}(A+I_3)=\operatorname{det}(A+2I_3)$. Then is $2\operatorname{det}(A+I_3)+\operatorname{det}(A-I_3)+ 6 =3 ...
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1answer
30 views

LDU matrix decomposition

Let $A$ be a matrix that can be written as $LDU$ for some lower unitriangular matrix $L$, some diagonal matrix $D$ and some upper unitriangular matrix $U$. Then, are the eigenvalues of $A$ the same as ...
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78 views

Minimum and maximum determinant of a sudoku-matrix

Let $A$ be a sudoku-matrix. Assume that its determinant is positive. What is the lowest, what the highest possible value for the determinant of $A$ ? $A$ must have the dominant eigenvalue $45$, but ...
0
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0answers
27 views

Eigenvalue bounds for a positive semidefinite matrix

I have a symmetric $(p\times p)$, positive semi definite matrix $\Omega$. If somebody says: find the eigenvalue bounds of the matrix such that $$w_1I \le \Omega \le w_2I$$ where $I$ is the identity ...
0
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1answer
33 views

Equality in the Collatz-Wielandt-formula

Let A be a matrix with positive entries. The perron-frobenius-theorem states that A has a positive dominating simple eigenvalue, called the perron-frobenius-eigenvalue. I denote it with p(A). The ...
0
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1answer
57 views

Basic Eigenvalue Question

The rotation matrix $$T=\left[\begin{array}{c c}\cos\theta&-\sin\theta\\ \sin\theta&\cos\theta\end{array}\right]$$ has no eigenvectors as an operator $T:\mathbb{R}^2\to\mathbb{R}^2$. Here ...
3
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2answers
68 views

Matrix with all eigenvalues $0$ but not triangular?

Is the situation described in the title achievable? I am looking for a $3\times 3$ case specifically.
0
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0answers
33 views

eigenvalue of symmetric matrix where some diagonal elements dominate

A symmetric matrix $M$ has the following properties: $$ M_{ii}\gg M_{ij} ~~~~~~ i\neq j ~~~~~~~~~~~~\text{for}~ i>i_0~~~~~~ $$ and all the dominated diagonal elements are equal. My ...
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1answer
86 views

Relationship between eigenvectors of matrices

I am investigating parameter estimation in reduced-rank regression and have come across the following linear algebra result which I haven`t been able to prove. Suppose, $A \in \mathbb{R}^{nxm}$ of ...
3
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0answers
64 views

Simultaneous diagonalization of commuting matrix

I have 3 diagonalizable matrices $A,B,C$. They commute with each other $[A,B]=[B,C]=[A,C]=0$ [edit] The matrix $A$ is Hermitian but $B$ and $C$ have no properties. [/edit] I can get the eigenvalues ...
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2answers
48 views

Matrix with eigen values given find [closed]

Let$$ P=\begin{bmatrix} 0&-2&-3 \\ -1&1 &-1 \\ a&2 &b \end{bmatrix},$$ for some $a,b \in \mathbb{R}.$ Suppose that $1$ and $2$ are eigenvalues of $P$ and $$ ...
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1answer
60 views

Prove or disprove: if a set of 2d-points have symmetry axises, then at least one of the axises is eigenvector

Let's assume we have a set of 2D-points. My claim is that if that group has at least one valid symmetry axis, then at least one of those axises is equivalent to an eigenvector of the covariance matrix ...
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56 views

Reference for simplicity of the principal eigenvalue of the Laplacian

i'm currently searching for a proper reference or proof to see that the first eigenvalue $\lambda \in \mathbb{R}$ of \begin{equation*} - \Delta u = \lambda u \text{ in } \Omega, \\ u \in ...
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0answers
26 views

Proof that all nonreal eigenvalues of real coefficient linear transforms come in conjugate pairs?

I know how to prove what the title says using determinants, but let's say I wanted to use another approach. In Axler's Linear Algebra Done Right, he seems to avoid using determinants for proving ...
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1answer
27 views

Why does an algebraic multiplicity of n imply an n-dimensional eigenspace for a Hermitian matrix?

I want to prove that given any Hermitian operator, we can find an orthonormal eigen basis for it. It is obvious there are $n$ eigenvalues counting multiplicities, and it is easy to prove that any two ...
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2answers
42 views

Eigenvalues of $I_n-A$

Is there a simple relationship betweeen the eigenvalues of a $n\times n$ matrix $A$ and the matrix $(I_n-A)$? I beg your pardon if this questions has already been answered.
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1answer
57 views

Proof wanted that there is no positive integer matrix with positive integer eigenvalues u,v,w, if $0<u<v$ and $1\le w-v\le 2$

I have the following conjecture : If u,v,w are integers with $0<u<v<w$, then there is a POSITIVE INTEGER 3x3 - matrix A with eigenvalues u,v,w if and only if $w-v\ge 3$. I approved the ...
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2answers
37 views

Need help with eigenvectors and eigenvalues

Let $A$ be an $n\times n$ matrix with $v \neq 0$ being it's eigenvector and $\lambda$ being the eigenvalue that $v$ is associated with. I need to prove this: 1) $\lambda$ is an eigenvalue of $A$ if ...
0
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1answer
24 views

In what cases are the eigenvalue equal to the pole points?

I have a transfer function in form of a matrix and want to determine the stability of the whole system. Now I'm wondering if I need to calculate the pole points or the eigenvalue. A friend of mine ...