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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a proof question regarding to eigenvalues and diagonalization

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
54 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 ...
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2answers
48 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
22 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
18 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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31 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
26 views

Show that g(S) and S have same eigenvectors when g(.) is an isotropic function

Given $\boldsymbol{S}$ is a positive definite, symmetric second order tensor and $\boldsymbol{Q}\boldsymbol{g}(\boldsymbol{S})\boldsymbol{Q}^T = \boldsymbol{g}(\boldsymbol{QSQ}^T)$ $\forall ...
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4answers
149 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 ...
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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
31 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 ...
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0answers
13 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
63 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 ...
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0answers
47 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
82 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 ...
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0answers
48 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
318 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
53 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
29 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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65 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 ...
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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 ...
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1answer
32 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 ...
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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 ...
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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.
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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 ...
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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
47 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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0answers
55 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
41 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
56 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 ...
2
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2answers
36 views

Eigenvalues of $A:\;A^2 +2A=0$

Let $A_n$ be square matrix where $n \geq 2$ and $A^2 +2A=0$. Then A is singular A is nonsingular 0 and -2 are eigenvalues of A either 0, or -2 is not an eigenvalue of A (1)-(4) are ...
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2answers
39 views

Eigenvectors of two matrices whose sum is an identity matrix? [closed]

What can I say about eigenvectors of two matrices whose sum is an identity matrix?
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1answer
36 views

Effective way of checking if all eigenvalues of a matrix are integers

Given A matrix with integer entries, it should be checked if all its eigenvalues are integers. Of course, the characteristic polynomial could be calculated, but is there any faster (or easier) ...
2
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0answers
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Which n-tuples of positive integers can be the eigenvalues of some matrix with positive integer entries?

This question is closely related to some questions I already asked Given a tuple of positive integers (such as (1,2,5) ), is there a matrix A with positive integer entries such that the integers in ...
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1answer
55 views

Norm of symmetric positive semidefinite matrices

I have been researching a lot trying to find an answer to my question and didn't find any so I would appreciate it if anyone can help. If we have 2 symmetric, positive semi-definite matrices $A$ and ...
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2answers
95 views

When is the dominant eigenvalue of this matrix greater than one?

So I am trying to figure out when this matrix $\left[\begin{matrix} a_1 & 0 & b \\ a_2 & a_3 & 0 \\ a_4 & a_5 & a_6 \end{matrix}\right]$ $b, a_i\geq0$ for all $i$, and ...
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2answers
37 views

Calculating all potencies of a Matrix

I've stumbled across this problem while reading my textbook (chapter eigenvalues) Calculate all potencies of $A$ and $A+aE$ $ a \in K$ and $A \in K-Vectorspace$ $ A= \begin{pmatrix} 0 & 1 & ...
3
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1answer
26 views

Eigenvalues of real square matrix with non-negative off diagonal?

Suppose I have a matrix $A$ with real entries such that the off-diagonal entries of $A$ are positive or zero. (The diagonal entries may be positive, negative or zero.) Is this a sufficient condition ...
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1answer
25 views

Control principal eigenvector of a row stochastic matrix

I am just trying to consider the classical discrete-time Markov Chain problem. Consider the transition matrix P, which transforms state vector $x(k)$ to $x(k+1)$, satisfying: $x(k+1)$ = $P*x(k)$ It ...
2
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64 views

Criteria for a solvable septic equation

I have a certain 7x7 matrix whose elements are all symbolic, and I want to know the eigenvalues. I have to solve a septic equation, but it is generally impossible. However, I am only interested in the ...