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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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
66 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
38 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
45 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 ...
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1answer
22 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 ...
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3answers
310 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
44 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 ...
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1answer
47 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
28 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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0answers
55 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
26 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
31 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
67 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
32 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
62 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
59 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
53 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
24 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
55 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
35 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 ...
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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 ...
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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) ...
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0answers
31 views

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
53 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
89 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
36 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 & ...
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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 ...
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0answers
62 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 ...
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1answer
26 views

Lower bound for the spectralradius of a matrix

Any submultiplicative norm (for example the row-sum-norm) is an upper bound for the spectralradius of a matrix A. But is there a way to get a suitable LOWER bound for the spectralradius ? ...
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Shape operator and orthogonality of eigenvectors

When studying differential geometry (at a hobby level) I always run into problems when it comes to the varying notations and statements about the shape operator ...
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1answer
18 views

Difference between matrices with altered eigenvalues

Given two p.s.d. matrices $X_1$ and $X_2$ with eigen decomposition $X_1 = U_1V_1U_1^T$ and $X_2 = U_2V_2U_2^T$ and a constant $\lambda > 0$ Now consider an altered version of the eigenvalue ...
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0answers
28 views

Numerical Computation for K smallest eigenvalues of a large Real Symmetric Matrix with restricted methods

I'm writing some code on a distributed platform, using some programming language like Hadoop, and now I need to calculate the K smallest eigenvalues for a Large Matrix. K is a small constant at most ...
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2answers
30 views

vector question assistance

let there be 2 lines: $(2,-3,1) + s(3,-2,1)$ and $(2,-1,-3) +t(3,-2,1)$ which are parallel to each other. find the formula of the plane determined by them. my try: a vector perpendicular to ...
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1answer
12 views

Change of Eigenvalues of Ellipsoid Tensor during Rotation

I have an ellipsoid defined by the semiaxes $a,b,c$ and the orthonormal vectors $v_x, v_y, v_z$ describing the directions in which the axes point. The matrix ...
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2answers
27 views

Finding two matrices that permit a change of variable eliminating the crossed term.

Learning about eigenvalues/vectors. Here is an exercise which I guess is about that, but I am not really sure how to get started with it: For $5x_1^2-4x_1x_2+5x_2^2=21$ find a diagonal matrix $D$ ...
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1answer
37 views

Constructing regular integer matrices with distinct integer eigenvalues

How can I construct matrices with positive integer values and distinct integer eigenvalues (not necessarily positive, but 0 should not be an eigenvalue). The standard-method to construct matrices ...
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0answers
19 views

Triangularisation of a linear map

The Calculation of the Char Poly is wrong but it's the method I am not able to understand In this example why does the eigenvector of A give the required eigenvector that is contained in the basis? ...
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1answer
29 views

Triangularisation of a linear transformation

I understand that Upper triangular matrices must have at least one eigenvector, but why does this mean that the basis of $[T]_B$ must contain an eigenvector for $[T]_B$ to be upper triangular?
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1answer
44 views

$A,B$ matrices , prove $Bv = \Lambda v$

$A,B$ are $n \times n$ matrices and $AB = BA$ Also, there is an eigenvalue $\Lambda$ in $A$ which its geometric complexity is $1$. Also there is $ v \ne 0 $ that $v$ is an eigenvector of $A$. ...
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Order of eigenvectors in jblas?

I am using jblas to compute eigenvectors of a double symmetric matrix. Using symmetricEigenvectors(myMatrix)[0], I can get a matrix which columns are the eigenvectors of my matrix. However I need them ...
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0answers
85 views

Eigenvalues of $\pmatrix{1&1\\1&2}$

I use maxima for calculation eigenvalues of this matrix: $$ \begin{bmatrix} 1 & 1 \\ 1 & 2 \end{bmatrix} $$ and I get $\frac{3\pm\sqrt{5}}{2}$ and then $[1,1]$ for some reason. Namely: ...
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26 views

relation between eigen values

Let $W$ be a finite subgroup of $GL(V)$ and hence it acts on $V$. Now consider the contra gradient action of $W$ on $V^*$. Now how to show that the eigen value of this action is the reciprocals of the ...