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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Movement of Horse Position during a race

I am trying to determine how to trace a horses position in running during a race and sort them in order of the horses have the fastest foot speed. Here is a sample of the data: ...
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1answer
28 views

Space of continuous functions linear operator eigenvalues

Let $V$ be the vector space of continuous functions from $\mathbb R$ to $\mathbb R$. Let $T$ be the linear operator on $V$ defined as $$(Tf)(x)=\int_0^x f(t)\,dt$$ Prove that $T$ doesn't have ...
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Questions about eigen vectors calculation in eignfaces

I am calculating the eigen vectors from a set of M grayscale faces images and I am using two methods. I would expect to get the same results but calculation gave me the different outcomes. Suppose I ...
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2answers
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Why does the set of an hermitian operator's eigenfunctions spans the functions space

During a discussion about linear hermitian operators, my professor claimed that if a linear operator $M$ is hermitian under a certian set of conditions, then genrally any function that fulfills these ...
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1answer
53 views

How do i find eigen vector

I need to find corresponding eign vector forthis problem Any hints for this .Thanks
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20 views

Change of base - Hermitic matrices

This exercise comes from a university exam (http://www.ubacs.com.ar/foro/viewtopic.php?f=67&t=3079, link in spanish). I'll copy it in english for everyone. It's #3: We define in $C^{n×n}$ the ...
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31 views

Proving matrix similarity for given matrices

Let $\mathbb{F}$ be a field, and let $A=(a_{ij})_{i,j=1}^n$ and $B=(b_{ij})_{i,j=1}^n$ be matrices in $M_n(\mathbb{F})$ such that: a. $b_{ij}=0 \iff a_{ij}=0$ $ \forall 1 \leq i,j \leq n$ b. ...
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1answer
28 views

Two square matrices with the same minimial polynomial are similar for $n=5$ or $n=6$

Let $\mathbb{F}$ be a field, $\lambda \in \mathbb{F}$ and $A,B \in M_n(\mathbb{F})$ such that $m_A(x)=m_B(x)=(x-\lambda)^k$ and such that the geometric multiplicity of $\lambda$ in $A$ equals to the ...
2
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2answers
24 views

Comparing complex eigenvectors if they are the same.

You have two normalized eigenvectors $V_1$ and $V_2$ in $C^n$. Even if $V_1 - V_2 \neq 0$, the two vectors may still represent the same eigenvector, since $a V$ for any complex number $a$ with unit ...
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1answer
43 views

How to simplify $\det(M)=\det(A^T A)$ for rectangular $A=BC$, with square diagonal $B$ and rectangular $C$ with orthonormal columns?

Assume a real, square, symmetric, invertible $n \times n$ matrix $M$ and a real, rectangular $m \times n$ matrix $A$ such that $m \geq n$ and $M = A^T A$. Also assume that $A = B C$, where $B$ is ...
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2answers
41 views

Eigenvalues of hermitian plus skew-hermitian PSD matrix

I was wondering, suppose you have a matrix of the form $A=B+iCC^\dagger$ where $^\dagger$ denotes the hermitian conjugate. $B$ is hermitian and $CC^\dagger$ is obviously hermitian positive ...
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+50

Maximizing a Concave function over a Non convex constraint set

I have been working on a problem which involves maximizing a concave function over a non convex constraint set. The problem is the following $$max. \frac{1}{2}x^TBx\\ s.t.\ x_i(1-x_i)=0,\ \ ...
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1answer
21 views

Eigenvalue of altered matrix: $pI_n + qA$

As a part of an exercise I have to prove the following: Let $p,q \in \mathbb{R}$. Let $A$ be an $(n \times n)$ matrix. Let $I_n$ be the $(n \times n)$ identity matrix. If $A$ has an eigenvalue ...
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1answer
9 views

Replacing Singular Values of a Matrix with Complex Ones

Is there a procedure to replace singular values of a real valued matrix according to: s1 -> i*s1 s2 -> i*s2 ... without going through any singular value decomposition (change singular values and ...
0
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0answers
25 views

Lax Pairs and constant eigenvalues

Can someone tell me whether the following is true, and if so a hint the proof? If we have a Lax Pair $\dot{L} = [A,L]$ then the eigenvalues of $L$ are constants of the motion. (The opposite ...
0
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1answer
22 views

Hermitian Matrix M(x) is continuous on x. Is its eigenvalue also continuous on x? [duplicate]

Each element of matrix $M(x)$ is a continuous function of $x$. Does this imply that all the eigenvalues are continuous function of $x$ too?
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1answer
26 views

find the eigenvalue of $A^m$

Let $$A = \pmatrix{7&9\\-3&-5},$$ it is a $2\times 2$ matrix. For every integer $m$, find all eigenvalues of $A^m$, and bases for the corresponding eigenspaces How to get it?!!
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2answers
38 views

How to find eigenvalues of this matrix

How to find eigenvalues of this matrix: $\left( \begin{array}{ c c } 2 & 0 & 0 \\ 0 & 2 & 4 \\ 0 & -1 & 2 \end{array} \right) $ ATTEMPT: $2-λ [(2-λ)(2-λ) ...
0
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1answer
25 views

Eigenvectors of multiplied matrices?

I have the review question if the vector u is an eigenvector of A and and eigenvector of B, then is also an eigenvector of AB, and BA, true or false, and explain why? I just have a feeling its true, ...
2
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3answers
49 views

If $v_1,…,v_r$ are the eigenvectors that correspond to distinct eigenvalues, then they are linearly independent.

Prove: If $v_1,...,v_r$ are the eigenvectors that correspond to distinct eigenvalues $\lambda_1, ...,\lambda_r$ of an $n \times n$ matrix $A$, then the set $\{v_1,...,v_r\}$ is linearly ...
5
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4answers
73 views

Whether a $2 \times 2$ matrix of rank $1$ has a zero eigenvalue

"Does $A = \begin{bmatrix}1&2\\2&4\end{bmatrix}$ have a zero eigenvalue?" Well, it would be a funny question to ask if the asker didn't state that he wants us to explain without computing the ...
3
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2answers
57 views

Find Jordan form of a $3\times 3$ matrix

$$\left( \begin{array}{ccc} 0 & 1 & 2 \\ -5 &-3 & -7 \\ 1 & 0 & 0 \end{array} \right) $$ I figured out the eigenvalues are all -1 from the characteristic polynomial, but I'm ...
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1answer
30 views

Finding eigenvvalue and eigenspace

I am given a matrix $A= \bigg({} \matrix{10 & 7 \\-14 &-11} \bigg{)}$ and eigenvalue $3$. My elite mission is to find the treacherous basis for the eigenspace. I used the $(A -eI)=v$ where ...
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2answers
28 views

Question about diagonalization and projections

Let a finite dimensional vector space $V$ above $\mathbb{F}$. Let $T:V\to V$ a diagonlizable transformation. We denote $a_1 \ldots a_r$ the $r$ different eigenvalues of $T$. By diagonalization, we ...
2
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2answers
43 views

$A$ be a $10*10$ matrix with complex entries s.t. all eigenvalues are non negative real and at least one eigenvalue is positive.

Let $A$ be a $10*10$ matrix with complex entries s.t. all eigenvalues are non negative real and at least one eigenvalue is positive. Then which of the following statements is always false? A. ...
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2answers
39 views

How to solve a linear system in matrix form using Laplace transform?

How to solve this linear system using Laplace transform? $$\mathbf X'(t)=\left[\begin{array}{r,r,r}-3&0&2\\1&-1&0\\-2&-1&0\end{array}\right]\mathbf X(t); ~~~~~~~~\mathbf ...
3
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1answer
23 views

Norm of an operator and eigenvalues

I have $K\colon L^2(0,T) \to L^2(0,T)$ a Hilbert-Schmidt integral operator (and so $K$ is linear, bounded, compact and self-adjoint) and I have obtained its eigenvalues and eigenvectors. From them, I ...
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3answers
56 views

Symmetric Matrix Transformation

Here's the question, Let $T$ be the transformation of 2 by 2 real symmetric matrices defined by: \begin{bmatrix}a&b\\b&c\end{bmatrix}>>>>\begin{bmatrix}c&-b\\-b&a\end{bmatrix} ...
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1answer
41 views

Prove or disprove the statement: if all the eigenvalues of a matrix are 0, then the matrix must be the zero matrix?

Prove or disprove the statement: If all the eigenvalues of a matrix are $0$, then the matrix must be the zero matrix. What I know : If the matrix is a upper or lower triangle matrix with the ...
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Extremal singular values of $P\Phi D$

Let $A=P\Phi D$ be a matrix where $P$ is a projection matrix such that $R(P)\subset R(\Phi)$ and $D$ is a non-singular diagonal matrix. Is there any relation between $\sigma_{min}(A)$ and ...
2
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2answers
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Let $A$ be a non-zero linear transformation on a real vector space $V$ of dimension $n$.

Let the subspace $V_o \subset V$ be the image of $V$ under $A$. Let $k = \dim (V_o) \lt n$ and suppose that for some $\lambda \in \mathbb{R}$, $A^2 = \lambda A$. Then which of the following are true? ...
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1answer
37 views

eigenvalue problem [closed]

Prove the following statement: $\max_i{\lambda_i(A^TA)}=\max_i{\left|\lambda_i(A^T)\right|^2}$ where matrix A is a N-by-N circulant matrix and $\lambda_i(X)$ denotes the $i$-th eigenvalue of matrix ...
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1answer
42 views

What is the linear space of Eigenvectors associated with a certain Eigenvalue?

The following matrix $A$ has $\lambda=2$ and $\lambda=8$ as its eigenvalues $$ A = \begin{bmatrix} 4 & 2 & 2 \\ 2 & 4 & 2 \\ 2 & 2 & 4 \end{bmatrix}$$ let $P$ be the ...
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2answers
60 views

Eigenvalues of a nilpotent matrix can only be $0$ [duplicate]

Prove that the eigenvalues for a square Nilpotent matrix A can only be $0$. Definition of nilpotent A $^n$=$0$ n is a positive whole integer
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2answers
65 views

Under what assumptions is it correct to say “a matrix is diagonalizable if and only if its eigenvalues are real”?

A $2\times 2$ matrix is diagonalizable if and only if its eigenvalues are real. Which statement is most correct: The proposition is true only if the eigenvalues are all greater than zero. The ...
0
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1answer
28 views

$p(x)$ divides the minimal polynomial iff $\exists v\ne 0: p(T)(v)=0$

Let $V$, a finite dimensional space. Let $T:V\to V$ a linear transformation. Show that $p(x)$, an irreducible polynomial divides $m_T$ (The minimal polynomial of $T$) iff there is a $V\ni v \ne 0$ ...
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0answers
55 views

Finding complex eigenvectors of $n \times n$ matrix, $n\geq 3$

An example: $$ \begin{pmatrix} 1 & 2 & 0 \\ 2 & -3 & 4 \\ 4 & -8 & 7 \\ \end{pmatrix} $$ Has eigenvalues $3$, $1+2i$, $1-2i$. How ...
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1answer
28 views

If $T^k = Id$ for $k\ge 1$ then $T$ is diagonalizable [duplicate]

Let $V$ a finite dimension space over $\mathbb{C}$ and $T:V\to V$, a linear transformation such that $T^k = Id$ for $k\ge 1$. Prove that $T$ is diagonalizable. I'd be glad for an hint. How do I ...
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33 views

Generalized Eigenvector Problem

I am reviewing a paper in which the solution to $$x =\max_\bf{v}\frac{\alpha \bf{v}^\dagger \bf{h}\bf{h}^\dagger \bf{v}}{\beta+\gamma\bf{v}^\dagger \bf{D} ...
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How does pointwise multiplication of two matrices affect their eigenvectors?

More specifically, suppose I have a known matrix $X\in\mathbb{R}^{d\times n}$ and an unkown vector $\alpha \in \mathbb{R}^n$. What can be said about the eigenvectors of $\alpha\alpha^T \odot X^T X$ ...
0
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2answers
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Prove that $T,S$ are simultaneously diagonalizable iff $TS=ST$. [duplicate]

Definition: We say that $S,T$ are simultaneously diagonalizable if there's a basis, $B$ which composed by eigen-vectores of both $T$ and $S$ Show that $S,T$ are simultaneously diagonalizable iff ...
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2answers
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If A and B have same eigenvector, then AB has an same eigenvector? [closed]

It says A and B have same eigenvector, call it V Does it follow that AB has same eigenvector? I think this is false statement?
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1answer
38 views

How to determine if a 3x3 matrix is diagonalizable?

The matrix is given as: $A=\begin{bmatrix} 0 & 1 & 1 \\0 & 0 & 4 \\ 0 & 0 & 3 \end{bmatrix}$ So the matrix has eigenvalues of $0$ ,$0$,and $3$. The matrix has a free ...
2
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0answers
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Polynomial Matrix Eigenvalue problem. Conditions under which there are only two complex eigenvalues?

I'm solving the polynomial matrix eigenvalue problem $(A\lambda^2+B\lambda+C)v=0 $. This is what I want the eigenvalues to look like. Are there any conditions on the matrices A,B,C such that there are ...
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0answers
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Eigenvectors and Generalized Eigenvectors

I've wondered whether someone could calrify me what are Generalized Eigenvectors, and why can I use them to find triangular form of a matrix. Say I have a $3\times3$ matrix, and I want to bring it to ...
0
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1answer
24 views

Linear combination of eigenvectors corresponding to the same eigenvalue

Suppose that $A$ is an element of $M_n(\mathbb{R})$ and $v_1,\dots,v_k\in\mathbb{R}^n$ are eigenvectors of $A$ corresponding to the same eigenvalue $\lambda$. Prove that if $u$ is a linear combination ...
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1answer
63 views

Eigenvalues of Left Shift + Right Shift in $l_2([0,\infty))$

This question appeared on an old final exam and I am having difficulty completing it for practice. Let $S_r$ and $S_l$ be defined on the hilbert space $l_2[0,\infty)\to l_2[0,\infty)$ as the ...
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Definiteness of matrix given definiteness of principal submatrix

I have a symmetric real $(J+1) \times (J+1)$ matrix $H$, with structure $$H=\begin{pmatrix} H_1 & H_2 \\ H_2^T & H_3 \end{pmatrix}$$ where $H_1$ is a symmetric $J \times J$ matrix, $H_2$ is ...
2
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43 views

Compute the Eigenvectors & Show A is diagonalizable

$A = \begin{bmatrix} 1&2&1 \\ 0&1&0 \\ 1&3&1 \\ \end{bmatrix} $ I computed the eigenvalues: $λ_ 1 = 1$ $λ_ 2 = 0$ $λ_ 3 = 2$ The ...
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1answer
21 views

Proof that triagonal matrices with distinct diagonal elements are similar

I'm trying to prove that if $A,B$ are triangular matrices with distinct elements along their main diagonals then the matrices are similar. I have been interpreting this to mean that the elements ...