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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What it means…eigenvalues of cubic

I've solved the flux Jacobian ($\frac{\mathrm dF}{\mathrm du}$) for a conservation equation ($\frac{\mathrm dU}{\mathrm dt} + \frac{\mathrm dF}{\mathrm du}\cdot\frac{\mathrm dU}{\mathrm dx}$ = Source) ...
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1answer
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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 ...
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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 ...
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1answer
21 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
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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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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) ...
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1answer
21 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, ...
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3answers
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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 ...
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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 ...
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2answers
54 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
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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 ...
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2answers
40 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
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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 ...
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1answer
21 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
55 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
39 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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0answers
5 views

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 ...
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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
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eigenvalue problem [on hold]

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
58 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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1answer
55 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 ...
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1answer
26 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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54 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
26 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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32 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 $\alpha\alpha^T \odot X^T X$ ...
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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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If A and B have same eigenvector, then AB has an same eigenvector? [on hold]

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 ...
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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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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 ...
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1answer
22 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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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 ...
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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
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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 ...
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1answer
42 views

Linear algebra problem about eigenvalues and eigenvectors

Please help with the following linear algebra problem. Suppose the following information is known about a matrix A: $$ A\begin{bmatrix} 1\\ 2\\ 4 \end{bmatrix} = 9\begin{bmatrix} 1\\ 1\\ 1 ...
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1answer
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Can Eigen vectors be different with the same normalization proceduce?

I calculated Eigen vectors of two badly-conditioned symmetric matrices of $K$ and $M$ ($M$ is positive definite). I employed two algorithms, the 1st algorithm is ...
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Find a representation of a matrix(M) expressed in the canonical basis in terms of the eigen-basis.

A 3*3 matrix (M) is given and I had a different 3*3 matrix which I used to find 2 corresponding eigen values and eigen vectors. My question is since I only have 2 eigen vectors, can I find -a ...
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1answer
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Proof that real symmetric negative matrix is negative definite

I have a reasonably simple symmetric $p \times p$ matrix $H$, where the $(j,k)$th element is given by $$h_{j,k} = -\sum_{i=1}^n \frac{a_i}{b_i^2} x_{ij} x_{ik}$$ and we know that all $a_i \geq 0$ ...
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1answer
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How to find the algebraic multiplicity given the eigenvalues and eigenspaces?

Let A be a 4x4 matrix with eigenvalues $\lambda$ = 2,3 and eigenspaces $E_{\lambda=2} = \operatorname{span} \left\{ {\begin{bmatrix} 1\\0\\0\\1\end{bmatrix}, \begin{bmatrix} ...
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1answer
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Eigenvalue problem $y'' + \lambda y = 0,$ $y'(0) = 0$, $y(1) = 0$

Find the eigenvalues of $$y'' + \lambda y = 0, \; y'(0) = 0, y(1) = 0$$ For $\lambda >0$, $$y(x) = c_1 \cos(\sqrt{\lambda} x) + c_2 \sin(\sqrt{\lambda}x)$$ We get that $y'(0) = 0 \implies ...
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Explain proof that any positive definite matrix is invertible

If an $n \times n$ symmetric A is positive definite, then all of its eigenvalues are positive, so $0$ is not an eigenvalue of $A$. Therefore, the system of equations $A\mathbf{x}=\mathbf{0}$ has no ...
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Maximum eigenvalue of product of two matrices

Let $A$ and $B$ be two Hermitian matrices. I wanted to know if there is any relation between the maximum eigenvalue of $AB$ and that of $A$ and $B$. Is the following relation true? ...
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Determine whether A is diagonalizable using these three steps.

Here is the question: A is the matrix $$ \begin{bmatrix} 3 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 1 & 2\end{bmatrix} $$ (a)find eigenvalues of A ...
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The Courant Min-Max theorem of elliptic pdes.

This is an exercise function Evans PDE book, Chapter 6. The theorem states that for $Lu:=-\text{div}(A\cdot\nabla u)+cu$ where $c\geq 0$, we have the eigenvalue of $L$ can be written in the following ...
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Why is it so difficult to obtain the spectral properties related to infinite matrices, especially when they are not symmetric?

I believe most of the spectral theory is revolving around the bounded self-adjoint linear operators being analogous to real symmetric infinite matrices. Whereas, there are cases when the matrices are ...
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Why are singular values of a positive

I read this in my textbook but couldn't understand why this is true: For a real positive semi-definite matrix A, the singular values are the same as the eigenvalues. Could someone please explain ...