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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Bounding lower triangular perturbation

Suppose $A\in R^{n\times n}$ is a matrix equal to sum of the Identity matrix and a lower triangular matrix $L$. Diagonal entries of $L$ are $0$. \begin{equation} A=I+L \end{equation} Define spectral ...
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2answers
23 views

Proving that if $A$ is diagonalisable then $\chi_A(A) = 0$

This could be a very simple question to answer, but I'm unsure how to prove this. If you have a diagonalisable matrix $A$, prove that $\chi_A(A)$ is the zero matrix. (where $\chi_A(x)$ is the ...
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1answer
13 views

Characteristic polynomial: Identity permutation?

This concerns the characteristic polynomial of a matrix. http://www.math.umn.edu/~olver/num_/lnv.pdf p. 7 (or p. 92). every term is prescribed by a permutation π of the rows of the matrix ...
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1answer
23 views

$A \in SO(3,\mathbb R)\setminus\{I\}$ , then there are exactly two points in $S^2:=\{(x,y,z)\in \mathbb R^3:x^2+y^2+z^2=1\}$ which are fixed by $A$?

Let $A \in SO(3,\mathbb R)\setminus\{I\}$ , then is it true that there exist exactly two points in $$S^2:=\{(x,y,z)\in \mathbb R^3:x^2+y^2+z^2=1\}$$ which are fixed by $A$? Or equivalently we ...
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14 views

Principal axis form (quadratic) to hyperbola/ellipse form?

How is the principal axis form (quadratic) of a symmetric orthogonally diagonalized matrix e.g. $$y^2+6yx+x^2=1$$ transformed into the equation of a hyperbola (or ellipse as well)? Completing the ...
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1answer
32 views

If the characteristic polynomial of $A,B$ is the same does it mean that $A,B$ are similar?

If the characteristic polynomial of $A,B$ is the same does it mean that $A,B$ are similar? So I read that it's true only if $A,B$ are diagonalizable, but why? if the characteristic polynomial is ...
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23 views

What is the maximum value of coefficient $f_v$ with the constraInt that the matrix is positive semi-definite?

I am trying to solve this equation my self with my knowledge about characteristic polynomials, etc but I have placed it here earlier because I'm not a mathematician and maybe you give me ideas to ...
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2answers
30 views

$A$ is diagonalizable, if $A,B$ have then same eigenvalues, then $B$ is also diagonalizable

Given $A_{n\times n},B_{n\times n} \in \mathbb R$ such that $A$ is diagonalizable then: if $A,B$ have then same eigenvalues, then $B$ is also diagonalizable over $ \mathbb R$. if $A,B$ ...
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16 views

Transform in eigenvetor space

Hello I have a squared matrix C C = 0 2.2361 63.7887 2.2361 0 61.6117 63.7887 61.6117 0 and I calculate its ...
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1answer
21 views

If $A^4=4A^2$ then $m_A(x)=x^2-4$ and if it isn't diagonalaziable over $\mathbb R$ then $0$ is an eigenvalue

Given $A_{n\times n} \in \mathbb R$ such that $A^4=4A^2$ then if $A$ is invertible and isn't of the form $cI, c\in \mathbb R$ then $m_A(x)=x^2-4$. if $A$ isn't diagonalaziable over ...
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28 views

Let $A$ be a real symmetric matrix with rank $1$ , then can all the diagonal entries of $A$ be $0$ ?

Let $A$ be a square real symmetric matrix with rank $1$ , then can all the diagonal entries of $A$ be $0$ ? I know that real symmetric matrices are diagonalizable . Also if all the diagonal entries be ...
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1answer
35 views

Given a normal $A_{n\times n}$ matrix, then $\lVert A^*v \rVert = \lVert Av\rVert$ and $\langle Av,v\rangle = \langle A^*v,v\rangle$

Let a normal $A_{n\times n}\in \mathbb C^n $ matrix, then: $\forall v \in \mathbb C^n:\lVert A^*v \rVert = \lVert Av\rVert $ $\forall v \in \mathbb C^n : \langle Av,v\rangle = \langle ...
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39 views

Second eigenvector of double eigenvalue matrix

$\begin{bmatrix}\frac{1}{2} & -\frac{1}{2} \\ \frac{1}{2} & -\frac{1}{2}\end{bmatrix}\begin{bmatrix}x \\ y\end{bmatrix} = \begin{bmatrix}0 \\ 0\end{bmatrix}$ $\lambda_{1} = \lambda_{2} = -1$ ...
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20 views

Matrix shear transformations [on hold]

If you know the line of a shear transformation (the invariant line), how word you go about finding the shear factor? Also, funny as it may sound, what is the shear factor - what does it show?
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41 views

Showing that if $A$ is diagonalizable then $A^2-4A+8I$ is diagonalizable

Let $A_{n\times n}$ be a real matrix then: if $A^4 = 8A$ then $A$ is not invertible. if $A$ is diagonalizable over $\mathbb R$ then $A^2-4A+8I$ is diagonalizable over $\mathbb R$. ...
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1answer
59 views

The eigenvalues of $AB$ and of $BA$ are identical for all square $A$ and $B$ …a different approach

The eigenvalues of $AB$ and of $BA$ are identical for all square $A$ and $B$. I have done the proof in a easy way… If $ABv = λv$, then $B Aw = λw$, where $w = B v$. Thus, as long as $w \neq 0$, it is ...
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2answers
34 views

Finding Eigenvectors for $3 \times 3$ matrix with rows of zeros.

For a $3 \times 3$ matrix: $ $[A]$ = \begin{bmatrix} 6 & -2 & 2 \\ -2 & 3 & -1 \\ 2 & -1 & 3 \end{bmatrix} $ I have the eigenvalues: ...
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1answer
44 views

True/false questions about minimal and characteristic polynomials of a matrix

We have the matrix $A= \begin{pmatrix} 0 &2 &2 \\ 2& 0 &2 \\ 2& 2 & 0 \end{pmatrix}$, then one of the following is true: $f_A(x)=m_A(x) $ The matrix ...
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1answer
9 views

Not getting dot product of Eigen vectors in MATLAB aslo in LAPACK to zeros

I know that theoretically eigenvectors of real symmetric matrix are orthogonal to each other. So for each pair, dot product will be zero. But when I am calculating eigenvectors from real symmetric ...
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1answer
30 views

Eigenvalue perturbation theory for $(A^TA)(B^TB)^{-1} + (B^TB)(A^TA)^{-1}$

Let $A, B$ be $n \times n$ matrices with full rank. I'm interested in getting a bound on how the smallest eigenvalue of $S = (A^TA)(B^TB)^{-1} + (B^TB)(A^TA)^{-1}$ changes when I perturb $A$ and $B$. ...
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18 views

LTI system: solving for the time at which a system state reaches a given value

Suppose I have the following Linear Time Invariant (LTI) system: \begin{equation} \dot{x}(t) = Ax(t) + Bu(t) \end{equation} where $x(t)=\begin{bmatrix}x_1(t) & x_2(t) &\ldots ...
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23 views

Repeated Eigenvector/Eigenvalue matrix method

So I am having trouble with finding the generalized solution and I am not sure why my answer is interpreted as incorrect and I wanted to double check. $$ \overrightarrow{y'} = \begin{pmatrix} -6 ...
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2answers
25 views

Eigenvalues and vectors of a Linear Transformation

I am kinda lost here. All I did until now was finding eigenvalues and vectors for a matrix but as far as I can understand the question it asks me to find the eigenvalues of a Linear Transformation? ...
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20 views

Eigenvalues and Eigenvectors of an hyperbolic partial differential equations $\partial_t W + A \partial_x W = 0$

I read in a article dealing with a hyperbolic partial differential equations this statement : For any system of hyperbolic partial differential equations (pde), expressed as (1) ...
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3answers
45 views

Understanding the method to find Eigenvectors

For a matrix: $ $[A]$ = \begin{bmatrix} 5 & 4 \\ 1 & 2 \\ \end{bmatrix} $ I have the eigenvalues: $\lambda = 6, 1$ Now for each value I need to find ...
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2answers
55 views

If A and B are diagonalizable then so is AB

When we have to n×n matrices that can be made diagonal (maybe not in the same basis), is it true that the same works for their product?
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1answer
17 views

Eigenvector with matrix amlost full with zeros

Hi i have weird problem with calculate eigenvector from simplest matrices. So have something like this: $A = \begin{bmatrix} \frac{1}{2} & 0 \\ 2 & \frac{1}{2} \end{bmatrix}$ Eigenvalues are ...
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Posch-Teller potential

I made a matlab code to find numerical solutions to the Schrödinger equation independent of time, and it works very good, the eigenfunctions always are what they should be, the problem is in the ...
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1answer
38 views

Calculate complex eigenvector

Hi i have problem i hope that someone can make this for me more clear: So i have matrix $A = \begin{bmatrix} -2 & 1 \\ -2 & 0 \\ \end{bmatrix}$ I have to calculate eigenvector as matrix $P$ ...
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47 views

condition number of matrix plus constant times identity

I saw this post on the eigenvalues of a matrix plus a constant times the identity matrix. Say $A$ is an $n\times n$ matrix (real and non-singular) with eigenvalues $\lambda_1,\ldots,\lambda_n$, then ...
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29 views

rayleigh quotient of eigenvalue problem (sturm liouville theory and partial differential equations)

I am reading "A First Course in Partial Differential Equations with Complex Variables and Transform Methods" (Weinberger, p. 168). if we have the eigenvalue problem $$ (pu')'- qu + \lambda \rho u = 0 ...
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1answer
59 views

An exercise from vector spaces

I do not even know how to start. Find a symmetric matrix $A \in \mathbb R^{3\times 3}$ with the following properties. Both $[1,2,2]^T$ and $[2,1,-2]^T$ are eigenvectors. It has three distinct ...
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1answer
17 views

Finding null space of symmetric matrix generated by outer product

Let $p, q \in \mathbb{R}^n$ such that $||p|| = ||q|| = 1$ and define $A = pq^T + qp^T$. I am trying to find the null space of $A$, but am not having very much luck. I have managed to show that $p + q$ ...
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1answer
25 views

Proving that $V = U \oplus W$ where $W$ and $U$ are sets of eigenvectors of $S: V \to V$

Let $V$ be a finite dimensional real vector space, $S : V \to V$ be a linear map such that $S^2 = I$. Show that $V = U \oplus W$ where $U = \{u \in V : Su = u\}$ and $W = \{ w \in V : Sw = -w\}$. ...
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11 views

Which numerical method gives the most accurate solutions of Helmholtz equation for arbitrary domains?

There are many numerical methods for the solutions of PDE's such as FDM, FEM, SEM, Meshfree methods etc. I'm wondering which method gives the most accurate Dirichlet eigenvalues (and corresponding ...
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12 views

What is the effect on the spectrum by addition of a matrix with that of a rank 2 matrix?

Let $A$ and $B$ be two $n\times n$ matrices with rank of $B$ equal to $2$. Then how is the spectrum of $A$ and $A+B$ related? Or whether we can say something about - which of the eigenvalues of $A$ ...
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1answer
49 views

Finding a matrix given eigenvalues and eigenvectors.

I am asked to construct a $4 \times 4$ symmetric matrix, with given eigenvalues and eigenvectors. I understand how to actually get $A$ as a product of $P^T, D$ and $P$, when $D$ is the diagonal ...
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22 views

What has been already done on spectrum of Hermitian matrices?

Could anyone suggest some books/articles related to the determination of eigenvalues and eigenvectors of some special complex Hermitian matrices?
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32 views

Solve system of two homogeneous first-order ordinary differential equa0ti0ns by eigenvectors. (7.16-1)

Please check my work and I shall have a few questions along the way. I am working out of the textbook Advanced Engineering Mathematics by Erwin Kreyszig, 1988, John Wiley & Sons. The problem to ...
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257 views

Eigenvector and eigenvalue for exponential matrix

$X$ is a matrix. Let $v$ be an eigenvector of $e^{X}$ with corresponding eigenvalue $a$. Show that $v$ is also an eigenvector of $e^{X}$ with eigenvalue $e^{a}$ If $X$ is diagonalizable, then we can ...
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If $D$ is a $3\times 3$-matrix of order $6$, then the geometric multiplicity of eigenvalue $-1$ in $D^3$ is two

Consider the subgroup $H \le \operatorname{GL}(3,3)$ generated by the two matrices $$ A = \begin{pmatrix} 0 & 0 & 1 \\ 0 & -1 & 0 \\ 1 & 0 & 0 \end{pmatrix} \quad\mbox{ and ...
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Davenport's Q-method (Finding an orientation matching a set of point samples)

I have an initial set of 3D positions that form a shape. After letting them move independently, my goal is to find the best rotation of the original configuration to try to match the current state. ...
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1answer
43 views

Can we say anything about the relation of the first eigenvectors of a matrix and its rank one updated?

Let $R=S+xx^t$ where $x\in \mathbb{R}^n$ and $R$ and $S$ are $n\times n$ symmetric positive semidefinite matrices. Is there anything I can say about the difference of the first eigenvectors of $R$ ...
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3answers
47 views

Solutions of $\begin{pmatrix} 0&1&0 \\ 0&0&1 \\ 0&0&3\\ \end{pmatrix}x=0$

I'm having a bit of confusion here. What are the solutions of $\begin{pmatrix} 0&1&0 \\ 0&0&1 \\ 0&0&3\\ \end{pmatrix}x=0$ Clearly, $x_2=0$ $x_3=0$ $3x_3=0$ ...
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24 views

vector decomposition using eigenvectors

The eigenvalues of the matrix $$ \begin{pmatrix} 0 & 3 & 0 \\ 3 & 0 & 4 \\ 0 & 4 & 0 \\ \end{pmatrix} $$ are $5$, $0$, and $‐5$. ...
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43 views

Eigenfunctions of integral operator

I am faced with the problem of calculating the eigenfunctions for an operator of the form: $(Kf)(x) = \int_{-\infty}^{\infty} K(x-\alpha y) f(y)dy $ Does anyone know for which functions (or types of ...
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16 views

For what operators are $sin(ax)$ and $cos(bx)$ eigenfunctions?

So it is clear that the operator $\frac{d^{2}}{dx^{2}}$ is an eigenfunction of $sin(ax)$ and $cos(ax)$. For what other operators are sins and cosines eigenfunctions?
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59 views

How to compute the exponential of this given matrix?

I have a problem and can't find any solution. I have the matrix $A= \begin{bmatrix} 0 & 2 \pi \\ -2 \pi & 0 \\ \end{bmatrix}$ and I must compute the matrix $e^A$. I remeber that there was ...
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16 views

What can we say about the eigenvalues and diagonalization of this $2N\times2N$ matrix $A$?

There is a $2N\times2N$ matrix $A$, which is of the form: $A=\left(\begin{array}{cc} B & C\\ -C^{*} & -B^{*} \end{array}\right),$ where $B$ is a hermitian matrix, and $C$ is a symmetric ...
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1answer
33 views

Fundamental set of solutions for First Order Differential Equation

I am a bit unsure about this question and how to approach it and have tried numerous times. It is as follows: $$\overrightarrow{y_1}(t) = \begin{pmatrix} 2e^{3t}-4e^{-t}\\ ...