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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How to show linear independence of complex eigenvector solutions

I'm having difficulty with a showing linear independence... Let $t = a + ib, b \neq 0$ be an eigenvalue of real matrix $A$ with associated eigenvector $z = p + iq$ Then the two real solutions ...
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14 views

matrix functions that preserve a specific property

Let $A\in \lbrace 0,1\rbrace^{n,n}$ be a symmetric matrix with $diag(A)=0$. Suppose there exists $i$ and $j$ such that $$\; \forall k\not\in \{i,j\}: \; A_{ik}\geq A_{jk} \quad (*)$$ $Let \; f: ...
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16 views

Differential/Bessel integration show that question

Given $y_k=J_m(\sqrt{\lambda_k}x)$ and let $y(x,\lambda)=J_m(\sqrt{\lambda}x)$. I can't seem to compute this integration and show $\int^1_0({{\dfrac{d}{dx}(xy'_k)y-\dfrac{d}{dx}(xy')y_k}} ...
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1answer
33 views

what is the eigenvalue of shift operator?

Show that shift operators have no eigenvalues. the shift operator or translation operator is an operator that takes a function $f(x)$ to its translation $f(x+a)$.let $α$ be an eigenvalue of the shift ...
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12 views

Is is possible to define a sign convention for eigenvectors calculated with a small uncertainty?

I'm working with a numerical method that involves the diagonalization of a real, symmetric $n \times n$ matrix $H$. Now obviously the sign of the (normalized) eigenvectors $\phi_i$ is not well ...
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1answer
25 views

Sturm-Liouville and Bessel function identity

Given S-L equation $\dfrac{1}{x}[\dfrac{d}{dx}(xy')+(\dfrac{-m^2}{x})y]=-\lambda y$ Say $\mathcal{L}$ is the Sturm-Liouville operator, $y_k$ is eigenfunction $J_m(j_{mk}x)$ where $J_m$ is Bessel ...
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Show that the space of all eigenvectors corresponding to one particular eigenvalue of a compact operator is finite dimensional. [on hold]

Show that the space of all eigenvectors corresponding to one particular eigenvalue of a compact operator is finite dimensional.
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10 views

what are the applications of linear transformations in civil engineering [on hold]

i wanted to know the applications of linear transformations, Eigen value problems and singular value decomposition in civil engineering
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21 views

The eigenvalues after a row and a colum has been deleted from a matrix.

Now I have a zero row sum matrix $L$, and a diagonal matrix $H$, where $L$ can be reviewed as a Laplacian matrix of a directed graph. That is, the off-diagonal elements of $L$ are either $0$ or $-1$, ...
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1answer
15 views

When PSD, singular value is equal to eigenvalue

It is known that If a matrix is PSD (symmetric), then its eigenvalues are equal to its singular value. How to prove it? Hope for a hint. thanks,
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singular eigenproblem due to using scatter matrices

I am facing a problem and I need urgent help :( I am using eig(A,B) and A and B are singular matrices (det=0 or very small number, and cond=Inf or a large umber). Consequently, all the returned ...
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37 views

A picture of the generalized eigenvectors

Just finished reading the proof of the existence of Jordan normal form in Artin's Algebra. I find it useful to have the following "picture" in mind to help me understand what Artin is doing, though I ...
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3answers
37 views

Need help with linear transformations (with projection and reflection)?

Let $L$ be the line given by the equation $4x − 3y = 0$. Let $S : \mathbb{R}^2 \rightarrow \mathbb{R}^2$ be reflection through that line, and let $P : \mathbb{R}^2 \rightarrow \mathbb{R}^2$ be ...
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36 views

Exponential decay estimate

Assume $u$ is a smooth solution of $$\begin{cases} u_t - \Delta u = 0 & \text{in }U \times (0,\infty) \\ \qquad \quad u=0 & \text{on }\partial U \times [0,\infty) \\ \qquad \quad u = g ...
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2answers
47 views

$Rank(A)=$number of non-zero eigenvalues then is $Rank(A)=Rank(A^2)$?

Let $A$ be an $n$ by $n$ matrix on some field. If $Rank(A)=$number of non-zero eigenvalues of $A$ then can we say that $Rank(A^2)=Rank(A)$? I believe we can say this (thinking about idempotent ...
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1answer
11 views

Orthogonalization of right/left eigenvectors of non-hermitian, matrices

For a non-hermitian matrix, that had a complex diagonal, but is otherwise symmetric (not hermitian), there are different eigenvectors for the left and right associated with the same (approximate) ...
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1answer
23 views

Calulate the eigenvalues and the eigenstates

An observable is given by $$\sum\limits_{n= 1}^N a_n|a_n\rangle\langle a_n | $$ Here $\langle a_n |a_m\rangle = \delta_{nm}$. What are the possible measurement results corresponding to the operator ...
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13 views

How can we use the symmetry of this complex matrix?

Find the Jordan normal form of $A\in \mathbb C^{4,4}$ if A is symmetric, $A^2=A$ and $\operatorname{rank} A=3$. So $A^2=A$ implies that the only eigenvalues are $0$ and $1$. From ...
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45 views

When can $|AB-I|=|BA-I|$?

Prove or disprove that for ANY two matrices $A$ (of dimension $m$ by $n$) and $B$ (of dimension $n$ by $m$), $\det(AB-I)=\det(BA-I)$. The answer is easily false as I found a counter example. ...
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3answers
14 views

Find the eigenvalues of the operator

A projection operator $P$ is defined as $P^2$=$P$. Use this definition to find the eigenvalues of this operator. In this question is it necessary to define what the projection operator ...
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38 views

Finding an eigenvectors and eigenvalues to a matrix

I got a question : Given A a matrix which the sum of all elements in each row equals to a constant $\alpha$, find eigenvector and eigenvalue it is belong to. I have no clue from where to start, ...
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2answers
51 views

$T^3-4T^2+4T=0$: inference about $T$

Suppose $T$ is a linear operator on $\mathbb R^2$ such that $T^3-4T^2+4T=\theta$ where $\theta$ is the null transformation. Then, describe $T$, given that $T$ is diagonalizable. My approach: ...
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2answers
28 views

Difference between these two eigenvectors?

i have a simple question regarding eigenvalues and eigenvectors. Consider the following matrix: $\left(\begin{array}{ccc} 1 & 1 & 3 \\ 1 & 5 & 1 \\ 3 & 1 & 1 ...
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59 views

Do the statements hold in an inner product space over $\mathbb R$ as well?

Let $V$ be an $n$-dimensional inner product space over $\mathbb C$ and $f\in \mathcal L (V)$ normal. Show that: $f^2=f^3 \implies f=f^2 \implies f = f^*$ $f$ nilpotent $\implies f=0$ ...
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26 views

A real and normal matrix with all eigenvalues complex but some not purely imaginary?

I'm trying to construct a normal matrix $A\in \mathbb R^{n\times n}$ such that all it's eigenvalues are complex but at least one of then also has a positive real part (well, at least two then, since ...
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32 views

Given a Positive Definite Matrix, find conditions of elements inside the matrix

I have a question that asks me to use the following symmetric positive definite matrix of order $n + 1$ $$B = \begin{bmatrix} \alpha & a^T \\ a & A \end{bmatrix} $$ With this matrix, I ...
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14 views

relationship of eigenvalues of a matrix with conjugate gradient method

Assume that $Q$ has all its eigenvalues in the two intervals $[a,b]$, $[a+\delta,b+\delta]$, while $a,b,\delta>0$. Show that for every start point $x_0$, after two steps of conjugate gradient ...
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1answer
36 views

Finding eigenvalues of a matrix with two unknowns

I've been asked to find the eigenvalues of the following matrix: $$ \begin{bmatrix} 0&1&1\\ 0&0&1\\ 216k^3&-108k^2&18k \end{bmatrix} $$ I'm just not sure how to work it out as ...
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1answer
55 views

Show that $ \lVert A \rVert_2^2 \leq \lVert A \rVert _1 \lVert A \rVert _ \infty $

With the definition of $ \lVert A \rVert_2$ and $\lVert A \rVert_1$ and $\lVert A \rVert_ \infty$ that is: \begin{gather} \lVert A\rVert_1 = \max_{j} \sum_{i=1}^m \lvert a_{ij}\rvert\\ \lVert ...
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73 views

Show $T$ is diagonalizable if $T-\lambda I$ is idempotent

Suppose $V$ is a finite dimensional vector space of dimension $n$ and $T$ is a linear operator on $V$ such that the characteristic polynomial of $T$ splits. Let $\lambda_1,\lambda_2,...,\lambda_k$ ...
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249 views

Eigenvalues that are complex numbers

Have a square matrix problem that involves complex numbers and am at a loss. $M$ is a square matrix with real entries. $\lambda = a + ib$ is a complex eigenvalue of $M$, show that the complex ...
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1answer
36 views

How to find eigenvalues of $n \times n$ real symmetric matrix?

So am trying to find signature of a bilinear form that satisfies $ f\mathrm (e_i\mathrm , e_j \mathrm)$ for all $i, j$ where $\mathrm e_i$ are the standard unit vectors. So the only way I can think ...
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2answers
22 views

the eigenvectors of two different square matrices that have the same eigenvalue

I have two square matrices $Y$ and $Z$ size $n$, and matrix $M = Z^{-1}YZ$ eigenvalue is the same as Matrix $Y$'s eigenvalue. I have been able to prove that the eigenvalues are the same, and thus the ...
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3answers
39 views

How to diagonalize this matrix??

So I have this matrix. $$\begin{bmatrix} 1 & 2 & 0 \\ 2 & 1 & 0 \\ 0 & 0 & -3 \\ \end{bmatrix}$$ $\det(A-λI)=0$ I get $$\begin{vmatrix} 1-λ & 2 & 0 \\ 2 & 1-λ ...
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29 views

Recover a specific solution from the general solution of the Riccati equation.

Consider the equation $XAX - AX = 0$, where $A,X$ are square $n \times n$ real matrices. We know $A$ and assume for simplicity it is diagonable. We want to solve the equation for $X$. We have $XAX - ...
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26 views

what is the eigenvalue and eigenvector of a matrix with the same value as diagonal but the rest of the matrix is 1

I have a question I'm trying to solve and have some intuition but need help with framing and formalizing it. I have a matrix $$ A = \begin{pmatrix} d& 1& 1&...&1\\ 1& ...
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1answer
24 views

Eigenvalues of special matrix with ones on the diagonals and constant $c$ on off diagonals

I came across this question in my research. If I have a p by p matrix $X$, with constant $c$ $X_{p\times p} = I_{p\times p} + c\mathbf{1}_{p\times p} - c diag(\mathbf{1})$, how do I analytically ...
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Problem with understanding theorem on Riccati Equation.

`The matrices $A,B,C,D,X$ are real, square, $n \times n$. I have trouble understanding theorem 7.1.2 from Lancaster & Rodman "Algebraic Riccati Equations". The part that I understand is as ...
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25 views

Largest positive eigenvalue of a matrix

I am dealing with the Capacity of constrained noiseless communication channels. It has been said that the channel capacity of such a channel is $\log{\lambda}$, which $\lambda$ is the largest positive ...
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2answers
31 views

the eigenvalues of matrix with a real number as the diagonal and 1 around it

I have a question I'm trying to solve and have some intuition but need help with framing and formalizing it. I have a matrix $$ A = \begin{pmatrix} d& 1& 1&...&1\\ 1& ...
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solving for eigenvalues & eigenvectors of the product of a column vector and row vector

I have a problem in my studies and am trying to prove it. I have worked so far, but need some advice as to whether I'm handling it correctly: Let $v$ be a column vector of size $n$ and let $A = ...
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Trying to prove that if two matrices with the same Eigenvectors are summed, the result has the same eigenvectors

This is a proof I am tyring to work out: A and B are square matrices, of same size. I am trying to show that if the eigenvector V of both and A and B, then v is also an eigenvector of $ M = c_1 A + ...
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Is the characteristic equation in ODE the same characteristic equation in linear algebra?

Can someone show me whether this "characteristic equation" thing in ODE is the same characteristic equation that we derive for a matrix? For example, given $y'' + 2y = 0$, the characteristic equation ...
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34 views

eigenvectors and eigenvalues of a symmetric positive semi-definite matrix

Given a symmetric, positive semi-definite matrix $M$ with $p$ dimensions, and its eigenvalues are $\lambda_1=\lambda_2>...>\lambda_p$, how to show that the corresponding eigenvectors $u_1$ and ...
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35 views

Cayley Transform and Eigenvalues

I have a particular operator, namely $A=-i\frac{d}{dx}$ that I would like to Cayley transform. $A$ is defined on the Hilbert space $L^{2}[0,1]$ and has domain $\mathcal{D}_{\alpha}=\{g:g \in ...
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1answer
43 views

Let $A$ be a $3$X$3$ matrix whose eigenvalues are $1$, $2$, $3$. Find $\det(B)$ where $B = A^2 + A^T$. [duplicate]

Let $A$ be a $3$X$3$ matrix whose eigenvalues are $1$, $2$, $3$. Find $\det(B)$ where $B = A^2 + A^T$. I know that $\det(A) = 6$, but I cannot proceed after $|A^2 + A^T|$. Any hints as to how to ...
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1answer
31 views

A pattern among the singular values of a matrix with a pattern

This question asked for an intelligent way to find $$ \det \begin{bmatrix} 2 & -1 \\ -1 & 2 & -1 \\ & -1 & 2 & -1 \\ && -1 & 2 & -1 \\ &&& -1 & ...
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43 views

Eigenvalues of an operator induced in a quotient space

Give an example of a vector space $V$, an operator $T \in \mathcal L(V)$ and a $T$-$\space$invariant subspace $U$ of $V$ such that $T/U$ has an eigenvalue that is not an eigenvalue of $T$. Attempt: I ...
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Find $\det(A^{2}+A^{T})$ when eigenvalues are $1,2,3$

We have to find $\det(A^{2}+A^{T})$. It is given that eigenvalues of $A$ are $1,2,3$. My attempt: Since the question implicitly states that the answer would be same for all $A$ with eigenvalues ...
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33 views

Relating Wilks' Lambda Criterion to Hotelling's T^2

I'm having trouble showing the relationship between Wilks' Lambda Criterion defined as w = det(E)/det(E+H), where H = n(X_bar - mu_0)(x_bar -mu_0)^t and E = (n-1)S and Hotelling's T^2. Can anyone help ...