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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Eigenvector of a tridiagonal matrix

$A = \left[\begin{array}{cccc} 2 & -1 & 0 & 0 \\ -1 & 2 & -1 & 0 \\ 0 & -1 & 2 & -1 \\ 0 & 0 & -1 & 2 \end{array}\right]$ where $A$ is an $4 \times ...
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1answer
23 views

If a given matrix A have the same eigenvalues that another matrix say B, then A and B are similar?

I want to make an argument that uses that if a given matrix A have p eigenvalues and if I'm asked to check if A is diagonalizable, then I can take another matrix B that have the same p eigenvalues but ...
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1answer
10 views

Symmetric Positive Definite 2x2 matrix equal to an Upper Triangular * Transpose of Upper Triangular

I have a symmetric positive definite matrix $$ B = \begin{bmatrix} 41 & 12 \\ 12 & 34 \end{bmatrix}$$ I am trying to find an upper triangular matrix $U$ such that $B = U^t U$ $\lambda_1 = ...
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1answer
20 views

Calculating eigenstates of Pauli matrices

I need to find out the eigenvalues and the eigenstates of the Pauli matrices. I know that the eigenvalues of $\sigma_x$, $\sigma_y$, and $\sigma_z$ are all $\pm 1$. How do I find the eigenstates?
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1answer
27 views

Eigenspace versus Basis of Eigenspace

I was wondering if someone could explain me the difference between eigenspace and basis of eigenspace. Right now I have only been able to somewhat understand the latter. Let's say that the row ...
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1answer
33 views

Symmetric positive semidefinite matrix is the square of a symmetric matrix

I am trying to show that matrix $A$ is symmetric positive semidefinite if and only if there exists a symmetric matrix $B$ such that $B^2 = A$. Here is my solution, any comments? I have attempted ...
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1answer
28 views

how to solve for the eigenvectors of a tridiagonal matrix

I have a tridiagonal matrix $A = \left[\begin{array}{cccc} 2 & -1 & 0 & 0 \\ -1 & 2 & -1 & 0 \\ 0 & -1 & 2 & -1 \\ 0 & 0 & -1 & 2 ...
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1answer
23 views

Relation between minimal polynomial and divisibility

Let A be a $n$ x $n$ matrix with rational elements and $p$ a prime number such that $A^p = I$ with $p<n$. If $det(A-I)\neq0$ it is true that $p-1$ divides $n$? Here is what I've worked so far. ...
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2answers
37 views

Creative ways to show that a given matrix is diagonalizable?

I know the standard method of calculate the characteristic polynomial, then get the eigenvalues, and look for the dimension of the null space associated to each eigenvalue, then see if their algebraic ...
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15 views

What physical intuition the eigen values and eigen vectors of adjacency matrix and laplacian of a graph provide?

So I have a undirected graph and its corresponding adjacency matrix $A$ and laplacian $L = D -A$, where $D$ is a diagonal degree matrix. What physical intuition can the eigen values and eigen vectors ...
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1answer
27 views

How do I show that a given value is an eigenvalue of matrix A?

I have been given a 3x3 matrix 'A' and a value 'v'. I have to show that v is an eigenvalue of matrix A. How do I start? And also, how to I determine all the eigenvectors that corresponding to v?
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17 views

How to find the eigenvalue and eigenvectors of a matrix with zeros along diagonal and non-zeros in first and last row? [duplicate]

What are the eigenvalues and eigenvectors of the matrix $$ \begin{matrix} 0 & b & b & ... & b \\ c & 0 & 0 & ... & 0 \\ c & 0 & ...
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2answers
31 views

Find the eigenvalues, eigenvectors and determinant of a matrix with unknowns along diagonal, first row and first column?

How do I find the eigenvalues, eigenvectors and determinant of the matrix $$ \begin{matrix} a & b & b & ... & b \\ c & a & 0 & ... & 0 \\ ...
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3answers
45 views

How many orthogonal eigenvectors does a symmetric and positive semidefinite matrix $A_{n\times n}$ has?

Suppose $A_{n\times n}$ is a symmetric and positive semidefinite matrix, and Rank(A)=k. I know that $A$ has k nonzero eigenvalues and corresponding orthogonal eigenvectors $v_1,\ldots,v_k$. I have ...
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0answers
30 views

Confused about eigenvetor

I'm trying to find the eigenvectors of a matrix with first row of (1 1) and second row of (1 -1). Got the eigenvalues to be $\sqrt 2$ and $-\sqrt 2$. When I tried however to find the eigenvector of ...
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1answer
20 views

How to prove the relationship between the leading $k$ eigenvectors and remaining eigenvectors?

Suppose $\mathbf{A}_{m\times m}$ is real symmetric and positive definite matrix. We employ SVD on A to get all orthogonal eigenvectors $\mathbf{u}_1,\cdots,\mathbf{u}_m$. Assume that ...
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1answer
20 views

Eigenvalues and Eigenvectors of a Normal Matrix

W is a normal stochastic matrix which has non-negative elements and each row sums to 1. W can be represented by the factorization (a constraint that can be imposed on the particular system): W = ED ...
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1answer
26 views

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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18 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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1answer
19 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
34 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
28 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. [closed]

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

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

i wanted to know the applications of linear transformations, Eigen value problems and singular value decomposition in civil engineering
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24 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
16 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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9 views

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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0answers
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
38 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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1answer
38 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
49 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
24 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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1answer
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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46 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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1answer
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
52 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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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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2answers
61 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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2answers
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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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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3answers
74 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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4answers
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
37 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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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 ...