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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Decide if there exist supspaces $W,U$

Let $f(x_1, x_2, x_3, x_4)=(x_2+x_3,x_3,0,0) \ $ decide if there exists invariant, two dimensional subspaces $W,U$ of $\mathbb{R}^4$ such that $\mathbb{R}^4 = W \oplus U$ My question is if there ...
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Is there any trick to quickly find the eigenvalues of this matrix?

Is there any trick to quickly find the eigenvalues of this matrix: $$ \begin{pmatrix} 33 & 6 & 6 \\ 6 & 24 & -12 \\ 6 & -12 & 24 \\ ...
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52 views

What condition on $x$ makes the eigenvalues real or complex?

I have a $2 \times 2$ matrix $\begin{bmatrix}-i x& -1\\ -1& -i x\end{bmatrix}$. I computed the eigenvalues to get $\lambda = -ix -1$ or $-ix +1$. According to my teacher, $x$ must be greater ...
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If I have a given matrix A and I perform it a row/column operation on it to get a new matrix B, are A and B similar?

I wonder if the matrices are similar, and if this is true, then if I want to solve a minimal polynomial problem on a matrix A, if I can simplify the matrix A until it has the form of a diagonal matrix ...
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46 views

eigenvalues of given matrix $A$ and $B$

Let $$A=\begin{bmatrix} 0 & 1-i\\ -1-i & i \end{bmatrix} \qquad \text{ and }\qquad B=A^T\ \overline{A}.$$ Then (A) an eigenvalue of $B$ is purely imaginary (B) an eigenvalue of $A$ is zero ...
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28 views

Let $2 \times 2$ matrix $A=(2,-2,-2,5)$ Find a $2 \times 2$ matrix M such that $A=M^tM$

I was thinking to go this way: Determine if $A$ is positive definite, if each of the eigenvalues of $A$ are positive. Then we diagonalize $A$. $A=PDP^T$. Thus we can let $C$ be the diagonal matrix ...
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20 views

Need help to Compute a specific derivation

I have a function equal to: Where L is an orthonormal matrix of eigenvectors of a matrix S, and ...
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50 views

Let A be a symmetric positive definite matrix. Find a matrix B such that $B^2=A$

I believe this question is the same as asking find matrix B to be the square root of the matrix A. $B=\sqrt A$. Since the problem is not specific I am thinking to solve it in the general case by ...
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25 views

EigenVectors for Commuting Linear Operators

Prove that for any ( possibly infinite ) set of commuting linear operators on a finite Vector Space over $\mathbb{C}$ a) There Exists a common Eigen Vector b) There exists a basis in which the ...
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23 views

Prove that these are the eigenvalues of the Transformation

Consider the Linear Operator $f(x) \rightarrow f(ax+b)$ on the Space $\mathbb{R}[X]_n$ Show that the eigenvalues are $1,a,a^2,......,a^n$ There's a hint which says I should use the fact that ...
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20 views

Find width and height of a data set using eigenvectors?

I have a box represented by a set of points. I have found eigen vectors of those data set using principal component analysis. Is there a way I can measure the width(horizontal span) and ...
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1answer
49 views

Eigenvalues of $AB$ and $BA$ ${}\qquad{}$ [duplicate]

Are the eigenvalues of the matrices $AB$ and $BA$ identical? If yes, why? From the examples that I have tried I think they are identical but I just can't come up with a formal proof for this.
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7 views

Show the eigenvalue and eigenvector of a composed matrix

Could anyone help me with this question: if ${λ_i, v_i }, i=1, 2, . . . , n$ are the eigenpairs of the symmetric matrix A, show that the eigenpaires of $[I +P_k (A)A]^T A[I +P_k (A)A]$ are $λ_i ...
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1answer
27 views

Decomposition of inverse covariance matrix

If I have a covariance matrix $\Sigma$, and $x^T \Sigma^{-1} x = ||Ax||_2^2$. What exactly is the significance of $A$? I tried solving for $A$ with an eigenvalue decomposition of $\Sigma$ as ...
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20 views

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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25 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
17 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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26 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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28 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
35 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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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
24 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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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
28 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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18 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
33 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
48 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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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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25 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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22 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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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
20 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
35 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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15 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
31 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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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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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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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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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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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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42 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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$Rank(A)=$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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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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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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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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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. ...