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 get a basis for an eigenspace

I don't see how the book I'm using gets the following eigenvectors. I got two vectors for the basis while the book only got one for each eigenvalue. Can anyone explain how? I got $x = x_2[0;1;0] + ...
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Forcing Eigenvector elments to zero

I have a large sparse eigenvalue problem of the form: $A\mathbf{x}=\lambda\mathbf{x}$. The problem resembles an electromagnetic problem. Is there a general way of manipulating the system matrix to ...
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2answers
30 views

Basis of a 3x3 eigenspace

I'm currently in the middle of a question where I'm given a 3x3 matrix: $$\left(\begin{array}{rrr} 3 & 0 & 0\\ -2 & 7 & 0\\ 4 & 8 & 1 \end{array}\right).$$ and have been ...
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0answers
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Connection between power iterations and QR Algorithm

I am seeking an intuitive understanding of why the QR Algorithm solves the symmetric eigenvalue problem. In class, and also in Golub and Van Loan, it has been suggested that there is somehow deep ...
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1answer
32 views

How to find out if some eigenvalues of a matrix are the same?

I know that in order for a matrix to have two equal eigenvalues, one term in the characteristic polynomial must be in the power of two. Is there any way to tell if two eigenvalues are the same? I have ...
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number of zero eigenvalues of cyclic tridiagonal matrix

Prove that following cyclic tridiagonal matrix has two zero eigenvalues for $k=6r$ for any positive integer $r$. \begin{equation*} T_\lambda=\begin{bmatrix} -n_1 & n_2 & 0 &.&.&0 ...
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34 views

Are the only 2x2 real matrices with complex-conjugate eigenvalues the rotation matrices?

If so, how can I see this fact? I'm wondering if it's something fundamental that I am overlooking. Thanks,
-1
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1answer
47 views

Trouble to obtain eigenvectors of a matrix knowing its eigenvalues

The problem: Being given the matrix: $$ \begin{bmatrix} 0 & -1 & -1 \\ 1 & 2 & 1 \\ -1 & -1 & 0 \end{bmatrix}$$ and two of its eigenvalues ...
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1answer
31 views

eigenvaue of Sturm Liouville problem

Let the limit probem $$ \begin{cases} (P(x) y')' + q(x) y' + \lambda r(x) y=0\\ \alpha_0 y(0)+ \alpha_1 y'(0)\\ \beta_0 y(l) + \beta_1 y'(l) \end{cases} $$ with $\alpha_0^2 + \alpha_1^2 >0$ and ...
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0answers
16 views

Restriction of diagonalizable endomorphism to an invariant subspace is diagonalizable - another approach

There are some questions discussing the diagonalizability of a restriction of a diagonalizable endomorphism to an invariant subspace, however, I have a question regarding a certain approach, which ...
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1answer
42 views

Find the eigenvalues and eigenvectors of the following square matrix

Given the following matrix: $\begin{bmatrix}2 & 1\\2 &3\end{bmatrix}$ Find the Eigenvalues and Eigenvectors. So I found the Eigenvalues to be 2 and 3. Then I plugged in 2 to find the ...
7
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67 views

Find the eigenvalues of a 3 x 3 matrix

I have a question on determining eigenvalues for a given matrix A: $$ A= \begin{bmatrix} 2 & 1 & 2 \\ 0 & 2 & -1 \\ 0 & 1 & 0 \\ \end{bmatrix} $$ Here's what I have so ...
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8 views

Eigenvalue of Integral Operator and Gamma Function

$''$ Prove that the following integral operator $ Ku(x) = \int_{0}^{ \infty } \ e ^{-xy} u(y) dy $ has as eigenvalue the $ φ_α(x) = \sqrt {Γ(α)} x^{-α} + \sqrt {Γ(1-α)} x^{α-1} $ for $ ...
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0answers
15 views

Prove that the eigenvalues of a skew-symmetric matrix are purely imaginary [duplicate]

Proof idea: $A$ is a skew symmetric matrix. $A$ is similar to $A^t$ because every matrix is similar to it's transpose. $$A^t = -A $$ $A$ is similar to $-A$. Let $P_{(\lambda)}$ be the characteristic ...
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1answer
18 views

Range of linear operator restricted in generalized eigenspace

Let $T:V\to V$ be a linear operator on a finite-dimensional complex vector space $V$. Let $\lambda_1,\lambda_2,...,\lambda_k$ be the distinct eigenvalues of $T$ and $K_{\lambda_i}$ be the generalized ...
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Computing the covariance matrix from eigenvalues computed by L1-PCA

L1-PCA are PCA solutions that try to optimise the L1 distance. Generally by maximization of the L1-norm of the projected points. It is used for its robustness to outliers. And the output is a base of ...
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2answers
33 views

A matrix with the rank of 1 has only rational numbers. Are all the eigenvalues necessarily rational?

I know that all of the eigenvalues except for one are $0$. Can the last eigenvalue be non-rational?
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39 views
+100

How to prove this result about the interlacing of eigenvalues.

Let $A$ be a real symmetric matrix of order $n$ with eigenvalues $\mu_1\ge \mu_2,\ldots, \mu_{n}$. $B=\begin{bmatrix} x&x&x&x&x \end{bmatrix}$, where $x$ is non zero column vector in ...
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Spectrum of Linear Operator

The spectrum of a linear operator on a finite dimensional space is pure point spectrum, that is, both continuous and residual spectrums are empty. Or we can say that on a finite dimensional space, ...
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1answer
29 views
+50

Block matrix of order $m$ with three block matrices

How to find eigenvalues of following block matrices? $M=\begin{bmatrix} A & B & O & O & O & O & O & \cdots & O & O\\ B & A & B & O & O & O ...
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$T:R^3 \rightarrow R^3$. Show that there is a line L such that $T(L) =L$.

Let $T:R^3 \rightarrow R^3$ be a linear transformation. Show that there is a line L such that $T(L) =L$. My Idea: So the characteristic equation would be a cubic polynomial hence there is at least ...
2
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1answer
34 views

Find $A^{20}x$ using eigenvectors and eigenvalues.

Find $A^{20}x$ A is a 3X3 matrix with the following eigenvectors and eigenvalues: $V_1 = [1, 0, 0]... V_2 = [1, 1, 0]... V_3 = [1, 1, 1]$ corresponding to Eigenvalues.. $\lambda1 = -1/3, \lambda2 = ...
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Bessel-type singularity

The question I have to answer is \begin{align} -y''+\frac{1}{x^2}y&=\lambda y\quad x\in(0,1)\\ y(1)&=0\\ \lim_{x\to0}y(x)&=0 \end{align} Find the indicial equation and show ...
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1answer
32 views

Find the eigenvectors of a hermitian matrix as a function of angles

I have a problem which seems flawed to me. Consider the hermitian matrix M \begin{pmatrix} a/2 & b^* \\ b & -a/2 \end{pmatrix} with a>0 real and b complex. Let $\theta$,$\phi$ ...
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What are the eigenvalues of the following Hermitian matrix?

Let $\mathtt{i}=\sqrt{-1}$ and $p=1+\mathtt{i},q=1-\mathtt{i}$. Let $A$ be an $n\times n$ matrix such that $$A=\begin{bmatrix} 0 & p & p & \cdots & p & \color{blue}{q}\\ q & 0 ...
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0answers
21 views

Maximization of quadratic form on a sphere [duplicate]

I have to following problem $$\max_{x}x^TAx+b^Tx\quad \mathrm{s.t.}\quad x^Tx\leq c,$$ where $A$ is real, symmetric and positive semi-definite. Firstly I tried to solve the problem with the KKT, but ...
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1answer
14 views

Do automorphisms of infinite-dimensional vector spaces over algebraically closed fields always have eigenvalues?

Let $V$ be a vector space over an algebraically closed field $K$ and let $f:V\to V$ be an automorphism, i.e. a bijective endomorphism. If $V$ is finite-dimensional, we know that the characteristic ...
3
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1answer
62 views
+50

Eigenvalues of block matrix of order $m+1$

How to find eigenvalues of following matrix? $\begin{bmatrix} mkI-A & -A & -A & \cdots & -A\\ -A & kI-A & O & \cdots & O\\ -A & O & kI-A & \cdots & O\\ ...
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Do endomorphisms of infinite-dimensional vector spaces over algebraically closed fields always have eigenvalues?

Let $V$ be a vector space over an algebraically closed field $K$ and let $f:V\to V$ be an endomorphism. If $V$ is finite-dimensional, we know that the characteristic polynomial $\chi_f$ has a zero ...
2
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1answer
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Prove that for any $A \neq 0$ there is a matrix $B$ such that $A + B$ and $B$ have no eigenvalues in common.

I am not sure how to begin thinking about this problem. Could anyone provide a push in the right direction, or an explanation of what are the first things that go through their head when solving ...
2
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1answer
13 views

Showing squared sum of eigenvalues is an integer

I've been trying to improve my Linear algebra skills lately and I've run into this problem without not knowing where to start. Any suggestions/answers? Suppose a $4\times 4$ matrix of integers has ...
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1answer
19 views

Eigenvalues on the diagonal of a Hermitian(self-adjoint) matrix.

I appreciate any help that can be given on this, I just can't seem to get started. Let $A$ be an $n\times n$ Hermitian matrix with eigenvalues $\lambda_1 \leq \lambda_2\leq \cdots \leq \lambda_n$. ...
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Eigenvalues of a tridiagonal block matrix

When a tridiagonal matrix is also Toeplitz, there is a simple closed-form solution for its eigenvalues, namely $ a + 2 \sqrt{bc} \, \cos(k \pi / {(n+1)})$ , for $ k=1,...,n$. Now my question is that ...
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35 views

Characteristic polynomial of matrix

If I wanted to find the eigenvalues of a matrix $\mathbf{A}$, then I could use these two options. $$\lambda\mathbf{I}-\mathbf{A}=\mathbf{0}$$ $$\mathbf{A}-\lambda\mathbf{I}=\mathbf{0}$$ However, ...
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What is the eigenvalue and eigenvector for $T$ on $U$?

Assume that $U$ is a subspace of the space of infinitely differential-able (complex valued) functions of real numbers that is spanned by $f_1=e^t \cos(t)$ and $f_2=e^t \sin(t)$. What is the ...
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Symmetric block matrix related

How to find eigenvalues of following symmetric matrix $\begin{bmatrix} kI-A & -A & -A & \cdots & -A\\ -A & kI-A & -A & \cdots & -A\\ -A & -A & kI-A & ...
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What is relation between these two determinants?

Let $x\ne 0$ and $A$ be a square matrix of order $n$ and $$ B=\operatorname{diag}\begin{bmatrix} x & x & \ldots&x&x&x&x+\frac{2}{x} \end{bmatrix}$$ $$ ...
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1answer
64 views

Show that 1 + $\lambda$ is an eigenvalue of $I + A$

Show that if $\lambda$ is an eigenvalue of $A$, then 1+$\lambda$ is an eigenvalue of $I+A$. What is the corresponding eigenvector? What I have done so far (if it is correct at all...): ...
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1answer
36 views

The rank of a positive square matrix $A$ for $A^k = A$

Determine the rank of $A$ if $A$ is a positive $n\times n$ matrix with $A^2 = A$. Give a geometric interpretation of $A$. What happens in the case when $A^k = A$ for some integer $k\geq 3$? ...
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1answer
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Eigenvalues of block matrix where blocks are related [closed]

How to find eigenvalues of following block matrix $A$ in terms of eigenvalues of matrix $B$? $A=\begin{bmatrix} 4I-B & -B \\ -B & 2I \\ \end{bmatrix}$ Where $B$ is square matrix of order ...
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How to determine if a homogeneous integral equation has non-trivial solutions?

The equation is $f(p) = \int_{0}^{\Lambda} dk \; \mathcal{M}(k,p;E) \, f(k)$, where the kernel $\mathcal{M}$ is $\mathcal{M}(k,p;E) = \frac{2/ m\pi}{1 - \sqrt{E + 3p^2/4}} \frac{k}{p} \log\left( ...
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How to find the characteristic polynomial for the following graph G

What is the closed form of characteristic polynomial (adjacency matrix) for the following graph $G$: With the help of eigenvectors, I found that $4$ eigenvalues of $G$ are that of $P_4$ and $6$ ...
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how to interpret each variables's contribution to eigenvectors

Table of eigenvectors-values In the image attached you will find the eigenpairs i gt from one of the covariance matrix. The last row unattached to the table is the eigenvalues corresponding to each ...
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39 views

Solving generalized determinant related

How to solve following determinant by applying suitable elementary row/column transformations to obtain characteristic polynomial? \begin{align*} \left\vert \begin{matrix} -\lambda & 0 & 1 ...
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SVD - Decomposed Matrix Sizes

I had a question about SVD. Specifically about the size of matrices $U$, $\Sigma$ and $V$ decomposed from the $m\times n$ matrix $X$ using the formula $$X = U \Sigma V^T $$ Most of the the tutorial ...
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1answer
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Eigenvalues and Eigenvectors relating to orthogonal basis and diagonal matrices

Find the eigenvalues and eigenvectors of the matrix. $$A = \begin{bmatrix} 1 & 1 & 0 \\ 1 & 0 & -1\\ 0 & -1 & 1 \end{bmatrix}$$ As we have seen in the ...
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2answers
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How to find another eigenvector when I have same eigenvalues?

As in the topic. Let's say I have three eigenvalues, but two of them are equal. For example for $\lambda_1$ I got $v_1$ and how do I find another (I don't know how to write it in English) not ...
3
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1answer
39 views

Quantum Mechanics - Eigenvalue and Eigenvector of a Matrix

I'm attempting to find the eigenvalues and eigenvectors from the following matrix : \begin{pmatrix} -3\cos\theta&\sqrt{3}\sin\theta e^{iφ}&0&0\\\sqrt{3}\sin\theta ...
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Problem with Spectral Theorem Proof

Claim: Let A $\in \mathbb{R}^{n \times n }$ be symmetric. Then there is an orthonormal basis of $\mathbb{R}^n$ consisting of eigenvectors of A. Sketch of proof: Induction on n. Claim is clear for ...
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Hermitian Matrix and nondecreasing eigenvalues

I am studying for finals and looking at old exams. I found this question and am not sure how to proceed. Let $A$ be an $n\times n$ Hermitian matrix with eigenvalues $\lambda_1 \leq \lambda_2 \leq ...