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Questions tagged [eigenvalues-eigenvectors]

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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Principal components and loadings

Suppose we have a data matrix $X$ that contains $n$ observations with $p$ entries. Let $S$ be its covariance sample matrix. By eigendecomposition, $S=VDV^{T}$, where $V$ is called a loading matrix. ...
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proof that for eigenvalues of matrix A, (Coefficient of $\lambda^{N-1}$) = - Tr(A)

I am having a bit of trouble with this proof regarding the eigenvalues. My textbook merely stated it but didn't really provide the proof: Prove that for a matrix $A$ with eigenvalues $\lambda_{i}$: $$...
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when singular value decomposition is equal to eigenvalue decomposition

I've read in my textbook that the right singular vector $v_i$ is actually the eigenvector of $A^TA$ with eigenvalue $\sigma_i^2$, and the left one $u_i$ is the eigenvector of $AA^T$.So I guess if A is ...
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Insight on the polar decomposition of a shear?

I recently learned it, and really love the polar decomposition of a matrix, because it was the first time I actually could picture what it meant to "apply a transformation to space" (a phrase I kept ...
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Geometry induced through the characteristic polynomial for an eigenvalue optimisation problem

Suppose I solve the generalised eigenvalue problem: $(K+ \Delta K)v_i = \lambda_i(M+\Delta M)v_i, \quad v_i\in\mathbb{R}^n$ where $\Delta K = diag(a_0, ..., a_n), \Delta M = diag(a_{n+1}, ..., a_{...
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Relationship between singular values of $A$ and eigenvalues of $B:= \begin{bmatrix} 0 & A \\ A^\ast & 0 \end{bmatrix}$

Let $$B:= \begin{bmatrix} O_m & A \\ A^\ast & O_n \end{bmatrix}$$ where $A$ is an $m \times n$ matrix. Find the relationship between the two: Singular values and singular vectors of $A$...
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Eigenvalues of A+B are not of the form an eigenvalue of A + an eigenvalue of $B$

Let $A, B\in M_n (\mathbb{C})$ s.t. $AB=BA$. I know that the eigenvalues of $A+B$ are not of the form an eigenvalue of $ A$$+$ an eigenvalue of $B$, but I need an example.
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Check diagonalizability through the characteristic polynomial

Basically what is given is the characteristic polynomial of A $$p(x)=(2-x)^2$$ and I am asked if the matrix A is diagonalizable. I have already realized that $x=2$ is a root (eigenvalue) with ...
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Eigenvalues of rank 2 perturbation of the identity

Let $I$ be the identity matrix, and $u, v\in\mathbb{R}^n$. What can we say about the eigenvalues of the following matrix? $$2I+vu^{\mathsf{T}}+uv^{\mathsf{T}}$$ I'm particularly interested in the ...
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Given an SPD matrix, any diagonal submatrix of full rank must be SPD.

I need help with the following proof: Given a symmetric positive-definite matrix, show that any diagonal submatrix of full rank must also be symmetric positive-definite. Thanks
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Getting an eigenvector from a particular matrix using an RREF algorithm

Using Wolfram's software, I have been debugging some code that returns the eigenvalues of a matrix and the eigenvector corresponding to the largest of them. My code works by using a QR algorithm (...
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Proving that a specific Volterra integral operator is not positive

I want to prove that the operator $$ A: L^2[0,1] \to L^2[0,1], \quad A(u)(s) = \int_0^1 |t-s| u(t) dt $$ is not positive, i.e. $\langle Au, u \rangle \geq 0$ does not hold for every $u \in L^2[0,1]$. ...
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At least one eigenvalue among all roots

Let $V$ be a vector space and $f \in \text{End}V$. Let $p$ be a polynomial over a field $K$ so that $p(f)=0$. Also we have deg $p = m$ and $c_1,c_2,\dots,c_m$ be all roots of $p$. Prove that at least ...
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Sum of sqrt of eigenvalues without computing all eigenvalues

Let $A$ be a positive-definite matrix with eigenvalues $e_1, ..., e_n$. I want to compute $\sum\limits_{i=1}^{n} \sqrt{e_i}$ without calculating all eigenvalues first (or rather: with a method faster ...
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Is this simple symmetric matrix positive semi-definite?

Let the $n\times n$ symmetric matrix $A$, where $n\geq 9$ be given by \begin{equation} A_{i,j}= \begin{cases} 1.4, &\text{for } 1\leq i=j\leq 9\\ (0.9)^{|i-j|},&\text{for } 1\leq i\neq j\leq 9\...
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Show that a linear map on a finite dimensional complex vector space always have an eigenvalue.

What is an alternative proof that a linear map $T$ on a finite dimensional complex vector space $V$ with dimension $n$ always has an eigenvalue? Here is the original proof idea: We take a no zero ...
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Eigenvalues and eigenvectors for the moment of inertia matrix

Find the eigenvalues and eigenvectors for the moment of inertia matrix given by $$I={m\over 2}\left(\begin{matrix} 1 & -1 & 0\\ -1 & 1 & 0\\ 0 & 0 & 2\end{matrix}\right)$$ ...
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Eigenvector of two rotation matrices

I am having difficulty in understanding a geometry problem which contains geometric-transformation, rotation and reflection. Background In this image, a camera with camera center $O_c$ is presented. ...
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Eigendecomposition proof

Let $X$ be an $(n \times n)$ matrix. Let $V$ be the $(n \times n-k)$ be the matrix of eigenvectors of $X$ which correspond to non-zero eigenvalues of $X$. Let $E$ be the $(n-k \times n-k)$ diagonal ...
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Eigenvalues and eigenvectors of $A_{1}$ and $ A_{2}=A_{1}^{T}$

We have a positive integer $n$ and two $n\times n$ matrices of real numbers, $A_{1}$ and $A_{2}$. For $j=1, 2$, we have the eigenvalues and eigenvectors $\lambda _{j}$ and $x_{j}$ of $A_{j}$. Show ...
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Characteristic polynomial and eigenvector of Frobenius matrix

Consider the following $n \times n$ matrix (I believe this is similar to companion matrix): $$ A = \begin{pmatrix} 0 & -1 & 0 & \cdots & 0 & 0 & 0 \\ 0 & 0 & -...
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Matrix perturbation and eigenvector

$Ae=\lambda e$ $e^Te=1$ $A$ is a real matrix. $\lambda, e$ are real. $(A+\Delta A)(e+\Delta e)=(\lambda+\Delta \lambda)(e+\Delta e)$ Neglecting small terms, $\Delta Ae +A \Delta e=\Delta \lambda ...
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Why are singular values of “complex” matrices always real and non-negative?

I've already read the following related questions on math.SE: Why can't singular values be complex numbers? Clarification on the SVD of a complex matrix Why are singular values always non-negative? ...
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What can be said about the definiteness of the following inequality?

Given a Hurwitz matrix $R\in\Re^n$ which has all the eigenvalues located in the closed left-half plane. For a positive-definite matrix $Q$, we know that there exists a unique solution $P$ to the ...
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QR-algorithm complexity on a symmetric tridiagonal matrix

Why does the QR algorithm (for calculating eigenvalues) only require O(m) calculations per step when performed on a symmetric tridiagonal matrix?
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How to prove that a $3\times 3$ matrix has only $2$ eigenvectors?

I am working through a problem in Riley, Hobson and Bence (Mathematical Methods for Physics and Engineering) that revolves around the following matrix: $$ A= \begin{pmatrix} 2 & 0 & 0 ...
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Way of predicting change in gradient of line following matrix transformation?

I recently learnt about dominant and repulsive eigenvectors. I noticed that the farther a line is initially from the dominant (though still closer than to the repulsive eigenvector), the more dramatic ...
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1answer
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Let $V$ be a real vector space and $E$ be an idempotent linear operator on $V$. Prove that $I + E$ is invertible.

Let $V$ be a real vector space and $E$ be an idempotent linear operator on $V$, that is a projection. Prove that $I + E$ is invertible. Find $(I + E) ^{-1}$ My teacher taught me the following proof ...
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How to project a vector $x$ onto the eigen-space, i.e. $x = \sum_{i=1}^{n} \langle w_i, x \rangle v_i$?

Assume $A$ is a complex-valued square matrix, i.e. $A\in \mathbb{C}^{n\times n}$, and $A$ has a full set of eigenvectors denoted as $V=[v_1, v_2, \cdots, v_n]$. Then we known the following facts \...
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Matrix Normalization

From the eigenvectors matrix: I did normalization but I think there's an error I could not find.
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Matrix Orthogonality

I have the eigenvector matrix like this $\begin{bmatrix} 1 & 1 & 1 \\ 0 & \frac{-b + \sqrt{(b^{2} + 8a^2)}}{2a} & -\frac{b + \sqrt{(b^2 + 8a^2)}}{2a} \\ -1 & 1 & 1 \end{...
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If $P(x)$ is characteristic polynomial of $A$ then is $P(A) = 0$?

I'm a student and I've just read the Characteristic polynomial on Wiki. I have a feeling that: If $P(x)$ is characteristic polynomial of $A$ then is $P(A) = 0$ thanks to the Matrix calculator I've ...
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1answer
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Eigenvalues decrease with power

Take $n \in \mathbb{N}$, and consider a square matrix $A$ of size $n \times n$, with real and positive entries, and such that $\|A\|_2 \leq 1$. I think the following statement holds from simulation, ...
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Can the eigenvectors of a linear operator in an infinite-dimensional space span the space and be linearly dependent at the same time?

Consider a vector space $V$ over the complex field which is infinite-dimensional with a Euclidean inner-product. Let $L$ be a linear operator on $V$. Say a subset of eigenvectors of $L$ forms a ...
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Relation between leading eigenvlaue and eigenvector of individual blocks with the leading eigenvector of non-negative symmetric block matrices

For a 2 by 2 block non-negative symmetric matrix $\mathcal{M}$, $\mathcal{M}= \left[ \begin{array}{c|c} \mathcal{A} & \mathcal{C} \\ \hline \mathcal{C}^T & \mathcal{B} \end{array} \right]$ ...
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Calculate the eigenvalues and eigenvectors of 2 x 2 matrix

Its $2 \times 2$ matrix and having square-root value
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1answer
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Can an orthogonal non-symmetric 3x3 matrix have 3 real eigenvalues?

I was wondering if a non-symmetric orthogonal matrix can have his 3 eigenvalues in the real numbers. All the 3 real eigenvalues orthogonal matrix i've found are symmetric. Can someone give me a ...
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Find eigen vectors of a “subsetted” matrix

I have a symmetric matrix, $A$, with dimension 100 x 100, of which I know the eigen vectors and eigen values ($A = U'VU$ ). Now I want to know the eigen vectors and eigen values of $B$, which is just ...
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Spectral radius of a hollow symmetric block matrix

Let $B$ be a $2 \times 2$ matrix. Suppose we have a $4 \times 4$ real symmetric matrix of the following form $$A = \begin{bmatrix} O_2 & B \\ B^T & O_2\\ \end{bmatrix}$$...
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1answer
31 views

How to find the eigen values of the following matrix:

Is there any way to find the eigen values of the following matrix: $A_{2n\times 2n}=$ \begin{bmatrix}\textbf{0} & E_{n\times n}\\E^T&\textbf{0}\end{bmatrix} where $E=$ \begin{...
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1answer
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Eigenvalue multiplication endomorphism

When considering the multplication endomorphism \begin{equation*} \begin{split} [\times z]_{K/Q}: & \:\: K \rightarrow K \\ & \:\: x \:\mapsto xz \end{split} \end{equation*} for $Q$ a field ...
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Find first eigenvector of Hadamard division of $AA^T$ and $BB^T$ using power method

For an $m \times n$ matrix $A$, it is possible using the power method to find the eigenvector corresponding to the largest eigenvalue of $AA^T$ by factoring it into a matrix vector product $(AA^T)v = ...
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Show that the characteristic polynomial is the same as the minimal polynomial

Let $$A =\begin{pmatrix}0 & 0 & c \\1 & 0 & b \\ 0& 1 & a\end{pmatrix}$$ Show that the characteristic and minimal polynomials of $A$ are the same. I have already computated ...
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Proving a matrix iteration converges

Consider the sequence $x_{n + 1} = Mx_{n} + b$. Suppose the matrix $M$ is symmetric and for any $x \neq 0$, $$-1 < \frac{x^{T}Mx}{x^{T}x} < 1$$ holds. Prove that $x_{k}$ converges. ...
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Maximize the smallest nonzero singular value

I want to maximize the smallest nonzero singular value of (non-square) matrix $X$. This is equivalent to maximizing $\lambda_{\min}(X^\top X)$, which can be reformulated as follows $$\begin{array}{ll}...
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How to create matrix with high ratio of first two eigenvectors and equal row sums

To create a matrix where the ratio between the first two eigenvectors $\frac{\lambda_1}{\lambda_2}$ is large, I can set a matrix $A = \lambda_1 u u^T + \lambda_2 v v^T$ with orthonormal $u,v$ and the $...
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1answer
37 views

eigenvalue of a graph

What does the eigenvalue of a graph mean? I know how to compute the eigenvalues from the adjacency matrix representation of a graph but am interested in its physical significance. If two graphs have ...
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How to measure repulsion between numbers

How is repulsion measured between two eigenvalues or any two numbers for that matter? Assume that repulsion is $1,$ ($100$ percent) when the two numbers have zero space between them, and repulsion is $...
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Eigenvalues of $Q=I+2P$

I have tried to do it evaluated option (a). I think it is correct.Can not get the other options.Please help me.
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1answer
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How to show for a normal matrix $A$, $(A,\lambda,x) \Leftrightarrow (A^*,\bar{\lambda},x)$?

A matrix $A$ is normal if $AA^*=A^*A$. Suppose $(\lambda,x)$ is an eigenpair of $A$, i.e., $Ax = \lambda x$. Proof for a normal matrix $A$, $(\lambda,x)$ is an eigenpair of $A$ if and only if $(\bar{\...