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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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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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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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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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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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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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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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2answers
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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{\...
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1answer
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How can we decompose determinant of an upper triangular matrix? [on hold]

Let $T= \begin{bmatrix}A & B \\ 0 & C \end{bmatrix}$. How to show $\text{det}(T-\lambda I)=\text{det}(A-\lambda I) \text{det}{(C-\lambda I)}$ for square $A$ and $C$?
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Showing $| \lambda_i | < 1 \implies \operatorname{tr}(A) - 1 < \det(A) <1$

Prove that every eigenvalue of $A \in \mathbb{R}^{2 \times 2}$ has modulus less than 1 if and only if $$ \operatorname{tr}(A)-1 < \det(A) <1. $$ I can prove the forward direction, but not ...
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Quantum Mechanics - why should Probability of a degenerate eigenvalue be independent of the choice of an eigenvector in Euclidean space En?

Given $$|\psi_n> = \sum_{i=1}^{g_n}|u_{n}^{i}><u_{n}^{i}|\psi>$$ we get a projection $P_n|\psi>$. Incidentally, the square of this statement gives us the probability of finding a ...
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How do I solve the eigenvalue problem by matlab

My question is how to solve the eigenvalue equation $$(\frac{d^2}{dx^2}+\frac{d^2}{dy^2})u(x,y)+b\frac{d}{dx}u(x,y)+c\frac{d}{dy}u(x,y)+sin(x) cos(y)u(x,y)=\lambda u(x,y)$$ with periodic boundary ...
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Eigenpairs of a sparse symmetric block tridiagonal matrix with diagonal blocks on the outer diagonals

I'm looking at methods to find the eigenpairs of symmetric block tridiagonal matrices, with sparse blocks on the main diagonal and diagonal blocks on the outer diagonals. Has any research been done on ...
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0answers
37 views

Canonical Jordan form contradiction

I am faced with the following problem: Given endomorphism $f$ whose characteristic polynomial is $$P_c(x) = (x+1)^{10} (x-1)^{10} x^{10}$$ and whose minimal polynomial is $$P_m (x) = (x+1)^5 (x-1)^...
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Proof for a property of graph laplacian and its complement?

Here it says L(G) be the laplacian of an undirected graph. $L(G)+L(G^c)=n I_n-J_n$ Where $J_n$ is a matrix with all entries 1. Then how do we prove this: $\lambda_{n-i}(G^c)=n-\lambda_i(G)$
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What is the meaning of $\frac{(Ay,x)}{(y,x)}$?

$x^TAy$ is the inner product of a matrix A. If x,y are unit vectors, then what is the meaning of $\frac{x^TAy}{x^Ty}$? What does it do to the inner product? The orthogonal component of y wrt x is ...
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eigenvalues of a direct sum of matrices

According to Bacher's article, the eigenvalues of the adjacency matrix of Cayley graph of the symmetric group of order $n$ are $2-2\cos(\pi/n)$; my question is: If we know that this adjacency matrix ...
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(real symmetric matrices) does orthogonality of eigenvectors (distinct eigenvalues) depend on choice of basis?

For some reason I cannot wrap my head around this one. My schoolbook begins the chapter on eigendecomposition of real symmetric matrices by stating that eigenvectors from distinct eigenspaces are ...
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Invariant subspace SO(n)

Suppose $v\in\mathbb{C}$ is an eigenvector of $R\in SO(n)$ with nonreal eigenvalue $\lambda$. Let $V\subset \mathbb{R}^n$ be the two dimensional space spanned by $(v+\bar{v})/2$ and $(v+\bar{v})/(2i)$....
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Sturm-Liouville differential equation eigenvalue problem

If we have a Sturm-Liouville differential equation of the form $$ \frac{d}{dx}[p(x)\frac{dy}{dx}]+q(x)y=-\lambda w(x)y $$ and define the linear operator $L$ as $$L(u) = \frac{d}{dx}[p(x)\frac{du}{...
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Eigenvectors and invariant subspaces

If I have a 3 eigenvectors within $\mathbb{R}^3$ and I wanted 2 dimensional invariant subspaces of $\mathbb{R}^3$, could I just take the span of any 2 of of the 3 eigenvectors and they would give me ...
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$(A − \lambda_iI)$ is spanned by $e_1, …, e_{i-1}$

In a lecture, a proof of the Cayley-Hamilton theorem was preceded by the following lemma: Let $\mathbb{F}$ be a field with basis $e_1, ..., e_n$. Let $A \in M_n(\mathbb{F})$ be an upper triangular ...
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1answer
73 views

Eigendecomposition of a matrix (what assumptions need to be made)

For the sake of brevity, I'm given a 3x3 symmetric matrix with real entries with no further information as to what the rows and columns encode. (eg. this not need to be the case but the columns may ...
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Schur form of matrix with semisimple eigenvalue

I'm reading the paper The Canonical Schur Form of a Matrix with Simple Eigenvalues. In the very first equation, the author seems to assume that if $A$ has a semisimple eigenvalue $\lambda$ with ...
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1answer
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Given a scalar field, how to show that the eigenvalues of its Hessian are bounded?

Show that the eigenvalues of the Hessian of $$f(x_1,x_2) := x_1^2+x_1x_2+x_2^2+\ln (1+2e^{x_2})$$ are bounded, i.e., $$1 \leq \lambda_{\min} \left(\nabla^2 f(x)\right) \leq \lambda_{\max}\left(\nabla^...
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1answer
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Find a rank one non negative matrix $C$ such that the Matrix $B+C$ will have eigenvalues $13,2,-1$

I have been given the matrix $$B = \begin{pmatrix} 1 & 3 & 3 \\ 2 & 3 & 2\\ 2 & 1 & 4\end{pmatrix}$$ and I've been given that its eigenvalues are: $7, 2, -1$ Firsly I was ...
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A problem with eigenvectors of a Hermitian(?) Matrix

I have this matrix: $$A=\begin{bmatrix} 1 & -1 & 0 \\ -1 & 2 & -1 \\ 0 & -1 & 1 \end{bmatrix}$$ Eigenvalues are $\lambda_i=0,1,3$ ...
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Simplify the characteristic polynomial

I am taking my first math exam in a long time in linear algebra tomorrow. I have a feeling theres for sure going to be a question to find eigenvalues and their eigenvectors. I can do almost all of ...
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Inner products of of two vectors which are both orthogonal to a third vector.

If, in an $R^n$ space, $(x,y)=0$ $(x,z)=0$ Then what about $(z,y)$? What if $(z,y)\approx 1$ (although z,y are two different vectors with different elements), then what can we say about y and z? ...
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Why do you need to make sure the eigen vectors are orthonormal when trying to diagonalise a hermitian?

In an answer to this question I asked:Issue finding a unitary matrix which diagonalizes a Hermitian, it was said that without normalising my eigen vectors I wouldn't get the corresponding diagonal ...
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1answer
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What is the intuition behind having upperbound on eigenvalues of Hessian?

Suppose $f: \mathbb{R}^n \rightarrow \mathbb{R}$ and is a $C^1$, i.e., $\nabla f $ is a continuous vector valued function. Show if the $\lambda_{max}$ of $\nabla^2f$ is bounded, then $\nabla f$ is ...
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1answer
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Determining eigenvalues of sum of 2 matrices, and then evaluating whether the limit exists

I'm studying for an exam on Tuesday and have been stumped on this question for a little while. Any hints or help at all would be appreciate! I'm given 2 matrices, $A$ and $R$ as shown below. I'm also ...
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2answers
53 views

Show that the largest eigenvalue of $A$ lies in the given interval

Show that if the given matrix $A$ is positive semi-definite then the largest eigen value of $A$ lies in the interval $(6,7)$. $$A=\begin{bmatrix} 5&1&1&1&1&1\\ 1&2&1&...
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Is the condition number of a 2x2 block symmetric matrix greater than the condition number of its upper left hand block?

Is there any known relation between cond(M) and cond(Q) when $$M=\begin{bmatrix}Q&A^T\\A&0\end{bmatrix}$$ and Q is symmetric positive definite and A is rectangular full row rank? From the ...
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1answer
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Does the inner product of a matrix, $\frac{x^TAy}{x^Ty}$ stay the same or fall in a range for any x,y?

Is there any bound on $\frac{x^TAy}{x^Ty}$, for any vector $x$? I am observing that $\frac{x^TAy}{x^Ty}$ is approximately the same even when I change $x$. Why is that? Is there any property for the ...
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If $\frac{x^TAy}{x^Ty}\approx\frac{z^TAy}{z^Ty}$ then what can we say about x and z?

If $\frac{x^TAy}{x^Ty}\approx\frac{z^TAy}{z^Ty}\approx\frac{y^TAy}{y^Ty}$, where x is the eigenvector of A. Then what is the relationship between z and x? P.S.: In my case, $M=AB$ and $y$ is the ...
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Solutions of $A^2+A^t=I_n$ in $M_n(\mathbb{R})$

I would like to prove that, if $A\in M_n(\mathbb{R})$ satisfies $$A^2+A^t=I_n$$ then neither 0 nor 1 can be in $Sp(A)$. Such an $A$ satisfies $A(A-I)(A^2+A-I_n)=0$, hence $Sp(A)\subset \{0;1;a;b\}$, ...