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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Eigenvalues of a Hermitian matrix and a Herminitian form

Need some help and hints on how to prove this one: Let $F=\mathbb{R}$ or $\mathbb{C}$, and $_FV=M_{n,1}(F)$. Let $A \in M_n(F)$ be Hermitian (i.e $A^* = \bar{A}^T=A$) and $f(x,y)=x^*Ay$, for all $x,y ...
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Find Eigen values of given matrix with nonfactorable polynomial

I'm having trouble finding the Eigen values for this matrix: $$ A =\begin{pmatrix} 0&1&-2 \\ 1&3&0 \\ -2&0&5 \end{pmatrix} $$ I did $A - \lambda I $ and ended up with this ...
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Understanding a proof conceptually

Let's assume that $V$ and $W$ are vector spaces over a field $\mathbb{K}$, $\lambda\in\mathbb{K}$, $\lambda\neq0$. $S: V\rightarrow W$ and $T: W\rightarrow V$ are linear maps. Prove, that $\lambda$ ...
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Eigenvalue of a linear map (proof)

Let's assume that $V$ and $W$ are vector spaces over a field $\mathbb{K}$, $\lambda\in\mathbb{K}$, $\lambda\neq0$. $S: V\rightarrow W$ and $T: W\rightarrow V$ are linear maps. Prove, that $\lambda$ ...
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Eigenvalues of antisymetric matrix

Let $A \in \mathbb{R}^{n \times n}$ be a antisymetric matrix (that is $A^T = -A$. Then $A$ han a purely imaginary eigenvalues. How to prove it? I wanted to give that as a exercise, but I am not able ...
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$v^TAv\ge \left\|A\right\|\cdot\left\|v\right\|_2$ for all positive definite $A\in\mathbb{R}^{n\times n}$

Let $A\in\mathbb{R}^{n\times n}$ be positive definite and $v\in\mathbb{R}^n$. Let $\left\|\cdot\right\|_2$ be the Euclidean norm. Can we prove $$v^TAv\ge \left\|A\right\|\cdot\left\|v\right\|_2$$ for ...
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Matrix Differential Equations

I am working on a practice problem with the following equation: $$ \frac{d^3 x}{dt^3} + (k + 1)\frac{d^2x}{dt^2} + (k+1)\frac{dx}{dt} + kx = 0 $$ I understand the first part which is to convert to a ...
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26 views

What does does [H1,H2] = 0 mean if each H is a hamiltonian of a quantum system

What does does [H1,H2] = 0 mean if each H is a hamiltonian of a quantum system I'm trying to get through a research paper on theoretical quantum biology and I just want to make sure I'm interpreting ...
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Exercise 2.36 in 'Vector Calculus, Linear Algebra and Differential Forms' (Hubbard)

Let $A$ be an n by n diagonal matrix with diagonal entries $\lambda_1$ to $\lambda_n$, and suppose that one of the diagonal entries, say $\lambda_k$, satisfies $inf_{k\neq j}|\lambda_k - \lambda_j| ...
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Convergence of QR algorithm to upper triangular matrix

Sorry for asking really silly question. I guess the answer will be very simple. The question I am doing is: Does QR method always converge to a upper triangular matrix? I think the answer is not. ...
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Bounding cosine of angle between vectors

Let $M$ be a symmetric, positive definite matrix such that $0\lt c_1 \le \lambda_{min}(M)\le\lambda_{max}(M)\le c_2$. I am trying to show that $\dfrac{v^TMv}{||Mv||||v||}\gt 0$ for $v\ne 0$ I ...
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Reduction of matrix $A$ to $B$ to find eigenvalues by Power method [duplicate]

How to reduce matrix $A$ to $B$ such that it has all eigenvalues and eigenvectors of $A$ but the dominant eigenvalue (eigenvalue with largest magnitude) is replace by $0$ ? I am using Power method to ...
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Name for this possible mathematical structure? [on hold]

I'm thinking of a latent variable as an nth (each representing a variable) dimensional object - it can either be for a correlation matrix, or for a factor structure in factor analysis - that's ...
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How to reconstruct a symmetric matrix given the eigenvalues and eigenvectors.

I am trying to reconstruct a symmetric 3 x 3 matrix from just its eigenvalues and eigenvectors. I think the solution involves orthogonalizing two of the eigenvectors using the Gram-Schmidt ...
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15 views

finding corresponding eigen vector to eigen value for linear system

I have a question where I am trying to find the general solution of a linearised system, which I have linearised. I am just having difficulty obtaining the correct corresponding eigenvectors to my ...
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61 views

Real matrix without real eigenvalues commutes with some matrix of square $-I$.

Let $A\in M_n(\mathbb R)$ be such that, the minimal polynomial of $A$, has not any real root. Prove that there exist some $B\in M_n(\mathbb R)$ which: $B^2=-I_n$ and $AB=BA$. Suppose that ...
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60 views

Symmetric matrix eigenvalues

Let $A$ be an $n\times n$ matrix, with $A_{ij}=i+j$. Find the eigenvalues of $A$. A student that I tutored asked me this question, and beyond working out that there are 2 nonzero eigenvalues ...
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Behavior of components of a self adjoint linear transformation split over an orthogonal sum.

Let $V$ be a finite dimensional real vector space with an inner product $<\cdot,\cdot>$. Suppose $V=A\oplus B$ and where $A$ and $B$ are orthogonal subspaces. Let $T:V\rightarrow V$ be a self ...
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What is meant by eigen spaces are non-orthogonal?

$M$ is a square matrix $M$ ( matrix representation of a linear operator $L$ acting on a hilbert space $H$ , $L: H \to H$ ) with eigen values $\lambda_i$ and corresponding eigen spaces $V_i$. I know ...
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Origin of the word “Characterstic values” of a Matrix or a Linear Operator

I found the synonyms of the mathematical term "characteristic values" of a square matrix/ linear operator. One of them is "eigen value", which itself (that I learnt) derived from a German word ...
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Eigenvalue spectrum of backward Fokker-Planck operator

I encountered a paper in physics, in which the author states that an operator of the following form (backward Fokker-Planck) $\Lambda = P(x)\frac{d}{dx}+Q\frac{d^2}{dx^2}$ has an eigenvalue 0 and ...
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inverse rayleigh quotient iteration

For the inverse iteration we solve (A-uI)w = v(k-1). So if u is close to eigen value the A-uI is poorly ill conditioned. But why this apprant pitfall in inverse iteration causes no trouble. It would ...
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Prove if $A$ is a symmetric matrix with real entries, then the eigenvalues of $A$ are real.

Given a matrix $$A=\begin{bmatrix}a & b \\ b& c \end{bmatrix} $$ then let $\lambda = p+qi$ be a complex eigenvalue of $A$. I used the characteristic equation to get a factorizacion of ...
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How to find the other eigenvalues of a matrix when all rowsums are equal (where the rowsum provide the first eigenvalue)

If you have a situation where every row-sum is equal in a matrix A, this sum equals one of the eigenvalues of the matrix. Using this fact, is there any easy procedure/shortcut for finding the rest of ...
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62 views

Eigen value and Regular Graph (not Strongly Regular graph).

$A,B$ are 2 adjacency matrices of $d$ Regular graphs(not Strongly Regular graphs). I would like to know- 1.Results/ information related to Eigen values of A,B. There is a formula for ...
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Eigenvalues and eigenvectors for earthquake modeling

My instructor explicitly stated that, because we are asked to find eigenvalues and eigenvectors of a $7\times 7$ matrix, MATLAB would be easiest to use. The equation $(1)$ is intended to resemble ...
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1answer
25 views

Newtons Law of Cooling Differential Equations

We have two differential equations, $$\begin{cases} {dT\over dt} = -\alpha(T-B)\\ {dB\over dt} = -\beta(B-T)\end{cases}$$ If $T(0) = 7$ and $B(0) =3$, determine the equilibrium temperature of the ...
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Inhomogeneous eigenvalue problem, the shooting method and constraints

In trying to solve a problem occurring in QM calculations I've encountered the following pickle, with which I hope you could help me. I am trying to solve an inhomogeneous eigenvalue differential ...
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Proof if $A$ is normal then it is nondefective

What is the proof that if $A$ ($m\times m$ Matrix) is normal i.e $(AA^{\ast} = A^{\ast}A)$ then $A$ is non defective i.e (for each eigenvalue of $A$, its algebraic multiplicity is equal to the ...
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How to compute determinant (or eigenvalues) of this matrix?

Let us have the $n \times n$ circulant matrix given by \begin{equation} C(c_0,c_1,\cdots, c_{n-1}) =\begin{bmatrix} c_0 & c_1 & c_2 &\cdots & c_{n-1}\\ c_{n-1} & c_0 & c_1 ...
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65 views

what does it by raising a matrix to the power of $1/2$?

I came across the following which I did not understand at all. Let $A$ be a positive semi-definite. If $A(I-B)$ is positive definite, then the eigenvalues of $$A^{1/2}(I-B)A^{-1/2} = I ...
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number of **distinct** real roots of $f(x)=0$?

Let $a$ be a non zero real number. Define $$f(x) = \begin{vmatrix} x & a & a & a\\ a & x & a & a\\ a & a & x & a\\ a & a & ...
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Eigen value of principal submatrix.

I was studying "interlacing property" and trying to find out the below fact- $A$ is an adjacency matrix of a $r$ regular graph $G$. $u,v \in G $;$u,v$ are not similar vertices. $B$ is the ...
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Solve Ax=0 using Single Value Decomposition

Trying to solve Ax=o when $A=\begin{bmatrix}2&1&-1\\1&2&1\\ \end{bmatrix}$ using single value decomposition. I have the s,v,u and was thinking that x was as simple as $x=s*s^t$ but ...
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Eigenvalues and eigenvectors in $\mathbb{R}^\mathbb{N}$?

$\mathbb{R}^\mathbb{N}$ is the vector space of all real sequences $x=(x_n)_{n\in\mathbb{N}}$. The operations are defined in this manner: $(x_n)+(y_n):=(x_n+y_n)$ and $\lambda * (x_n):=(\lambda x_n)$. ...
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Invariant subspace (Proof)

How do I prove, that the eigenspaces of $T^n$ are invariant in regard to $T$, assuming T is an endomorphism in a real vector space V $(T: V\rightarrow V)$? That's how I started: Let $E_\lambda$ be ...
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Condition for largest eigenvalue being strictly larger than largest variance

Under which condition is the largest eigenvalue of a positive semi-definite matrix strictly larger than the largest of the matrix's diagonal entries?
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Is this matrix negative definite?

The questions if the following matrix is negative definite? $$ \mathbf{I}_N - \mathbf{X}(\mathbf{Z}'\mathbf{X})^{-1}\mathbf{Z}' - \mathbf{Z} (\mathbf{X}'\mathbf{Z})^{-1} \mathbf{X}' + \mathbf{Z} ...
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Shared eigenvectors between $A$ and $A^k$

$\newcommand\la{\lambda}$ Thanks to the spectral mapping theorem, we know that if $\la_1,\ldots,\la_n$ are the eigenvalues of a $n\times n$ complex matrix $A$, then $\la_1^k,\ldots,\la_n^k$ are the ...
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Is there a general way to determine the Laplacian of the eigenvalues of a real symmetric matrix?

I have a real symmetric $3\times3$ matrix $\mathbf{M}(\mathbf{r}$) which depends on $\mathbf{r} \in \mathbb{R}^3$. Each eigenvalue can be considered a scalar field $e_i(\mathbf{r})$ over ...
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Linear transformation of eigenspace is subset of eigenspace

Let $V$ be a vector space over a field $\mathbb{F}$ and let $L$, $M$ be two linear transformations from $V$ to itself. a. Show that the subset $W= {x ∈ V : L(x) = M(x)}$ is a subspace of $V$ b. ...
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show linear transformation bijective

Can you please help me prove this? Let $T:\mathbb{R}^7\to\mathbb{R}^7$ be a linear transformation such that 9 is an eigenvalue of $T$ and $dim(E_9)=6$ Prove that either T-4I or T-5I is a bijection ...
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1answer
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Different results while calculating eigenvectors with Gaussian elemination

Regarding this matrix $\begin{matrix} 1 & 1 \\ 1 &-1 \\ \end{matrix}$. In the end I have to solve this equation system: $(\sqrt2-1)x_1-x_2=0$ $-x_1+(\sqrt2+1)x_2=0$ While the ...
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25 views

Prove that for every $k$ there's an invariant subspace

Let $V$, a vector space above $\mathbb{C}$ and let $T:V\to V$, a linear transformation. Show that for every $0\le k \le n$ there is an invariant subspace of $T$ with a dimension $k$. It seems ...
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Eigenvectors of the matrix

I'm getting frustrated with a question. I'm trying to find the eigenvector of [[1/2,0], [0,2]] and it's been a few good years since I've had to touch eigenvectors. I get the characteristic polynomial ...
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Why is this the eigenvector?

For the eigenvector how are they getting \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix} when you have \begin{bmatrix} 0 & -1 & -1 \\ 0 & -1 & -3 \\ 0 & 0 & -2 \end{bmatrix} ...
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Matrix with eigenvalue that should equal 1.

I have the matrix: $$A = \begin{bmatrix}4 & -2 & 3\\0 & -1 & 3\\-1 & 2 & -2 \end{bmatrix}$$ and I need to find out if $\lambda = 1$ is an eigenvalue. So I solved the equation ...
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A real $2 \times 2 $ matrix $M$ such that $M^2 = \tiny \begin{pmatrix} -1&0 \\ 0&-1-\epsilon \\ \end{pmatrix}$ , then :

A real $2 \times 2 $ matrix $M$ such that $$M^2 = \begin{pmatrix} -1&0 \\ 0&-1-\epsilon \\ \end{pmatrix}$$ (a) exists for all $\epsilon > 0$. (b) does not exist for any ...
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Find a and b such that the matrix is diagonalizable

Find a and b such that the matrix $$ \left( \begin{array}{ccc} 1 & a \\ 0 & b \\ \end{array} \right) $$ is diagonalizable. I know that $$ D = S^{-1} A S $$ where S is a matrix made of the ...