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 positive linear combination of p.d. matrices

I want to prove a property on the eigenvalues of a positive linear combination of p.d. matrices. I have the following: $$ z \in \mathbb R^m_{++} $$ $$ A(z) = \Sigma z_i A_i $$ $$A_i \in S^n_{++} ...
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3 views

Lower bound on the sum of singular values for a sum of Hermitian matrices

Suppose $\mathbf{A}$ is a Hermitian $n\times n$ matrix with eigenvalues $\lambda_i(\mathbf{A})$, $i=1,\ldots,n$. Suppose $\mathbf{B}$ is an $n \times n$ complex-valued matrix and $b\neq 0$ is a ...
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11 views

Eigenvalue interpretation with direction field

I am running into some trouble with respect to some direction field plots of different eigenvalues. I am working with a system given as follows: $$ \overrightarrow{y'} = \begin{pmatrix} 0 & -5 ...
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12 views

Proving the basis of an eigenspace is not the same for a matrix and its transpose

With the information given in #18, prove that $A$ and $A^T$ need not have the same eigenspaces (I would use a 2x2 matrix, as #18 posits). Clarification: DO NOT solve #18. Using the information in ...
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1answer
34 views

Eigenvalues of $MA$ versus eigenvalues of $A$ for orthogonal projection $M$

Suppose that $M$ is symmetric idempotent $n\times n$ and has rank $n-k$. Suppose that $A$ is $n\times n$ and positive definite. Let $0<\nu_1\leq\nu_2\leq\ldots\nu_{n-k}$ be the nonzero ...
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4 views

Stability criterion for eigenvalues of an AR(2) process.

This is pretty much a question on linear algebra stemming from time series analysis. Essentially I want to find a stationarity criterion for an AR(2) process. It is easy to reduce this to the ...
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1answer
33 views

Does $A-\lambda I$ have rank smaller than $A$?

Consider $\lambda$ as one eigenvalue of $A$, can we say that $A-\lambda I$ must have rank smaller than $A$? Or equivalently, $A-\lambda I$ spans a space which is a subset of $A$?
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1answer
16 views

Multiple eigenvalue solutions problem

In a problem regarding multiple eigenvalue solutions (defective eigenvalues, complete eigenvalues, the like) I have a 4x4 matrix with one complete eigenvalue, and another incomplete eigenvalue with a ...
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0answers
22 views

Find only the real eigenvalues of a matrix.

If a matrix has many (thousands) complex and few (dozen) real eigenvalues is there a fast method for estimating only the real eigenvalues ?
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11 views

Perturbation theory for a symetric rank-one update

I know perturbation theory of the eigenspectrum/singular value decompostion of a symetric matrix $A$ under a symetric perturbation $E$, that besides being symetric has no other structure. Is there ...
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48 views

Eigenvalue problem for $−\psi''(x) − (ix)^ N \psi(x) = E\psi(x)$ in complex plane

To find the eigenvalue $E$ in the complex plane of $x$ for one dimensional Schrodinger equation $$ −\psi''(x) − (ix)^ N \psi(x) = E\psi(x). $$ where $N$ can be any real number, the boundary condition ...
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8 views

Bounding the perturbation between eigenvectors

Can somebody explain this part of the proof of a deduction from the Davis-Kahan $\sin \theta$ theorem? I understand how to get from: $||P_{u_1} - P_{v_1}|| \le \epsilon$ to $||P_{u_1}v_1 - v_1|| \le ...
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1answer
38 views

How to write this down in a rigorous manner?

I'm studying Quantum Mechanics and there's something I'm in doubt in how to write it down in a rigorous way. The idea is: consider a Hilbert space $\mathcal{H}$ and one hermitian operator $A\in ...
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27 views

Eigenvalues of the subtraction of a gram matrix and a psd rank $1$ matrix.

If $V$ is $p \times r$ matrix ($r<p$) with rank $r$, then $G=V^TV$ is a $r \times r$ gram matrix with rank $r$. Let $x$ be a $r \times 1$ vector, and $xx^T$ is a psd rank $1$ matrix. It's already ...
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1answer
34 views

Eigenvalue equation and separable solutions

Recently, studying Quantum Mechanics I found a doubt regarding separable solutions and eigenvalue equations for differential operators. Suppose we are considering some space $\mathcal{H}$ of functions ...
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1answer
28 views

Anticommuting matrices and their eigenvalues

Let $A,B\in \mathcal{M}_n(\mathbb{C})$. It is known that if $AB=BA$ and $\lambda_1, \lambda_2, \dots, \lambda_n $ are the eigenvalues of $A$ and $\beta_1, \beta_2, \dots, \beta_n$ are the ...
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2answers
34 views

Null space is an Invariant subspace

Let $\lambda$ be an eigenvalue of a square matrix $A$. Show the null space of $(A-\lambda I)^j$ is an A-Invariant subspace of $\mathbb{C}^n$ for all positive integers $j$. Proof without requiring ...
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30 views

Is class of graphs with eigenvalue $1$ of any particular importance?

Are graphs with eigenvalue $1$ of multiplicity more than $1$, important one? Please guide me to any book or article discussing such graphs.
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2answers
38 views

A variation on the $AB$ vs $BA$ nonzero eigenvalues question.

Let $A\in\mathbb{R}^{m\times n}$ and $B\in\mathbb{R}^{n\times m}$, so that $AB\in\mathbb{R}^{m\times m}$ and $BA\in\mathbb{R}^{n\times n}$ both exist. Thanks to Sylvester's determinant identity, we ...
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1answer
19 views

Number of eigenvalues for this operator

Say I have a F - vector space V and a subspace U given by U={va : a is in F}. Now suppose I have an operator defined by $Tv=av$. Clearly, U is invariant under T, since for any element of U, say bv, I ...
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1answer
59 views

Show that $ 4\times4$ matrix has real eigenvalues

I have a real $ 4\times4$ matrix of the form $$ C = \begin{pmatrix} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ c_{31} & c_{32} & 0 & c_{34} \\ c_{41} & c_{42} & ...
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3answers
38 views

Scaling a matrix to make its eigenvalues fall within a certain interval

Suppose I have a diagonalizable matrix $M$ which has all its eigenvalues between $a$ and $b$. Is it possible to scale $M$ to $M_S$ such that all the eigenvalues of $M_s$ lie in the interval $[-1,1]$? ...
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1answer
12 views

“Same range of values”, quadratic form transformation

A quadratic form in the variables $u_i$ is expressed as $u'Du$. Matrix $T$ consists of the n characteristic vectors (of matrix $D$): $T = [v_1\quad v_2\quad ...\quad v_n]$. The following ...
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1answer
56 views

Theorems restricting the eigenvalue of a matrix

I have a square matrix $C$, whose entries I will denote by $c_{ij}$, and I would like to bound the magnitude of its eigenvalues. Each $c_{ij}$ is defined in terms of $s_{ij}$ and $S_j$ as follows: ...
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1answer
25 views

Check if two square matrices are similar.

Check if matrices $A= \begin{bmatrix} 1 & 1 & 5 \\ 0 & 2 & 0 \\ 0 & 0 & 2 \\ \end{bmatrix}$ and $B=\begin{bmatrix} 1 & 7 ...
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0answers
27 views

$A$ is nonsingular. eigenvalues of $A$ are unequal to zero

Assume that $A$ is nonsingular. Show that all eigenvalues of $A$ are unequal to zero. Express the eigenvalues and eigenvectors of $A􀀀^{-1}$ in terms of those of $A$.
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31 views

In matrix algebra, what's the name for the inverse operation of pre- or post- multiplication?

For example, in this typical equation: $$\mathbf{Mv}-\lambda \mathbf{v}=\mathbf{0}$$ (where $\mathbf{M}$ is a symmetric matrix, $\mathbf{v}$ is a vector, $\lambda$ is a scalar, and $\mathbf{0}$ is a ...
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0answers
29 views

approximate projection into eigenvector space

Given a matrix A, $3 \times 3$, that is symmetric with zero diagonal, I calculate a matrix V, $3 \times 3$, whose columns are the corresponding right eigenvectors and a diagonal matrix D, $3 \times ...
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1answer
24 views

Computation of eigenvectors?

Given a matrix: $A = \begin{pmatrix} -\epsilon & tf_1 \\ tf_2 & -\epsilon \end{pmatrix}$ Compute the eigenvectors. I can easily find the eigenvalues to be $\lambda = -\epsilon \pm t\sqrt{f_1 ...
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1answer
27 views

graphs with smallest eigenvalue at least -1

Let $G$ be an undirected simple graph and let $A$ be its adjacency matrix. It is easy to see that $A$ is neither positive semidefinite nor negative semidefinite. I would like to know if there are ...
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0answers
24 views

Bounding lower triangular perturbation

Suppose $A\in R^{n\times n}$ is a matrix equal to sum of the Identity matrix and a lower triangular matrix $L$. Diagonal entries of $L$ are $0$. \begin{equation} A=I+L \end{equation} Define spectral ...
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2answers
28 views

Proving that if $A$ is diagonalisable then $\chi_A(A) = 0$

This could be a very simple question to answer, but I'm unsure how to prove this. If you have a diagonalisable matrix $A$, prove that $\chi_A(A)$ is the zero matrix. (where $\chi_A(x)$ is the ...
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1answer
16 views

Characteristic polynomial: Identity permutation?

This concerns the characteristic polynomial of a matrix. http://www.math.umn.edu/~olver/num_/lnv.pdf p. 7 (or p. 92). every term is prescribed by a permutation π of the rows of the matrix ...
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1answer
28 views

$A \in SO(3,\mathbb R)\setminus\{I\}$ , then there are exactly two points in $S^2:=\{(x,y,z)\in \mathbb R^3:x^2+y^2+z^2=1\}$ which are fixed by $A$?

Let $A \in SO(3,\mathbb R)\setminus\{I\}$ , then is it true that there exist exactly two points in $$S^2:=\{(x,y,z)\in \mathbb R^3:x^2+y^2+z^2=1\}$$ which are fixed by $A$? Or equivalently we ...
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1answer
19 views

Principal axis form (quadratic) to hyperbola/ellipse form?

How is the principal axis form (quadratic) of a symmetric orthogonally diagonalized matrix e.g. $$y^2+6yx+x^2=1$$ transformed into the equation of a hyperbola (or ellipse as well)? Completing the ...
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1answer
34 views

If the characteristic polynomial of $A,B$ is the same does it mean that $A,B$ are similar?

If the characteristic polynomial of $A,B$ is the same does it mean that $A,B$ are similar? So I read that it's true only if $A,B$ are diagonalizable, but why? if the characteristic polynomial is ...
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0answers
30 views

What is the maximum value of coefficient $f_v$ with the constraint that the matrix is positive semi-definite?

I am trying to solve this equation my self with my knowledge about characteristic polynomials, etc but I have placed it here earlier because I'm not a mathematician and maybe you give me ideas to ...
2
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2answers
34 views

$A$ is diagonalizable, if $A,B$ have then same eigenvalues, then $B$ is also diagonalizable

Given $A_{n\times n},B_{n\times n} \in \mathbb R$ such that $A$ is diagonalizable then: if $A,B$ have then same eigenvalues, then $B$ is also diagonalizable over $ \mathbb R$. if $A,B$ ...
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0answers
19 views

Transform in eigenvetor space

Hello I have a squared matrix C C = 0 2.2361 63.7887 2.2361 0 61.6117 63.7887 61.6117 0 and I calculate its ...
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2answers
36 views

If $A^4=4A^2$ then $m_A(x)=x^2-4$ and if it isn't diagonalaziable over $\mathbb R$ then $0$ is an eigenvalue

Given $A_{n\times n} \in \mathbb R$ such that $A^4=4A^2$ then if $A$ is invertible and isn't of the form $cI, c\in \mathbb R$ then $m_A(x)=x^2-4$. if $A$ isn't diagonalizable over ...
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2answers
31 views

Let $A$ be a real symmetric matrix with rank $1$ , then can all the diagonal entries of $A$ be $0$ ?

Let $A$ be a square real symmetric matrix with rank $1$ , then can all the diagonal entries of $A$ be $0$ ? I know that real symmetric matrices are diagonalizable . Also if all the diagonal entries be ...
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1answer
38 views

Given a normal $A_{n\times n}$ matrix, then $\lVert A^*v \rVert = \lVert Av\rVert$ and $\langle Av,v\rangle = \langle A^*v,v\rangle$

Let a normal $A_{n\times n}\in \mathbb C^n $ matrix, then: $\forall v \in \mathbb C^n:\lVert A^*v \rVert = \lVert Av\rVert $ $\forall v \in \mathbb C^n : \langle Av,v\rangle = \langle ...
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2answers
40 views

Second eigenvector of double eigenvalue matrix

$\begin{bmatrix}\frac{1}{2} & -\frac{1}{2} \\ \frac{1}{2} & -\frac{1}{2}\end{bmatrix}\begin{bmatrix}x \\ y\end{bmatrix} = \begin{bmatrix}0 \\ 0\end{bmatrix}$ $\lambda_{1} = \lambda_{2} = -1$ ...
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0answers
20 views

Matrix shear transformations [closed]

If you know the line of a shear transformation (the invariant line), how word you go about finding the shear factor? Also, funny as it may sound, what is the shear factor - what does it show?
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2answers
41 views

Showing that if $A$ is diagonalizable then $A^2-4A+8I$ is diagonalizable

Let $A_{n\times n}$ be a real matrix then: if $A^4 = 8A$ then $A$ is not invertible. if $A$ is diagonalizable over $\mathbb R$ then $A^2-4A+8I$ is diagonalizable over $\mathbb R$. ...
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1answer
65 views

The eigenvalues of $AB$ and of $BA$ are identical for all square $A$ and $B$ …a different approach

The eigenvalues of $AB$ and of $BA$ are identical for all square $A$ and $B$. I have done the proof in a easy way… If $ABv = λv$, then $B Aw = λw$, where $w = B v$. Thus, as long as $w \neq 0$, it is ...
2
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2answers
35 views

Finding Eigenvectors for $3 \times 3$ matrix with rows of zeros.

For a $3 \times 3$ matrix: $ $[A]$ = \begin{bmatrix} 6 & -2 & 2 \\ -2 & 3 & -1 \\ 2 & -1 & 3 \end{bmatrix} $ I have the eigenvalues: ...
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1answer
46 views

True/false questions about minimal and characteristic polynomials of a matrix

We have the matrix $A= \begin{pmatrix} 0 &2 &2 \\ 2& 0 &2 \\ 2& 2 & 0 \end{pmatrix}$, then one of the following is true: $f_A(x)=m_A(x) $ The matrix ...
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1answer
11 views

Not getting dot product of Eigen vectors in MATLAB aslo in LAPACK to zeros

I know that theoretically eigenvectors of real symmetric matrix are orthogonal to each other. So for each pair, dot product will be zero. But when I am calculating eigenvectors from real symmetric ...
2
votes
1answer
32 views

Eigenvalue perturbation theory for $(A^TA)(B^TB)^{-1} + (B^TB)(A^TA)^{-1}$

Let $A, B$ be $n \times n$ matrices with full rank. I'm interested in getting a bound on how the smallest eigenvalue of $S = (A^TA)(B^TB)^{-1} + (B^TB)(A^TA)^{-1}$ changes when I perturb $A$ and $B$. ...