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How can I obtain the eigenvalues of this matrix?

Consider the following $M \times 3$ matrix $$\mathbf F = [\mathbf h_1, \mathbf h_2, \mathbf h_3],$$ with distinct non-zero singular values $\sigma_1 >\sigma_2 > \sigma_3$, where $\mathbf h_k$'s ...
WPCN's user avatar
  • 31
1 vote
0 answers
179 views

QR algorithm for eigenvalues and eigenvectors of large symmetric matrices

I am trying to write a QR algorithm in Python for eigenvectors and eigenvalues finding for large symmetric matrices, My initial thought was to use Householder transformation with a Wilkinson shift ...
Daniel Belaish's user avatar
2 votes
1 answer
212 views

Similarity of two matrices

Consider the matrix, for some $\lambda \in \mathbb R$ . $$A=\begin{pmatrix} i \lambda & -1 & i & 0 \\ 1 & 0 & 0& 0 \\ i & 0 & - i \lambda & -1 \\ 0 & 0 & 1 ...
Pritam Bemis's user avatar
2 votes
0 answers
121 views

Eigenvalues of two positive-definite Toeplitz matrices

Consider two positive-definite Toeplitz matrices $M_1$ and $M_2$ both with dimension $2^j \times 2^j$. Their matrix elements are: $$M_1[x,y] = \frac{\text{sin}(\pi(x-y)/2^j)}{\pi(x-y)} \qquad M_2[x,y] ...
Chriscrosser's user avatar
3 votes
1 answer
2k views

Eigenvalues of a block matrix with zero diagonal blocks

Suppose $A$ is a $k_1\times k_2$ matrix with real entries, $k_1<k_2$. Let $M$ be the matrix \begin{equation} M:=\begin{pmatrix} 0_{k_1} & A\\ A^\top & 0_{k_2} \end{pmatrix}, \end{equation} ...
AdamNie's user avatar
  • 53
21 votes
5 answers
2k views

The middle eigenvalues of an undirected graph

Let $ \lambda_1 \ge \lambda_2 \ge \dots \ge \lambda_{2n} $ be the collection of eigenvalues of an adjacency matrix of an undirected graph $G$ on $2n$ vertices. I am looking for any work or references ...
Tomaž Pisanski's user avatar
1 vote
0 answers
331 views

Eigenvalues of an (almost) pentadiagonal symmetric Toeplitz matrix

I am looking for analytic expressions for the eigenvalues of matrices of the form $$A = \begin{bmatrix} 6 & -4 & 1 & 0 & 0 & 0 & 0 \\ -4 & 6 & -4 & 1 & 0 &...
E_Wijler's user avatar
0 votes
1 answer
263 views

Change in the largest eigenvalue due to perturbation of diagonal components of a symmetric matrix

Let $A\in \mathbb{R^{n\times n}}$ be a symmetric negative difinite matrix and $D\in \mathbb{R}^{n\times n}$ be a diagonal matrix $D = \mathrm{diag}\{d_i\}, (d_i < 0)$. From Weyl's inequality, the ...
seg nana's user avatar
-2 votes
3 answers
2k views

When is it possible to find the sum of all elements of inverse of a matrix?

Given sum of elements of each row of a positive definite square matrix $M$ of order $n$ all of whose entries are non-negative, when is it possible to find the sum of all elements of the matrix $M^{-1}$...
Learnmore's user avatar
  • 135
2 votes
0 answers
69 views

Unimodular eigenvalue of a H-self-adjoint matrix (indefinite innerproduct)

Let $A,H \in \mathbb{C}^{n \times n}$ be such that $H$ is Hermitian and invertible and $A = H^{-1} A^* H$. In this case, $A$ is said to be $H$-self-adjoint. This is due to the fact that if $\langle \...
Leo's user avatar
  • 175
2 votes
4 answers
293 views

Find a square, stochastic matrix of odd size, not a permutation matrix, with an eigenvalue other than 1 on the unit circle

...or prove that none exists. Note that such a matrix $M$ couldn't be primitive, so there would be at least one entry equal to zero in every power $M^k$ (Perron-Frobenius theory). Preferably the ...
tarski's user avatar
  • 21
26 votes
1 answer
5k views

Generalization of Cauchy's eigenvalue interlacing theorem?

Cauchy's Interlacing Theorem says that given an $n \times n$ symmetric matrix $A$, let $B$ be an $(n-1) \times (n-1)$ principal submatrix of it, then the eigenvalues of $A$ and those of $B$ interlace. ...
Hao's user avatar
  • 571
0 votes
0 answers
46 views

What kind of bounds for $\mathrm{Re}(\lambda(A))$ when $\lambda_{\mathrm{max}}(A + A^t) < 0$?

What can be said about the real parts of eigenvalues of $A \in \mathbb R^{n\times n}$ when $\lambda_{\max}(A + A^t) < 0$? I think the real parts of eigenvalues of $A$ will be negative, but I can't ...
seg nana's user avatar
2 votes
1 answer
236 views

How can I prove a randomly generated matrix has distinct non-zero eigenvalues?

Consider the following $M×M$ matrix $$ \mathbf A=\sum_{k=1}^K =a_k \mathbf h_k \mathbf h_k^H,(M≥K) $$ where $a_k$'s are real values and $h_k$'s are $M×1$ randomly generated vectors, e.g., complex ...
WPCN's user avatar
  • 31
2 votes
1 answer
74 views

Limitation through the singular values

Given matrix $X \in \mathbb{R}^{m\times n}$ and sequence $\left\{X^k\right\}_k$ converges to $X$ according to the Frobenius norm. I wonder that $\sigma_i(X^k)$ converge $\sigma_i(X)$ or not (where $\...
ohana's user avatar
  • 143
3 votes
1 answer
534 views

Spectrum of the sum of two general matrices

Are there any restrictions on the possible spectrum of the sum of two arbitrary matrices with given spectra other than the trace identity? In other words: Let $\alpha, \beta, \gamma$ be $n$-tuples (...
Dmitrii Korshunov's user avatar
5 votes
1 answer
241 views

Trace inequality under consideration of definiteness

Let $G \in \mathbb{R}^{3 \times 3}$ a symmetric, but indefinite matrix and $U \in \mathbb{R}^{3\times 3}$ a symmetric and positive definite matrix. I would like to prove the inequality $$ \text{Tr} \...
TARS's user avatar
  • 51
1 vote
2 answers
2k views

Eigenvalues of tridiagonal symmetric matrix

Could you tell me please, are there any analytical methods how to find eigenvalues of matrix such this one? $$ \begin{pmatrix} a_1 & b_1 & 0 & 0 & 0 & \ldots & 0 \\ b_1 & ...
MightyPower's user avatar
15 votes
1 answer
1k views

Existence of double eigenvalue

Let $A$ and $B$ be complex $4\times 4$ matrices. Assume both are Hermitian, and that they are linearly independent. Must there exist a nonzero real linear combination $aA + bB$ which has a repeated ...
Nik Weaver's user avatar
  • 42.8k
4 votes
1 answer
2k views

Relation of row sums to largest eigenvalue

I know that the largest eigenvalue of a graph is bounded between the minimal and maximal row sum of the matrix. If I have a $0-1$ symetric matrix (an adjacency matrix) and I know $k$ of the rows have ...
Yahav Boneh's user avatar
3 votes
2 answers
432 views

Maximum eigenvalue of a covariance matrix of Brownian motion

$$ A := \begin{pmatrix} 1 & \frac{1}{2} & \frac{1}{3} & \cdots & \frac{1}{n}\\ \frac{1}{2} & \frac{1}{2} & \frac{1}{3} & \cdots & \frac{1}{n}\\ \frac{1}{3} & \frac{...
Weiqiang Yang's user avatar
1 vote
0 answers
373 views

Upper bound on the sum of the smallest non-zero eigenvalues

Let $\mathcal A := \{ A_1, A_2, \dots, A_n \} \subset \Bbb R^{d \times d}$ be a set of symmetric and positive semidefinite matrices. For a matrix $A_k \in \mathcal A$, denote its (real) eigenvalues by ...
Apprentice's user avatar
2 votes
0 answers
81 views

Perturbed Gram matrix

Let $x_t \in \mathbb{S}^{d-1}$, $\forall t\in \mathbb{N}$ and let $e_1$ be the first canonical basis vector of $\mathbb{R}^d$, ie, $e_1 = (1,0,\cdots,0)$. Let us form a Gram Matrix $$\sum_{t=1}^T(x_t ...
rostader's user avatar
  • 215
1 vote
0 answers
618 views

Largest eigenvalue of matrix A smaller than 1, what about B when A=B+C? [closed]

Suppose I have a square matrix $A$ that only has non-negative real entries and is not symmetric and not primitive either. It has no "special" structure we could exploit. I know that the ...
blldt's user avatar
  • 11
5 votes
1 answer
335 views

Projecting a symmetric matrix onto the space of linear operators with a particular eigenvalue

Specifically, I am interested in the case where one eigenvalue is exactly $0$. Given an $n \times n$ symmetric matrix, I would like to find the closest $n\times n$ symmetric matrix that has one ...
Him's user avatar
  • 245
3 votes
0 answers
373 views

Eigenvalues of block matrix

Given scalars $\alpha, \beta \in \mathbb{R}$, a symmetric positive definite matrix $A \in \mathbb{R}^{n\times n}$ and a flat matrix $B \in \mathbb{R}^{m\times n}$, where $m < n$, can I say ...
Trb2's user avatar
  • 125
2 votes
1 answer
728 views

Maximum eigenvalue of Hadamard power of a positive semidefinite matrix

Let $K$ be a covariance matrix. It is positive semidefinite, its diagonal elements are all 1, and its off-diagonals are between -1 and 1. Let $K.^2$ be its element-wise power (Hadamard power). Can we ...
Soheil Feizi's user avatar
8 votes
1 answer
7k views

Upper bound on largest eigenvalue of a real symmetric $n \times n$ matrix with all main diagonal entries positive, everywhere else nonpositive

Is there a good analytic upper bound on the largest eigenvalue of a real symmetric n*n matrix with all main diagonal entries strictly positive, all other entries <=0 with typically many of them ...
equest's user avatar
  • 83
1 vote
1 answer
711 views

Eigenvalues of the product of traceless unitary hermitian matrices [closed]

As a follow up of the question raised in Determinant involving traceless unitary hermitian matrices, I would like to pose a similar question. If A and B are distinct traceless unitary hermitian ...
Gerson J Ferreira's user avatar
2 votes
2 answers
201 views

Eigen problem with constrained (equal) eigenvalues

Let $\Omega$ be a symmetric and positive definite matrix. From a test of hypothesis I know that some eigenvalues are likely to be equal (the test also suggests which eigenvalues). Do you have any ...
meie73's user avatar
  • 131
7 votes
0 answers
905 views

The Möbius function as eigenvalues

Let the $N$ by $N$ matrix $A$ be defined by the tetration: $$\Large \text{If } \gcd(n, k)=1 \text{ then } A(n,k)= \underbrace{e^{e^{\cdot^{\cdot^{e^{\Re\left(\frac{1}{n^s}\right)}}}}}}_m \text{ else }...
Mats Granvik's user avatar
  • 1,183
7 votes
3 answers
1k views

Checking positive semi-definiteness of integer matrix

Key Problem : Is there any theorem about eigenvalues or positive semi-definiteness of small size matrices with small integer elements? I have to check positive semi-definiteness of many symmetric ...
SIM2's user avatar
  • 73
13 votes
0 answers
809 views

Can one Gershgorin circle (only) contain all eigenvalues, when the other circles are not contained in it

In short, following a question from my students, I am trying to find a special case where all the eigenvalues of a matrix lie within only one circle, but not in the others, and the other circles are ...
Itay's user avatar
  • 673
1 vote
1 answer
125 views

A monotonicity property of eigenvalues

Let $A \in S^{n}_{+}$ be a positive semi-definite matrix and $D \in S^{n}_{+}$ a diagonal matrix with all the diagonal entries no smaller than one, i.e., $D_{ii} \geq 1$ for all $i \leq n$. I wonder ...
zxzx179's user avatar
  • 205
3 votes
2 answers
2k views

Completely positive matrix with positive eigenvalue

A matrix $A \in \mathbb{R}^{n \times n}$ is called completely positive if there exists an entrywise nonnegative matrix $B \in \mathbb{R}^{n \times r}$ such that $A = BB^{T}$. All eigenvalues of $A$ ...
JKay's user avatar
  • 133
19 votes
1 answer
2k views

Smallest eigenvalue of a tricky random matrix

While experimenting with positive-definite functions, I was led to the following: Let $n$ be a positive integer, and let $x_1,\ldots,x_n$ be sampled from a zero-mean, unit variance gaussian. Consider ...
Suvrit's user avatar
  • 28.6k
8 votes
2 answers
15k views

Upper bounds on eigenvalues of PSD matrix?

Suppose A is a symmetric positive semidefinite matrix. Is there a way to upper bound the largest eigenvalue using properties of its row sums or column sums? For instance, the Perron–Frobenius ...
Yisong Yue's user avatar
1 vote
0 answers
163 views

An estimation of the largest eigenvalue of a submatrix of $\left(\cos(\frac{kl\pi}{4n})\right)_{k,l=1}^n$

Let us consider the following matrix $A=(a_{k,l})$ where $$A=\left(\cos(\frac{kl\pi}{4n})\right)_{k,l=1}^n$$ Let us consider the submatrix $A_0$ of $A$ whose entries are those $a_{k,l}$ where $k\...
ABB's user avatar
  • 4,058
1 vote
1 answer
276 views

Spectral decomposition of a $4\times4$ real nonsymmetric matrix with unknown elements

I'm trying to eigendecompose the following matrix $A$, i.e. to find $Q$ and $\Lambda$ such that $$ A = \begin{bmatrix} -\alpha & \alpha & -\gamma^{-1} & 0\\ \beta &...
sound wave's user avatar
2 votes
1 answer
372 views

The maximal eigenvalue of average of positive matrices

Let $A$ and $B$ be two square real positive (all entries are positive) matrices that differ only in the first row. Let $\lambda_A$ and $\lambda_B$ be the maximal real eigenvalues of $A$ and $B$, ...
Eilon's user avatar
  • 745
5 votes
2 answers
249 views

Eigenvalue density of a symmetric tridiagonal matrix

Let $A_n\in\mathbb{R}^{n\times n}$ be defined as $$ A_n=\begin{bmatrix} a & b & 0 & \cdots & \cdots & 0 & 0\\ b & a & b & \cdots & \cdots & 0 & 0\\ 0 &...
Ludwig's user avatar
  • 2,712
1 vote
1 answer
350 views

A linear algebra question regarding the eigenvalues of the product of a diagonal matrix and a projection matrix

I need to prove a statement in my research. The statement seems to be fundamental linear algebra, and numerical studies in MATLAB supported this statement, but I wasn't able to prove it after a few ...
Ken's user avatar
  • 21
8 votes
3 answers
1k views

Does small Perron-Frobenius eigenvalue imply small entries for integral matrices?

Suppose that $M$ is an $n \times n$ matrix where each entry is a positive integer. Then $M$ is Perron-Frobenius and so has unique largest real eigenvalue $\lambda_{\textrm{PF}}$. Does an upper ...
Mark Bell's user avatar
  • 3,165
8 votes
2 answers
12k views

Relation between eigenvalues of $A$ and $A^TA$?

For an $n\times n$ diagonizable matrix $A$, is there a relation between the eigenvalues of $A$ and the eigenvalues of $A^TA$? I ask this because I am looking into the relation between $A$ and $A+cI$, ...
Eric S.'s user avatar
  • 181
2 votes
1 answer
8k views

Properties of eigenvalues of general nonnegative matrices

I am aware, that an answer to this question can be found via Perron-Frobenius theory or something very similar, but unfortunately I am far from being an expert in the field and I am unable to find the ...
042's user avatar
  • 83
0 votes
2 answers
588 views

Estimating the shift in the $\lambda_{\max}$ of a matrix under a diagonal perturbation

Given a matrix $A$ and a diagonal matrix $D$, how can we estimate $\lambda_{\max}(A+D) - \lambda_{\max}(A)$? Feel free to make other assumptions about the matrices that they are all symmetric and have ...
InterlacingStudent's user avatar
5 votes
1 answer
635 views

Largest Eigenvalue of a Matrix with Special Form in terms of n

In one step of solving a difficult problem, I would like to know the largest eigenvalue of a matrix with this pattern: $$A_n = \begin{bmatrix} 0 & 0 & 0 & 0 &\dots & 0 \\ ...
dave2d's user avatar
  • 103
-2 votes
1 answer
353 views

Can we attain the maximum and minimum of a Rayleigh quotient over any subspace? [closed]

Let $M\in\mathbb{C}^{n\times n}$ be a Hermitian matrix and let $E$ be a subspace of $\mathbb{C}^n$. $$\mbox{Are } \sup_{x\in E\\ x\neq0}\dfrac{x^*Mx}{x^*x}\mbox{ and }\inf_{x\in E\\ x\neq0}\dfrac{x^*...
Chilote's user avatar
  • 596
18 votes
1 answer
847 views

Showing that a certain matrix is not positive definite

Let $J_k$ be a $k \times k$ all ones matrix and $B$ any $k \times k$ binary matrix - that is $B$ only has entries from $\{0,1\}$. I would like to show that the matrix $$X_B = (J_k -I) - B (J_k - I)^{-...
Jernej's user avatar
  • 3,463
4 votes
0 answers
447 views

How to find eigenvalues of following block matrices?

Is there a procedure to find the eigenvalues of A? ‎ $$A=\begin{bmatrix}X & I &&&&&&&&& 0\\I & 0 & P &&&&&&&&\\& P^t ...
Maryam Hak's user avatar