# Tagged Questions

**0**

votes

**1**answer

161 views

### Existence of a real eigenvalue

I have a matrix $M \in \mathbb{R}^{(n+1) \times (n+1)}$ that is tridiagonal.
In numerical computations I found out that I always find a real eigenvalue. My question is: Is there a theorem that ...

**1**

vote

**1**answer

91 views

### Neighborhood overlap matrix for a bipartite graph

Let $G$ be an undirected, simple, bipartite graph with parts $V$ (having $n$ vertices) and $W$ (having $m$ vertices). Define the following $n$-by-$n$ matrix: for any $i,j \in V$, $$a_{ij} = |N_i \cap ...

**1**

vote

**0**answers

173 views

### An optimization problem on the sphere

Let $S$ be a sphere centered at origin in $\Bbb R^{2n}$ of radius $\sqrt{2n}$. Let $D$ be a diagonal matrix. Let $U$ be an orthogonal matrix. Let $r\in\Bbb Z_+$ be a fixed integer.
Let vector ...

**1**

vote

**0**answers

75 views

### “Almost orthogonalizing” matrices using a signature matrix

Suppose $A$ and $B$ are two real symmetric $n \times n$ matrices (If simpler, consider $A$ and $B$ to be 0/1 matrices, say, adjacency matrices of d-regular graphs).
Then $||AB||_{op} \leq ...

**5**

votes

**1**answer

209 views

### Separating the spectrum of a Hermitian matrix

Given Hermitian matrix $A$, I would like to perturbate it so that its eigenvalues become well-separated.
Specifically, let $A$ be some Hermitian matrix, and let $G$ be a Gaussian matrix, with each
...

**2**

votes

**1**answer

166 views

### positive semidefinite matrix condition

There is a great work of Alizadeh that in section 4 speaks about Minimizing sum of the first few(k-largest) eigenvalues of a symmetric matrix. Instead of a symmetric model we use the weighted ...

**0**

votes

**0**answers

94 views

### Perturbation of spectrum and eigenspaces

Let $A \in \mathbb{C}^{n \times n}$ be an $n \times n$ matrix. Consider the rank-$1$ perturbation $A'$ of $A$ given by replacing a column $v$ of $A$ by $\alpha \cdot v$, where $\alpha \in [0, 1)$. Can ...

**3**

votes

**1**answer

297 views

### numerical range of a column-zero-sum matrix

I am trying to produce an example of a (necessarily non-normal) matrix that has only eigenvalues with positive real part, but whose numerical range contains elements with strictly negative real part. ...

**1**

vote

**1**answer

366 views

### square root of a certain matrix [closed]

Hello,
I'd like to know the square root of the following $n$ by $n$ matrix, for $n > 2$ and $r>0$:
$R_{ii}=r+1$ for $i < n$
$R_{ij}=r$ otherwise
The $2$ by $2$ case is given by
...

**1**

vote

**0**answers

63 views

### Second eigenvalue of a weighted tree

Hello,
I am interested in upper bounding the second largest eigenvalue of the adjacency matrix of a graph $T$ with the following property:
1. $T$ contains self loops.
2. $T$ contains multiple edges ...

**1**

vote

**1**answer

309 views

### Eigenvalues of Sum of non-singular matrix and diagonal matrix

Suppose $D={\rm diag}(d_i)$ is a diagonal matrix with all diagonal entries $d_i=\pm 1$. This implies $D^2=I$.
Suppose $A$ is a non-singular Hermitian matrix. If we know that $A+A^{-1}+D$ has rational ...

**2**

votes

**0**answers

108 views

### Optimization over Spectral Laplacian in cycles and trees

Is there any idea on how one can deal with an optimization problem of sum of k largest eigenvalues(min) of Laplacian matrix of a simple cycle or tree?
I would like to use semidefinite programming for ...

**1**

vote

**0**answers

48 views

### Possible diagonal values of a product of matrices with some specific characteristics

Hello all,
This is a question that might or might not be related to my previous one.
Imagine you have two matrices:
Matrix $\mathbf{\Phi}=[\Phi_1,\ldots,\Phi_M]\in\mathbb{R}^{L\times M}$ where ...

**3**

votes

**1**answer

188 views

### A spectral radius inequality

Define $\rho(A)$ to be the spectral radius of a square matrix $A$. Let $S$ and $T$ be two non-negative square matrices and $h$ a real number such that $\rho(S+T) < h$. Show that $\rho((hI-S)^{-1}T) ...

**0**

votes

**2**answers

330 views

### Eigenvalues of principle minors Vs. eigenvalues of the matrix

Say I have a positive semi-definite matrix with least positive eigenvalue x. Are there always principal minors of this matrix with eigenvalue less than x?
(Here "semidefinite" can not be taken to ...

**19**

votes

**9**answers

3k views

### real symmetric matrix has real eigenvalues - elementary proof

Every real symmetric matrix has at least one real eigenvalue. Does anyone know how to prove this elementary, that is without the notion of complex numbers?

**10**

votes

**1**answer

335 views

### relationship between eigenvalues of (A-B) and eigenvalues of (A^2-B^2)

Let us suppose that $A_{n}$ and $B_n$ are sequences of positive definite matrices satisfying
$c\leq \lambda_{\min}(A_n)\leq \lambda_{\max}(A_n)\leq C$
and
$c\leq \lambda_{\min}(B_n)\leq ...

**1**

vote

**1**answer

236 views

### A is a nonnegative matrix; the only principal submatrix having spectral radius above 1 is A itself

Let $\rho(M)$ denote the spectral radius (modulus of the largest eigenvalue) of a square matrix $M$.
I am looking for a characterization or anything else interesting about the set of matrices $A$ ...

**3**

votes

**1**answer

156 views

### Effects of unitarian multiplication into the spectrum of a finite matrix.

I am interested in the following problem: Let $P$ be a $n\times n$ complex finite matrix such as $PP^\dagger =W$. Given $W$, what can I say about the spectrum of $P$?
This matrix "square-root" has ...

**1**

vote

**1**answer

108 views

### Morse index and permutation of diagonal entries of a symmetric matrix

Do there exist results concerning preservation or not of the Morse index of a symmetric matrix $A$, after permuting its diagonal entries, and keeping fixed the off--diagonal ones?
Thanks!

**5**

votes

**3**answers

411 views

### Has the largest-to-rest eigenvalue ratio of real symmetric matrices been researched before?

I'm investigating the eigenvalue ratios
$$
\frac{\lambda_1}{\sum_{j=2}^N\lambda_j}
\quad\mbox{and}\quad
\frac{\sum_{j=1}^N\lambda_j}{\sum_{j=2}^N\lambda_j}
$$
of the NxN matrix $B=AA^T$. $\lambda_1$ ...

**13**

votes

**2**answers

902 views

### Minimum off-diagonal elements of a matrix with fixed eigenvalues

Hello,
I am en engineer working in radar research. I came accross a problem I cannot seem to find math literature on it.
I can ask it in two different ways. Perhaps depending on the reader, the ...

**2**

votes

**1**answer

144 views

### Generalizing the spectral radius of a unistochastic matrix

Consider a square matrix $A$, and from it construct $B$ whose entries are the squared magnitudes of those in $A$. What can we say about the spectral radius of $B$? I know that for a unitary matrix ...

**3**

votes

**2**answers

2k views

### Interesting relationships between Cholesky decomposition and diagonalization

Let $\Sigma$ be a hermitian positive definite matrix and $L$ be it's Cholesky decomposition so that $LL^\ast=\Sigma$. Furthermore, let's diagonalize $\Sigma$ as $\Sigma = P\Lambda P^\ast$. $\Lambda$ ...

**8**

votes

**0**answers

447 views

### Bounding sum of first singular values squared for Kronecker sum of traceless matrices

Let $A$ and $B$ be $4\times4$ traceless matrices with Hilbert-Schmidt norms summing up to $1/4$, i.e.
$$\text{Tr}\left[ A\right]=\text{Tr}\left[ B\right] = 0,\qquad\text{Tr}\left[ A^\dagger A + ...

**6**

votes

**1**answer

569 views

### 0 eigenvalue for a symmetric tridiagonal matrix

Let $T\in \mathbb{R}^{n\times n}$ be a symmetric tridiagonal matrix having the off--diagonal entries equal to -1. The diagonal entries are all positive, $a_i>0$, $i=\overline{1,n}$, and there ...

**4**

votes

**1**answer

253 views

### Spectral Properties of $A(I-A)^{-1}$

I am working with a class of matrices $A$ which are non-negative-definite, not symmetric, and have maximum eigenvalue less than 1. I am interested in the spectral properties of the matrix $H = A(I - ...

**7**

votes

**2**answers

469 views

### Efficiently computing a few localized eigenvectors

Let $H = \triangle + V(x) : \mathbb{R}^2 \rightarrow \mathbb{R}^2$. I am interested in domain decomposition for an eigenproblem involving $H$.
The lowest 1000 eigenfunctions of $H$, $ \psi_i $, can ...

**1**

vote

**3**answers

913 views

### Generalization of eigenvalues/vectors to modules?

What is the generalization of eigenvalues/vectors to modules?
To be specific, given a "vector" v in a module over some ring, and a linear "operator" O from the module to itself (please feel free to ...

**4**

votes

**1**answer

542 views

### Tridiagonal Matrix

What is the most efficient way to calucate the dominant eigenvector of a real symmetric tridiagonal matrix, and what's the corresponding time complexity bound? Could someone give me a reference for ...

**2**

votes

**1**answer

214 views

### Singular values of differences of square matrices

Suppose $A, B \in \mathbb{R}^{n \times n}$. Let $\sigma_1(A),\ldots,\sigma_n(A)$ be the singular values of $A$, and let $\sigma_1(B),\ldots,\sigma_n(B)$ be the singular values of $B$. If I know these ...

**4**

votes

**1**answer

870 views

### dominant eigenvector

Hi, everyone! Is there any efficient way to simplify the following tensor product
$X \otimes X + X^T \otimes X^T$, where $X$ is a square $n \times n$ matrix.
My goal is to efficiently compute the ...

**14**

votes

**2**answers

577 views

### Eigenvalues of an “oblique diagonal” matrix

I am looking for guidance about the behavior of powers of a particular matrix (call it $A_n$ for $n\ge2$), which has come up in a counting problem about quantum knot mosaics (a good reference for ...

**27**

votes

**3**answers

2k views

### Perron-Frobenius “inverse eigenvalue problem”

The Perron-Frobenius theorem says that the largest eigenvalue of a positive real matrix (all entries positive) is real. Moreover, that eigenvalue has a positive eigenvector, and it is the only ...

**11**

votes

**2**answers

951 views

### Combinatorial proof of (a special case of) the spectral theorem?

The spectral theorem for a real $n \times n$ symmetric matrix $A$ says that $A$ is diagonalizable with all eigenvalues real. If $A$ happens to have non-negative integer entries, it can be interpreted ...

**14**

votes

**1**answer

2k views

### Fast Fourier Transform for Graph Laplacian?

In the case of a regularly-sampled scalar-valued signal $f$ on the real line, we can construct a discrete linear operator $A$ such that $A(f)$ approximates $\partial^2 f / \partial x^2$. One way to ...

**1**

vote

**2**answers

378 views

### Matrix Logarithms are Not Unique

In my ODE class, we proved that if exp(L) = exp(L') then the eigenvalues are congruent mod 2πi. Here L, L' are two nxn matrices. I wanted to know if something more precise was true.
In a way, we ...

**0**

votes

**0**answers

355 views

### iterated characteristic polynomials

If I have $N$ $M\times M$ symmetric positive definite matrices $A_i$ and an $N\times N$ positive semi-definite symmetric matrix B, let the $N\times N$ matrix $C_{ij}(\lambda)=B_{ij}$ for $i\ne j$ and ...

**2**

votes

**3**answers

1k views

### spectral radius of a matrix as one element changes

Here's my question --
Let $A$ be an $n \times n$ real matrix, and suppose that the spectral radius $\rho(A)$ is less than one (spectral radius = max eigenvalue). Let's choose some $1 \leq i \leq N$ ...