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4 votes
1 answer
90 views

Tight upper bound on a ratio involving symmetric PSD matrices and Kronecker products

Let $\mathbf{A}_i \in \mathbb{R}^{d \times d}$ ($i = 1, \dots, T$) be symmetric positive semidefinite (PSD) matrices. Define the quantity $$ m = \frac{\lambda_{\max}\left(\sum_{i=1}^T \mathbf{A}_i^2\...
Ran's user avatar
  • 73
2 votes
1 answer
299 views

Eigenvalues of a specific matrix

I have a block matrix $$M=\begin{bmatrix} I_0& I_1& \cdots& I_1\\ I_2& I_0& \ddots& \vdots\\ \vdots& \ddots& \...
Young Q's user avatar
  • 43
3 votes
1 answer
5k views

Relation between the eigenvalues of a block matrix and the eigenvalues of its diagonal blocks

Consider the $(m+n) \times (m+n)$ block matrix $$M = \begin{bmatrix} A & B \\ C & D \end{bmatrix}$$ I need references where they are talking about the relation between the eigenvalues of $M$ ...
GA316's user avatar
  • 1,269
3 votes
0 answers
538 views

Diagonalizing a block tridiagonal matrix

Is there an efficient way to diagonalize a block tridiagonal $N\times N$ matrix of the following form: \begin{matrix} A_0 & B & 0 & 0 & \ldots \\ B & A_1 & B & 0 & \...
Ritteraxt's user avatar
2 votes
2 answers
320 views

Eigenvalues and eigenvectors of k-blocks matrix

I'm trying to find the eigenvalues and eigenvectors of the following $n\times n$ matrix, with $k$ blocks. \begin{gather*} X = \left( \begin{array}{cc} A & B & \cdots & \\ B & A & ...
ari6739's user avatar
  • 21
0 votes
0 answers
149 views

Diagonalizing a specific case of symmetric block matrix

Let's consider the following block matrix $$ M = \begin{pmatrix}D&A^T\\A&-D\end{pmatrix},$$ where $A$ and $D$ are $n \times n$ matrices. The diagonal matrix $D$ is defined by $D_{kk} = k \...
Marin's user avatar
  • 1
0 votes
0 answers
232 views

How to analyse the range of eigenvalues of a symmetric and indefinite matrix?

Let $G$ be a symmetric and indefinite matrix defined as follows $$ G := S - \begin{pmatrix} I_n & I_n \\ I_n & I_n \end{pmatrix},$$ where $S$ is a symmetric positive definite matrix of size $...
Nxy's user avatar
  • 1
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
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
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
3 votes
1 answer
463 views

Spectrum of this block matrix

Consider the following block matrix $$A = \left(\begin{matrix} B & T\\ T & 0 \end{matrix} \right)$$ where all submatrices are square and matrix $B = \mbox{diag}\left(b_1 ,0,0,\dots,0,b_n \...
Sascha's user avatar
  • 536
3 votes
1 answer
3k views

Bounds for eigenvalues of block matrix

Let's say I have a block matrix of the form $$X = \begin{bmatrix} A & B\\ B^T & C\end{bmatrix}$$ where $A$, $C$, and $X$ are all positive definite. I have bounds on both the minimum and ...
bringingdownthegauss's user avatar
1 vote
0 answers
171 views

Eigenvalues of non-negative block matrices

$B$ is a non-negative irreducible block matrix as follows: $$B= \left[ \begin{array}{c|c|c} 0 &B_{12}&B_{13}\\ \hline B_{21}& 0& B_{23}\\ \hline B_{31}& B_{32}&0 \end{array} \...
afra's user avatar
  • 21
0 votes
1 answer
1k views

How do eigenvalues change if we duplicate a row and column of a symmetric matrix?

Let $\bf A$ be a $n \times n$ symmetric positive semidefinite matrix whose first column is denoted by ${\bf a}_1$. We define a new matrix, $$ {\bf B} = \begin{bmatrix} a_{11} & {\bf a}_1^T \\ {\bf ...
Jingxian Wu's user avatar
3 votes
0 answers
1k views

Eigenvalues of block-hermitian matrices with zero diagonal blocks

I have a matrix of the form $$D = \left( \begin{array}{cc} 0 & C \\ C^{\dagger} & 0 \end{array} \right)$$ where $C$ is not necessarily hermitian. In general, can we say anything about the ...
Unwieldy Bob's user avatar
5 votes
1 answer
2k views

Condition for block symmetric real matrix eigenvalues to be real

I have a $2n \times 2n$ block symmetric matrix that in the simplest case ($n=2$) looks like: $$ M_2 = \begin{bmatrix} a_1 & 0 & b_{1,2} & -b_{1,2}\\\ 0 & -a_1 & b_{1,2} & -b_{...
Giulia's user avatar
  • 151
0 votes
0 answers
224 views

Upper bound on matrix perturbation such that all eigenvalues lie within the unit circle

Consider the matrix $$N=\left[\matrix{\mathbb{I}_n-\epsilon L & X\\ \epsilon Y & Z}\right]$$ where $\epsilon>0$ is a small positive parameter and $Z$ is a square $m\times m$ matrix with ...
CTNT's user avatar
  • 101