-2
votes
0answers
18 views

Stability of a block matrix with a stable upper left corner

Given a $n\times n$ matrix $A$ is stable, can it be proven that $G=\left(\begin{array}{cc}A & B\\C & -d\end{array}\right)$, where ...
0
votes
0answers
10 views

How to find the rules by which eig(A) = diagonal entries

Suppose A = [a b c;d e f;g h i] how to find a series of rules such that eig(A) = a, e, i such as h = 0 or f = 0? Is there any rule for this? (I know we can find this by principal submatrix, but I ...
1
vote
0answers
24 views

A question about eigenvalues of a special block matrix

Thanks for anyone who views or answers this question! $N\times N$ matrix $L$ can be partitioned into $p\times p$ blocks, ...
3
votes
1answer
157 views

Eigenvalues of a submatrix

A self-adjoint matrix $A$ in a complex linear space has real eigenvalues bounded between $0$ and $1$. If $A$ is projected onto an arbitrary two-dimensional subspace, what would be the bounds on the ...
3
votes
1answer
79 views

eigenvalues of sum of matrices: $A$+block matrix are strictly less than eigenvalues of $A$+identity

Let $A$ whose sum for rows is 0, can I prove that the $ \lambda_i \left ( \begin{bmatrix} I & 0\\ 0 & 0 \end{bmatrix} +A\right ) $ are strictly less than $\lambda_i \left ( \begin{bmatrix} I ...
0
votes
1answer
127 views

Zeros diagonal element of a semidefinite matrix leads to zeros row/column. Why?

I have a similar problem as in this question. In short words: Assume a square, positive semidefinite matrix $A\in\mathbb R^{n\times n}$. Show that if a diagonal element of $A$ is zeros then the ...
1
vote
3answers
142 views

eigenvalue of block matrix in terms of original matrix

A is a $4*4$ matrix with eigenvalues $\lambda_A$. Consider a block matrix $B = \left( \begin{array}{ccc} A & I \\ I & A \end{array} \right) $. Then how can we find eigenvalue $\lambda_B$ of ...
6
votes
3answers
159 views

Eigenvalues of block matricies

If the eigenvalues of a matrix $A$ are $\lambda_1,\lambda_2,\dots,\lambda_n$, what are the eigenvalues of the matrix? $\begin{bmatrix}0 &A\\A&0\end{bmatrix}$ From some numerical examples I ...
4
votes
2answers
208 views

nonegative inverse eigenvalue problem

I need to construct a non-negative matrix with desired eigenvalues. To that end, I came up with a block matrix of the following form: $$ \mathbf{M} = \begin{vmatrix} \mathbf{A} & \mathbf{b} \\ ...
0
votes
2answers
121 views

Eigenvalues of a block matrix

For $X=\left(\begin{array}{cc} A & B\\ C & 0 \end{array}\right)$ , how are eigenvalues of X related to eigenvalues of A ?
1
vote
2answers
153 views

Eigenvalue of anti triangular block matrix (skew matrix?)

I have an real anti-triangular matrix $M=\left[ \begin{array}{cc} A & B \\ I & 0 \\ \end{array} \right]$ where I is an identity matrix. $A$, $B$, $I$, $0$ are all square real ...
1
vote
2answers
197 views

Eigenvalues and Jordan form

I have a $5\times 5$ matrix and I need to find the Jordan form and its inverse. I know how to find the inverse. But for the Jordan form I am screwed. The matrix is $$\begin{bmatrix}3 & 0 & 0 ...
4
votes
2answers
158 views

Eigenvalues of Block matrices with known eigenvalues

Let's have the following $(n+1) \times (n+1)$ matrix with block elements $\mathbf{Y} = \begin{bmatrix} \mathbf{A} & - \mathbf{w} \\ - \mathbf{w}^{T} & b \end{bmatrix}$ where $\mathbf{A}$ is ...
0
votes
1answer
83 views

Perturbing/sensitivity of the eigenvalues of a block tridiagonal matrix

This question is related to my former question: Checking if one "special" kind of block matrix is Hurwitz Given the next matrix $$ J = \begin{bmatrix}-(B+B^T) & B \\ 0 ...
1
vote
1answer
118 views

Why a (general) block matrix does not have the same eigenvalues of the first block?

Based on the next relation: $$\det\begin{bmatrix}A & B \\ C & D\end{bmatrix} = \det(A)\det(D - CA^{-1}B),$$ I have that for computing the eigenvalues of the block matrix: ...
3
votes
1answer
242 views

Checking if one “special” kind of block matrix is Hurwitz

I have the next block matrix $$ J = \begin{bmatrix}A & B \\ K &0\end{bmatrix} $$ all matrices are square, where $A < 0$ (definite negative), $B$ has all its eigenvalues with positive real ...
5
votes
2answers
573 views

Eigenvalue decomposition of block covariance matrix for Canonical Correlation Analysis (CCA)

Edited: My question is related to a tutorial I was reading. The covariance matrix is a block matrix where $C_{xx}$ and $C_{yy}$ are within-set covariance matrices and $C_{xy} = C_{yx}^T$ are ...
2
votes
1answer
102 views

Is eigenvectors/eigenvalues of symetric matrix related with eigenvectors/eigenvalues of its block?

$A$ is square symmetric matrix as: $$A=\begin{pmatrix} A_1&A_2 \\ A_3 & A_4 \end{pmatrix}$$ I have two points which need help to understand clearly: All blocks $A_1$, $A_2$, $A_3$, $A_4$ ...
0
votes
0answers
198 views

Eigen values of sum of two matrices

I have a following block matrix \begin{align*} M_1 &= M_a + M_b\\\ M_a &= \begin{pmatrix} a_{11} & 0 &b_{13}\\\ 0 & a_{22}& b_{23}\\\ b_{13}^T & b_{23}^T & d_{33} ...
1
vote
0answers
84 views

Eigen value of a complex block matrix

I have a following block matrix M1 =([a11 a12;a21 a22] [b13;b23];[b13;b23]' d33) + ([c -cr]'*[c -cr] 0;0 I) now what i observe is that whether i use ...
0
votes
2answers
126 views

Eigenvalue of 'extended block' matrix

Thanks guys for the previous answer, Now suppose if I have a matrix e.g $$M_1 = \begin{pmatrix} \begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix} & B \\ B' & D ...
5
votes
2answers
197 views

Eigen value of a block matrix

I have following block matrices: $$M_1 = \left(\begin{array}{cc}A & B\\B' & D\end{array}\right)$$ and $$M_2 = \left(\begin{array}{cc}A & -B\\-B' & D\end{array}\right)$$ I want to ...
5
votes
1answer
773 views

Eigenvalues of certain block hermitian matrix

Suppose I have a special block, Hermitian matrix $H = \begin{bmatrix} A & B \\ B^* & A^* \end{bmatrix}$, where * denotes conjugate transpose. The blocks $A$ and $B$ are themself Hermitian in ...
0
votes
1answer
78 views

bilinear equation OR diagonal matrix search

Having a real square $n \times n$ symmetric matrix $B$ (which consists of 2 square blocks of positive numbers and 2 rectangular blocks of negative numbers) I want a real diagonal matrix $D$ such that ...