0
votes
0answers
10 views

Issues with connecting the SVD and Eigenvalues for block matrix

In class, we have talked about the singular value decomposition and its connection to Eigenvalues. Specifically, for a matrix A, if the columns of a matrix contain linearly independent eigenvectors, ...
1
vote
0answers
33 views

Computations for LDA: Eigendecomposition

While reading the book Elements of Statistical Learning p. 113, the author used eigendecomposition of the covariance matrix $\hat{\Sigma}_k =\mathbf{U}_k\mathbf{D}_k\mathbf{U}_k^T$ where ...
1
vote
2answers
56 views

How to find eigenvalues of the following block circulant matrix

I have a block matrix of size PN x PN of the form: Where A and C are P x P matrices. I would like to find the eigenvalues of the matrix B, that is where
1
vote
0answers
81 views

Same eigenvalue spectrum with different matrices

There are two matrices $D$ and $T$ which provides eigenvalue spectrum (dispersion relation) according to $$ E(K) = \operatorname{eig}(D + T \exp(iK) + T' \exp(-iK)) $$ $$ K = 0:dK:\pi $$ Where K is a ...
0
votes
0answers
20 views

Diagonalization of sparse block matrix

I have a real symmetric matrix, \begin{equation} \left( \begin{array}{ccc} 0 & M & M' \\ M ^T & 0 & 0 \\ M ^{ \prime T} & 0 & 0 \end{array} \right) \end{equation} ...
1
vote
2answers
42 views

Estimate eigenvectors of symmetric matrix with almost vanishing diagonal

Is there a way to approximate the eigenvectors of a symmetric matrix with almost vanishing diagonal elements, i.e. with the block matrix form, \begin{equation} M=\left( \begin{array}{cc} ...
1
vote
1answer
88 views

Finding eigenvalues of a block matrix

I have a block matrix of size $2N \times 2N$ of the form $$B = \begin{bmatrix} A_N & C_N \\ C_N & A_N \end{bmatrix}$$ where $A_N$ and $C_N$ are both $N \times N$ matrices. Specifically, $$A_N ...
2
votes
1answer
83 views

Block diagonalizing two matrices simultaneously

There are two matrices $A$ and $B$ which can not be diagonalized simultaneously. Is it possible to block diagonalize them? What if the matrices have an special pattern? Physics of the problem is ...
0
votes
0answers
12 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
27 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
170 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
86 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
158 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
150 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
161 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} \\ ...
1
vote
2answers
127 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
168 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
203 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
163 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
85 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
121 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
250 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
620 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
104 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
200 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
85 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
127 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
200 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
795 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
79 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 ...