Spectral radius of the product of a right stochastic matrix and a block diagonal matrix

Let us define the following matrix: $C=AB$ where $B$ is a block diagonal matrix with $N$ blocks, $B_1$, $B_2$ … $B_N$, each of dimensions $M \times M$. I know that $B_k = I_M – \mu R_k$ with $R_k$ equals to a hermitian matrix and $\mu$ some positive constant. Moreover, I know that the the entries … Read more

How to find eigenvalues of following block matrices?

Is there a procedure to find the eigenvalues of A? ‎ A=[XI0I0PPt0II0PPt0⋱⋱⋱⋱⋱0II0PPt0I0k×y−kI00y−k×kYy×y] (All rows and columns except the last are k×k blocks.) where A is adjacency matrix of cubic graphs and X,P are circulant matrices of order k, Y is a matrix of order y, k≠y and I is a identity matrix and 0 is … Read more

The normalizer of block diagonal matrices

Let G=Un or GLn(C) and H the subgroup of block diagonal matrices respecting a partition n=n1+⋯+nk. Is the normalizer N=NG(H) computed anywhere in the literature? I guess, but haven’t proved, that it is generated by H and the permutations (“transpositions”) exchanging the partition’s same-length segments (ni=nj, if any). I also suspect this may be discussed … Read more

Determinant of block tridiagonal matrices

Is there a formula to compute the determinant of block tridiagonal matrices when the determinants of the involved matrices are known? In particular, I am interested in the case $$A = \begin{pmatrix} J_n & I_n & 0 & \cdots & \cdots & 0 \\ I_n & J_n & I_n & 0 & \cdots & 0 … Read more

Upper bound for ‖\|\textbf{D}^{-1}\|, where \textbf{D}\textbf{D} is a matrix with specific sparse pattern

Consider the block matrix given by \textbf{D} = \left[ \begin{array}{ccc} \left[ \begin{array}{ccc} D & \ldots & D\\ \vdots & \ddots & \vdots\\ D & \ldots & D\\ \end{array}\right] & \textbf{X} & \textbf{X}\\ \textbf{X} & \left[ \begin{array}{ccc} D & \ldots & D\\ \vdots & \ddots & \vdots\\ D & \ldots & D\\ \end{array}\right] & \textbf{X}\\ \textbf{X} … Read more

Partitioned inverse 3×3 block matrix

We know that matrices can be inverted blockwise by using the following analytic inversion formula: [ACTCD]−1=[A−1+A−1CTSD−1CA−1−A−1CTSD−1−SD−1CA−1SD−1] with SD=D−CA−1CT the Schur complement of the block D or alternatively, [ACTCD]−1=[SA−1−SA−1CTD−1−D−1CSA−1D−1+D−1CSA−1CTD−1] with SA=A−CTD−1C the Schur complement of the block A \ what about the case when we have a 3×3 partitioned matrix as following: [PCTCD]−1 where P=[XYYTZ]? To … Read more

Factorizing a block symmetric matrix

Let $X,Y\in\mathbb{R}^{n\times n}$ be symmetric matrices. You may assume that $X$ is positive semidefinite and $Y$ negative semidefinite, if needed, but not that they are invertible. I would like to find a way to factor the $2n\times 2n$ block matrix $$ \begin{bmatrix} X & I\\\\ I & Y \end{bmatrix} $$ into some form of the … Read more

Block matrices and their determinants

For n∈N, define three matrices An(x,y),Bn and Mn as follows: (a) the n×n tridiagonal matrix An(x,y) with main diagonal all y‘s, superdiagonal all x‘s and subdiagonal all −x‘s. For example, A4(x,y)=(yx00−xyx00−xyx00−xy). (b) the n×n antidigonal matrix Bn consisting of all 1‘s. For example, B4=(0001001001001000). (c) the n2×n2 block-matrix Mn=An(Bn,An(1,1)) or using the Kronecker product Mn=An(1,0)⊗Bn+In⊗An(1,1). … Read more

Off-diagonalize a matrix

Consider a self-adjoint matrix $M$ that has block form $$M = \begin{pmatrix} M_{11} & M_{12} \\ M_{12}^* & M_{11} \end{pmatrix}.$$ I am wondering if there exists any criterion to decide if this matrix can be transformed by some invertible matrix $T$ such that $$TMT^{-1} = \begin{pmatrix}0 & C \\ C^* & 0 \end{pmatrix}$$ for some … Read more