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

Sufficient conditions for invertibility of a block tridiagonal matrix

Let Mn∈RN×N be a block-tridiagonal matrix: Mn=[B1C100⋯0A1B2C20⋯00A2B3C3⋯000A3B4⋱⋮⋮⋮⋮⋱⋱Cn−100⋯0An−1Bn] where each Bi∈Rmi×mi is square and invertible, with varying sizes; Ai and Ci may not be square. Problem What are sufficient conditions on Ai, Bi and Ci for showing that Mn is invertible? Strategies The following is a list of strategies for approaching the problem; i.e. starting points. … 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

Iterated calculation of determinants

Given a $4 \times 4$ matrix $S$ over a commutative ring $R$. I want to consider it as a $2\times 2$ matrix over $M_2(R)$. Lets say $S=\left(\begin{array}{cc} A&B \\\ C&D\end{array}\right)$ with entries $A,B,C,D\in M_2(R)$. Asssume further, that they all commute. So my question is: Can one then compute the determinant stepwise via $\det(S)=\det(AD-BC)$. I checked … 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

A rank inequality

Suppose M:=[M11⋯M1d⋮⋱⋮Md1⋯Mdd] is a d×d block matrix such that Mjk=r∑i=1(Ai)jkBi for some Ai∈Md(C), Bi∈Mn(C) and d,n,r>2. Now, let M◻:=[MT11⋯MT1d⋮⋱⋮MTd1⋯MTdd] where MTjk=∑ri=1(Ai)jkBTi. Is the following inequality true? rank(M◻)rank(M)≤r For r=1,2 this statement is true! Answer AttributionSource : Link , Question Author : SMD , Answer Author : Community