I am interested in the properties of a matrix with elements different from zero only on the main diagonal and antidiagonal, like this:
$$\begin{bmatrix} a & 0 & 0 & h \\ 0 & b & g & 0 \\ 0 & f & c & 0 \\ e & 0 & 0 & d \\ \end{bmatrix}$$
For example this matrix is the result of a power series based on an antidiagonal matrix C:
$$Z = I + C + C^2 + C ^3 + \cdots$$
Has this kind of matrix some useful property? with respect to
- diagonalization
- eigenvalues eigenvectors
- inverse
Thanks, any information will come in handy.
Let $A$ be a matrix (with size $n$) of the above form. Let $P$ denote the permutation matrix whose columns are $e_1,e_n,e_2,e_{n-1},\dots$. We note that $$ P^TAP = \pmatrix{ A_1 \\ &A_2\\ &&\ddots\\ &&& A_k } $$ where $k = \lceil n/2 \rceil$. If $n$ is odd, then $A_{n}$ is simply a number. In all other cases, $A_k$ is a $2 \times 2$ square matrix.
What we find that is, for every eigenvector of $A_k$, there is a corresponding eigenvector of $A$. Also, for any $m \in \Bbb N$, we have $$ A^m = P \pmatrix{ A_1^m \\ &A_2^m\\ &&\ddots\\ &&& A_k^m } P^T $$ note that $A$ is invertible iff each $A_k$ is invertible. In this case, we may also take $m$ to be negative.
Suppose that $u$ is an eigenvector of $A_k$. Then define the block vector $$ v= \pmatrix{ 0\\ \vdots \\ 0\\ u \\ 0 \\ \vdots \\0} $$ The vector $Pv$ is an eigenvector of $A$. In fact, these give us a complete set of eigenvectors.