I have a block matrix T:
\begin{equation*} T=\begin{pmatrix} A & H \\ 0 & B \end{pmatrix} \end{equation*} \begin{equation*} A_{n,n}, B_{m,m}, H_{n,m}, 0_{m,n}. \end{equation*}
I need to find a function from a given matrix. For example, $2x+3$. I found the following formula: \begin{equation*} f(T)=\begin{pmatrix} f(A) & f^{[1]}(A,B)\diamond H \\ 0 & f(B) \end{pmatrix} \end{equation*} According to the task, you need to find the function from the matrix in two ways:
- Just calculate the function from the entire matrix (i.e. do the operation f(T))
- Do manipulations with each block as according to the formula.
And here I don't understand the formula $f^{[1]}(A,B)\diamond H$. Yes, this is a divided difference, that's understandable. But how can this divided difference be applied to matrices? Can someone describe how to do this? Thanks.
If you have a general polynomial matrix function $f(x) = \sum_{n =0}^p a_n x^n$ then you need a way to calculate $T^n$ for an integer $n \ge 0$.
You can use induction to find a recursive relationship.
Start with the following sequence
$$ \begin{aligned}T^{0} & =\begin{bmatrix}1 & 0\\ 0 & 1 \end{bmatrix}\\ T^{1} & =\begin{bmatrix}A & H\\ 0 & B \end{bmatrix}\\ T^{2} & =\begin{bmatrix}A & H\\ 0 & B \end{bmatrix}\begin{bmatrix}A & H\\ 0 & B \end{bmatrix}=\begin{bmatrix}AA & AH+HB\\ 0 & BB \end{bmatrix}\\ T^{3} & =\begin{bmatrix}A^{2} & AH+HB\\ 0 & B^{2} \end{bmatrix}\begin{bmatrix}A & H\\ 0 & B \end{bmatrix}=\begin{bmatrix}A^{3} & A^{2}H+\left(AH+HB\right)B\\ 0 & B^{3} \end{bmatrix}\\ \vdots \end{aligned} $$
and by designating $H_n$ the upper right block matrix of $T^n$ then we have
$$\begin{aligned}T^{n-1} & =\begin{bmatrix}A^{n-1} & H_{n-1}\\ & B^{n-1} \end{bmatrix}\\ T^{n} & =\begin{bmatrix}A^{n-1} & H_{n-1}\\ & B^{n-1} \end{bmatrix}\begin{bmatrix}A & H\\ & B \end{bmatrix}=\begin{bmatrix}A^{n} & A^{n-1}H+H_{n-1}B^{n-1}\\ & B^{n} \end{bmatrix} \end{aligned}$$
which leads to the following recurrence relationship, with initial conditions $H_0 = 0$ and $H_1 = H$
$$ \boxed{ H_{n}=A^{n-1}H+H_{n-1}B^{n-1} }$$
to be used in
$$ T^n =\begin{bmatrix}A^n & H_n\\ 0 & B^n \end{bmatrix} $$
or specifically
$$ f(T)=\sum_{n=0}^{p}a_{n}T^{n}=\sum_{n=0}^{p}\begin{bmatrix}a_{n}A^{n} & a_{n}H_{n}\\ 0 & a_{n}B^{n} \end{bmatrix} $$