|
|
Functions of Matrices: Theory and Computation page 5:
Definition 1.4 (matrix function via Hermite interpolation). Let $f$ be defined on the spectrum of $A \in \mathbb{C}^{n \times n}$ and let $\psi$ be the minimal polynomial of $A$. Then $f(A):=p(A)$, where $p$ is the polynomial of degree less than
$$
\sum_{i=1}^{s} n_{i}=\operatorname{deg} \psi
$$
that satisfies the interpolation conditions
$$
p^{(j)}\left(\lambda_{i}\right)=f^{(j)}\left(\lambda_{i}\right), \quad j=0: n_{i}-1, \quad i=1: s .
$$
There is a unique such $p$ and it is known as the Hermite interpolating polynomial. |
An example is useful for clarification. Consider $f(t)=\sqrt{t}$ and
$$
A=\left[\begin{array}{ll}
2 & 2 \\
1 & 3
\end{array}\right] \text {. }
$$
The eigenvalues are 1 and 4 , so $s=2$ and $n_{1}=n_{2}=1$. We take $f(t)$ as the principal branch $t^{1 / 2}$ of the square root function and find that the required interpolant satisfying $p(1)=f(1)=1$ and $p(4)=f(4)=2$ is
$$
p(t)=f(1) \frac{t-4}{1-4}+f(4) \frac{t-1}{4-1}=\frac{1}{3}(t+2) \text {. }
$$
Hence
$$
f(A)=p(A)=\frac{1}{3}(A+2 I)=\frac{1}{3}\left[\begin{array}{ll}
4 & 2 \\
1 & 5
\end{array}\right] .
$$
It is easily checked that $f(A)^{2}=A$. Note that the formula $A^{1 / 2}=(A+2 I) / 3$ holds more generally for any diagonalizable $n \times n$ matrix $A$ having eigenvalues 1 and/or 4 (and hence having a minimal polynomial that divides $\psi(t)=(t-1)(t-4)$ )-including the identity matrix. We are not restricted to using the same branch of the square root function at each eigenvalue. For example, with $f(1)=1$ and $f(4)=-2$ we obtain $p(t)=2-t$ and
$$
f(A)=\left[\begin{array}{cc}
0 & -2 \\
-1 & -1
\end{array}\right]
$$ |
|
