Binomial formula for matrix $(\mathbf{I}+\mathbf{A})^{-1/2}$.

Question

I am wondering if the following binomial expansion for the matrix is correct: $$ (\mathbf{I}+\mathbf{A})^{-1/2}=\sum_{n=0}^{\infty}\left(\begin{array}{c} -1/2 \\ n \end{array}\right) \mathbf{A}^n, $$ where $\mathbf{I}$ is a unit matrix. If right, is there any existing literature to support the result?

Answer 1

(I assume to be in $\mathbb{C}^N$.) If $A$ is normal and the greatest absolute value of the eigenvalues of $A$ is less than $1$, then the identity is true.

The left hand side is defined spectrally. If $A=\sum_l l P_l$ is the spectral decomposition of $A$, where $l$ are its eigenvalues and $P_l$ the orthogonal projectors on the associated eigenspaces, $$(I+A)^{-1/2}:= \sum_l (1+l)^{-1/2} P_l\:.$$ The proof of the wanted identity is trivial using $A= UDU^{-1}$ where $D$ is a diagonal matrix containing the eigenvalues of $A$ (with their multeplicities) on the diagonal and $U$ a suitable unitary matrix (it exists just because $A$ is normal). The series converges absolutely by trivial use of the uniform norm (this norm is the absolute value if the greatest eigenvalue of $A$), but all norms are equivalent in a finite dimensional complex linear space.

(The identity also holds in C* algebras for normal elements and thus also for bounded normal operators in Hilbert spaces, in all cases for $||A||<1$.)