Page 152 - Textos de Matemática Vol. 44

142 T. MORAN AND I. SPITKOVSKY
is unitarily similar to (1.1), in its turn A∗A−1 is (unitarily) similar to
 λ 0  λ −1 − β1  1 1 μλ1
 ... . ...
(6.9)  
β1
 0  
μ
 
=   
.  −1 βn−1 
λ n − 1 ... βn−1
−
λ n − 1 0 ... 0
11 μλ1
λn−1 /λn−1
β1/λ1 ... βn−1/λn−1 μ − 1 n−1 βj2
μλn−1 μ−1

−β1λ1 
λ /λ
...
. 
. − 
βn−1 λn−1 μλn−1 
A multiple eigenvalue ω of (6.9), if it exists, has to also be an eigenvalue of its left upper (n−1)-by-(n−1) submatrix, and thus coincide with one of λj/λj. By inspection it is easy to see that ω is an eigenvalue of (6.9) with geometric multiplicity m − 1, where m is the size of the cluster containing λj .
On the other hand, (6.9) is an arrow-head matrix, and it remains such af- ter subtracting a scalar multiple of the identity. Thus, for the characteristic polynomial we obtain:
μ μ j=1 λj
(6.10) det(A−1A∗ − ξI) = det(A∗A−1 − ξI)
n−1 n−1 2 
μ λj ξ βj λk =μ−ξ λ−ξ+μλ λ−ξ.
j=1 j j=1 j k̸=j k
The algebraic multiplicity of ω is therefore equal to m − 1 if condition (6.8) holds for the cluster containing λj, and is not smaller than m otherwise. Con- sequently, (6.8) is necessary and sufficient for geometric and algebraic multi- plicities of all the eigenvalues of A−1A∗ to coincide, that is, for the matrix to be diagonalizable.
Note that for any invertible A the characteristic polynomial of A−1A∗ is self-inversive, that is, satisfies the identity
f(z) = κzdeg f f(1/z) for some unimodular constant κ. Indeed,
det(A−1A∗ − z−1I) = det(A−1A∗ − z−1I)∗
= det(AA∗−1 − z−1I) = z−ndet A(det A)−1 det(A−1A∗ − zI).