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Extra info for Introduction to quantum computation and quantum information
Clearly p+ + p− + p0 = n. The integer sgn A = p+ − p− is called the signature of A. Note that p− + p+ is the rank of A, so that rank and signature of an Hermitian matrix determine its inertia. Two Hermitian n × n matrices A and B are called congruent if there is a nonsingular matrix C such that B = C ∗ AC, where C ∗ denotes the conjugate transpose of C. The following is Sylvester’s inertia law, which will frequently be applied in this paper. 1. Congruent matrices have the same inertia. We will often apply the following version of Sylvester’s inertia law.
Then the matrix T (uJ ) is nonsingular. 2,1 can be written in the form C = T (uJ )−1 Jn BezH (u, v) = T (v)T − T (vJ )T (uJ )−1 T (u)T . We see that C is the Schur complement of the left upper block in R = Res (u, v) O In In O = T (uJ ) T (u)T T (vJ ) T (v)T . Recall that the concept of Schur complement is deﬁned in connection with the factorization of a block matrix In A−1 B O In A O A B = , G= −1 −1 CA O In In O D − CA B C D where A is assumed to be invertible. Here D − CA−1 B is said to be the Schur complement of A in G.
A matrix B is called quasi-H-Bezoutian if rank ∇H B ≤ 2. We give a general representation of quasi-H-Bezoutians that is also important for H-Bezoutians. 4. 9) (t) are coprime and r ≤ n . 3). Proof. For B is a quasi-H-Bezoutian, there exist a, b, c, d ∈ Fn+1 such that (t − s)B(t, s) = a(t)d(s) − b(t)c(s) . Since for t = s the left-hand side vanishes, we have a(t)d(t) = b(t)c(t). Let p(t) be the greatest common divisor of a(t) and b(t) and q(t) the greatest common divisor of c(t) and d(t). Then a(t) = p(t)u(t) and b(t) = p(t)v(t) for some coprime u(t), v(t) ∈ Fr+1 (t) (r ≤ n).