Let H be a complex n x n Hermitian matrix.
(a) Verify that x*Hx is in real for all x in complex.
(b) Show that if a Hermitian matrix H can be written as H = A*A for some invertible complex matrix A, then x*Hx > 0 for all nonzero vectors x in Complex.
Question about Hermitian matrix
2006-11-22 4:04 pm
回答 (1)
2006-11-22 4:39 pm
✔ 最佳答案
Hermitian matrix means H = H*Let < , > denote the inner product, and ~ denote complex conjugate.
(a)
Then (x*Hx)~
= < Hx, x >~
= < x, Hx > (conjugate of inner product means switching the order)
= (Hx)*x
= x*H*x
= x*Hx (H is hermitian)
so the conjugate equals itself, and hence it is a real number.
(b) x*Hx = x*A*Ax = (Ax)*(Ax) = ||Ax|| >0
參考: PhD Math
收錄日期: 2021-04-26 18:00:31
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