Re: [線代] 線代(12)
※ 引述《PttFund (批踢踢基金只進不出)》之銘言：
: (a) Prove that the eigenvalues of a Hermitian matrix (A* = A) are
: all real.
consider that Ax=λx where λ is eigenvalue of A and x is the corresponding
eigenvector
take * ont the both side, ie, x*A*=λ*x* (a)
since A is Hermitian, x*A=λ*x*
multiply x on the both side, x*Ax=λ*x*x
x*λx=λ*|x|, λ|x|=λ*|x|
since |x|=\=0 , that is λ=λ*
ie, all the eigenvalue of a Hermitian matrix are all real
Q.E.D.
: (b) What can you say about the eigenvalues of a unitary matrix
: (A*A = I)? Prove your assertion.
according to (a), x*A^-1=λ*x*
multiply x on the both side, x*A^-1x=λ*x*x
x*(1/λ)x=λ*|x|
(1/λ)|x|= λ*|x|
since |x|=\=0, λ*λ=1
Q.E.D.
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