25.  Which of the following statements are correct?

(A) If Q is orthogonal, then det(Q) = ±1. 

(B) Let A be a real n x n matrix. Then A is symmetric if and only if A is orthogonally equivalent to a real diagonal matrix.

(C) Let A ∈ be a matrix whose characteristic polynomial splits over . Then A is orthogonally equivalent to a real upper triangular matrix. 

(D) Let T be a self-adjoint (Hermitian) operator on a finite-dimensional inner product space V. Then every eigenvalue of T is positive.

(E) Let T be a self-adjoint (Hermitian) operator on a finite-dimensional inner product space V. Then every eigenvalue of T is negative.

詳解 (共 1 筆)

助人為本

B1 · 2026/06/21

#7415430

所以你只要記住

如果A是可正交化的，則det值會是正負1

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如果是實對稱矩陣

代表是可以正交化且可對角化

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如果特徵多項式可以在R上完全分解

意思是所有的eigenvalues都是real

這時候A就可以被正交化矩陣化成實對稱上三角矩陣

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荷米特矩陣可以證明eigenvalues are real

不代表其是positive和negative

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