每週問題 December 21, 2015

這是計算一線性變換的特徵值與特徵向量。

Let A be an n\times n matrix, and T be the linear transformation defined by T(A)=\frac{A+A^T}{2}. For n>1, find the eigenvalues and corresponding eigenvectors of T.

 
參考解答:

寫出特徵方程 T(A)=\frac{A+A^T}{2}=\lambda A,等號兩邊取轉置,\frac{A+A^T}{2}=\lambda A^T,故 \lambda A=\lambda A^T。若 \lambda\neq 0,則 A^T=A,並推得 \lambda=1。若 \lambda=0,則 A^T=-A。令 \mathcal{S}\mathcal{K} 分別表示 n\times n 階對稱矩陣和反對稱矩陣形成的子空間。因此,T 有特徵值 1,對應的特徵空間為 \mathcal{S},重數為 \dim\mathcal{S}=\frac{n^2+n}{2},以及特徵值 0,對應的特徵空間為 \mathcal{K},重數為 \dim\mathcal{K}=\frac{n^2-n}{2}

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