Re: [線代] 幾個eigenvalue的觀念
※ 引述《mqazz1 (無法顯示)》之銘言：
: students A and B were asked to solve the eigenvalues of the same matrix M
: [a b c]
: = [0 d 1]. Unfortunately, Student A mistook the value of d and obtained the
: [0 2 e]
: eigenvalues 0, 1, 3. Student B mistook the value of e and obtained the
: eigenvalues 1, 1, -2.
: (1) find the value of a 請問為什麼可以看出a=1?
a一定是eigenvalue，而兩個人算出的eigenvalue中重疊的只有1
: ==========================================
: (2) If A is a 3*3 matrix with 3 distinct eigenvalues 0,1,2,
: then the matrix (A+I) must be invertible
: true 請問為什麼?
A+I的eigenvalue是1,2,3
det(A+I)=6 != 0
: (3) An n*n matrix with n linearly independent eigenvectors is invertible
: False 請問為什麼?
0方陣可以找到足夠多線性獨立的eigenvector
對於是否invertible，真正的重點是eigenvalue有沒有0
: (4) If A is an n*n diagonalizable matrix, then each vector in R^n can be
: written as a linear combination of eigenvectors of A
: true 請問為什麼?
可以被對角化的矩陣，一定有n個線性獨立的eigenvector
這些eigenvector會span整個空間
: 謝謝
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