Re: [線代] 幾個eigenvalue的觀念

[Vulpix (Sebastian)]

※ 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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◆ From: 111.248.5.30
推 handsboy    :第二題看不懂有人可以解釋一下嗎QQ
推 s3300046    :det(A+I)=6!=>不等於0，所以會有反矩陣
→ ert0700     :或者因為0不是特徵值 所以 ker(A)={0}
→ LiamIssac   :positive eigenvalues means invertibility
→ handsboy    :其實是看不懂det(A+I)=6!  QQ
→ ricestone   :他是寫不等於