Re: [線代] 對角化(台大資工)
※ 引述《mqazz1 (無法顯示)》之銘言：
: 1. Let A in R^(3*3)
: [1 1 1] [2 0 0] T [x] [x]
: P = [2 3 3], if AP = P[0 1 0] and A [y] = [y]
: [1 3 4] [0 0 -1] [1] [1]
: then (x,y) = ?
: [ 1 2 3 4 ] 3 2
: 2. H = [ 5 6 7 8 ]. suppose H = aH + bH + cI, a,b,c in R
: [ 9 10 11 12 ]
: [13 14 15 16 ]
: (a,b,c) = ?
: 這題應該是C-H 可是很難算
: 請問有人知道這兩題怎麼解嗎?
: 謝謝
rank H = 2, so c=0 and
[1 0]
H = [1 1][1 2 3 4] = A*B
[1 2][4 4 4 4]
[1 3]
BA= [10 20]
[16 24]
with minimal polynomial t^2 - 34t -80=0.
since H^k=A(BA)^(k-1)B, so H^3=34H^2+80H.
so (a,b,c)=(34,80,0)
若有另一組解(a,b,c), 則 H^2=kH for some k.
這個算幾項就知道不可能了
PS.這題也可以找null space的基底來做
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