1.If the real matrix A is m*n and has the property A^T*A=I, then for any vector x in Rm, x-AA^Tx is orthogonal to the column space of A.這題是true
這題看來是正交矩陣，但我推不大出來為何正交…因為是正交矩陣…所以x-AA^Tx應該是屬於N(A^T)或N(A)嘛？
2.If A is an n*n real matrix such that x^TAx = 0, for all x in Rn, then A=0. 這題是false
這是因為x有可能是0對嗎？若加上條件x≠0，應該就成立吧？
14 則留言:
1.
其實是因為A^T*A=I 這句話 才知道是正交矩陣 你可以翻翻筆記 老師那時候有畫一個表格的
1. A^T(x-AA^Tx)
= (A^T)x - (A^T)A(A^T)x
= (A^T - (A^T)A(A^T))x
= (A^T - A^T)x
= 0
=> x-AA^Tx is in N(A^T)
Since N(A^T) is the orthogonal complement of R(A),
x-AA^Tx is orthogonal to the column space of A
2. It's still false even when it's "for all x≠0", ex:
Given any x=[x1 x2]^t, let A=[0 1; -1 0]
then x^tAx = -x1x2+x1x2 = 0
請問怎麼發文阿
我找了好久
抱歉在這發問...
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1.感謝助教的回覆，我有想到垂直N(A^T)，卻沒想到拿來乘A^T，殘念…
2.請問第2題這個x^T*A*x，好像是在算正(半)定時常會用的東西，而A≠0有什麼定理或課本哪裡有相關資料可以參考嗎
sorry~~又新增一個問題，麻煩幫幫忙
1.Suppose A is symmetric positive definite and Q is an orthogonal matrix. T or F
(a)Q^T*A*Q is a diagonal matrix這題為何是false呢？
(b)Q^T*A*Q is symmetric positive definite這題是true，是指正定對稱矩陣乘正交矩陣仍為正定對稱矩陣嗎？書上好像找不到相關的定理
(c)Q^T*A*Q has the same eigenvalues as A這題是true，這應該就是可對角化，所以有eigenvalue吧，可是Q是正交矩陣又不一定是A的eigenvector
(d)e^-A is symmetric positive definite，這題也是True，這好像也是對角化的應用，麻煩高手幫忙解答一下，感謝
(a) A is orthogonal diagonalizable, but in here, the columns of Q might not be the orthonormal eigenvectors of A
(b) If you couldn't come up with something you've learned, you can try to prove it, just like both of the problems that you asked before
Since A is symmetric and positive definite => ((Q^T)AQ)^T=(Q^T)AQ and (x^T)((Q^T)AQ)x=((Qx)^T)A(QX)>0, therefore (Q^T)AQ is symmetric positive definite
(c) Let P=Q, B=(Q^T)AQ, then P^-1AP=B, thus A is similar to B, in this case they will have the same eigenvalues
(d) it's similar to (b) because of the property of similarity
不好意思 我也想請問一下
1.If the null space of A and B are the same, then A is row equivalent to B 為甚麼答案是False
2.If a vector space is spanned by a set of n vectors, then every set of more than n vectors must be linearli dependent.答案是True
我的想法是在R3*3中 V=span{e1,e2}但{e1,e2,e3}:LI 所以是false 請問為甚麼這題是True?
感謝助教！！
1. This one is TRUE, since N(A)=N(B) => R(A^T)=R(B^T)
2. I think what it means is that every set of more than n vectors "in V" must be linearly dependent, anyway you are right, the question is not clear
嗯嗯~~了解了~~下次我會試著自己證證看的~~謝謝助教
不好意思我想請問一下
資工所第二次模擬考數學的第三題
題目說要find all solutions
是否應該要寫通解
可是答案只有70跟37
我覺得有點疑惑
謝謝
這裡寫 x=70 (mod 71) 的意思 x 的解集合為 {...,-1,70,141,...}, 同裡, y 的解集合為 {...,-34,37,108,...}, 所以解答確實有把所有的解寫出來
那若是我的答案寫x=70+71t
y=37-71t
這樣是否可以?
可以
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