[線代] 關於rank(A^T A)=rank(A)的證明?
關於rank(A^T A)=rank(A)for any A m×n的證明如下:
Since elementary operations do not change the rank of a matrix. We have
rank(ATA)=rank(ETATAE)
, where E is a multiplication of several elementary operations which make
AE=[A1,A2], where A1 is a column full rank matrix with rank(A1)=rank(A).
Thus we can find a matrix B such that A1B=A2
and AE=[A1,A1P]=A1[I,P]
Thus rank(ETATAE)=rank(A1[I,P])T(A1[I,P])
In this equation, the four matrices are all full rank and the rank equals
rank(A) , so rank(ATA)=rank(A), completing the proof.
原文網址:http://goo.gl/A4Mg3l
我有疑問的是:
1.A是 m×n rank(A)可能小於min(m,n) AE=[A1,A2]這一定辦的到嗎?
2.為何rank(A1[I,P])T(A1[I,P])會等於rank(A)?
以上兩點請高手指點迷津
謝謝
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