[線代] Pseudoinverse of a Matrix
我想要問下面這個定理該怎麼證
給我一點trick也好
定理出自Friedberg 寫的Linear Algebra 第415頁
Thm6.29(證這個)
Let A be an m x n matrix of rank r with a singular value
decomposition A = UΣV* and nonzero singular values σ1≧σ2≧....≧σr
+
Let Σ be the n x m matrix defined by
+
Σij= 1/σi if i=k≦r
0 otherwise
+ + +
Then A = VΣ U* and this is a singular value decomposition of A
參考定理:Thm6.27(這個不用證)
Let A be an m x n matrix of rank r with a singular value
decomposition A = UΣV* and positive singular values σ1≧σ2≧....≧σr
Let Σ be the m x n matrix defined by
Σij= σi if i=k≦r
0 otherwise
Then exists an m x m unitary matrix U and an n x n unitary matrix V
such that A = UΣV*
謝謝!!
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