proof of existence and uniqueness of singular value decomposition
Proof of existence and uniqueness of SVDFernando Sanz Gamiz
To prove existence of the SVD, we isolate the direction of the largest action of , and then proceed by induction on the dimension of . We will denote hermitian conjugation by . Norms for vectors in will be the usual euclidean 2-norm and for matrix the induced by norm of vectors.
Let . By a compactness argument, there must be vectors with and . Normalize by setting and consider any extensions of to an orthonormal basis of and of to an orthonormal basis of ; let and denote the unitary matrices with columns and respectively. Then we have
so that . But and are unitary matrix, hence ; it therefore implies .
is an SVD of . completing the proof of existence.
For the uniqueness let a SVD for and let denote the i-th, vector of the canonical base of . As and are unitary, , so each is uniquely determined.
|Title||proof of existence and uniqueness of singular value decomposition|
|Date of creation||2013-03-22 17:07:46|
|Last modified on||2013-03-22 17:07:46|
|Last modified by||fernsanz (8869)|