374 CHAPTER 13. MATRICES AND THE INNER PRODUCT
Corollary 13.15.4 Let A be an m× n matrix. Then the rank of both A and A∗equals thenumber of singular values.
Proof: Since V and U are unitary, they are each one to one and onto and so it followsthat
rank(A) = rank(U∗AV ) = rank
(σ 00 0
)= number of singular values.
Also since U,V are unitary,
rank(A∗) = rank(V ∗A∗U) = rank((U∗AV )∗
)= rank
((σ 00 0
)∗)= number of singular values. ■
13.16 Approximation In The Frobenius NormThe Frobenius norm is one of many norms for a matrix. It is arguably the most obvious ofall norms. Here is its definition.
Definition 13.16.1 Let A be a complex m×n matrix. Then
||A||F ≡ (trace(AA∗))1/2
Also this norm comes from the inner product
(A,B)F ≡ trace(AB∗)
Thus ||A||2F is easily seen to equal ∑i j∣∣ai j∣∣2 so essentially, it treats the matrix as a vector in
Fm×n.
Lemma 13.16.2 Let A be an m×n complex matrix with singular matrix
Σ =
(σ 00 0
)
with σ as defined above, U∗AV = Σ. Then
||Σ||2F = ||A||2F (13.31)
and the following hold for the Frobenius norm. If U,V are unitary and of the right size,
||UA||F = ||A||F , ||UAV ||F = ||A||F . (13.32)
Proof: From the definition and letting U,V be unitary and of the right size,
||UA||2F ≡ trace(UAA∗U∗) = trace(U∗UAA∗) = trace(AA∗) = ||A||2F
Also,||AV ||2F ≡ trace(AVV ∗A∗) = trace(AA∗) = ||A||2F .