12.10. APPROXIMATION IN THE FROBENIUS NORM 315

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 0

0 0

)= number of singular values.

Also since U, V are unitary,

rank (A∗) = rank (V ∗A∗U) = rank((U∗AV )

∗)= rank

((σ 0

0 0

)∗)= number of singular values. ■

12.10 Approximation in the Frobenius Norm

The Frobenius norm is one of many norms for a matrix. It is arguably the most obvious ofall norms. Here is its definition.

Definition 12.10.1 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∑

ij |aij |2so essentially, it treats the matrix as a vector

in Fm×n.

Lemma 12.10.2 Let A be an m× n complex matrix with singular matrix

Σ =

(σ 0

0 0

)

with σ as defined above, U∗AV = Σ. Then

||Σ||2F = ||A||2F (12.15)

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 . (12.16)

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 (AV V ∗A∗) = trace (AA∗) = ||A||2F .

It follows∥Σ∥2F = ||U∗AV ||2F = ||AV ||2F = ||A||2F . ■

Of course, this shows that

||A||2F =∑i

σ2i ,

the sum of the squares of the singular values of A.