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Unformatted text preview: ECE 3250 SINGULARVALUE DECOMPOSITION Fall 2008 In what follows, I’ll be relying on a few essential facts from linear algebra. Some of them are easier to prove than others. As always, F denotes R or C . By F m × n I will mean the set of all ( m × n ) matrices with entries in F . If A ∈ F m × n , A H is the conjugate transpose, or Hermitian conjugate, of A . Note that A H ∈ F n × m . A necessarily square matrix Q is a Hermitian matrix if and only if Q H = Q . If A ∈ F m × n , the nullspace of A is the set of all x ∈ F n such that Ax = 0. Note that the nullspace of A is a subspace of F n . The range of A is the set of all y ∈ F m such that y = Ax for some x ∈ F n . The range of A is the range of the linear mapping x 7→ Ax . The range of A is a subspace of F m . Now for the facts we’ll need. Fact 1: If A ∈ F m × n , then rank of A = n (dimension of the nullspace of A ) . One way to prove this fact is by analyzing the equation Ax = y by means of Gauss elimi nation. Fact 2: If A ∈ F m × n , then the rank of A H A is the same as the rank of A . To see this, first observe that A and A H A have the same nullspace because A H Ax = 0 = ⇒ x H A H Ax = 0 ⇐⇒ k Ax k 2 = 0 ⇐⇒ Ax = 0 , so the nullspace of A H A is contained in the nullspace of A . The reverse inclusion is obvi ous. Now apply Fact 1 to conclude that the rank of A and the rank of A H A are equal. Fact 3: If Q ∈ F n × n is Hermitian, then all the eigenvalues of Q are real. To see this, let x ∈ F n be an eigenvector of Q corresponding to eigenvalue λ o . Because Q is Hermitian, x H Qx = ( x H Qx ) H = x H Q H x = x H Qx . Meanwhile, x H Qx = λ o k x k 2 and x H Qx = λ o k x k 2 , from which it follows that λ o = λ o because x 6 = 0. Fact 4: If A ∈ F m × n , then all the eigenvalues of A H A are real and nonnegative. To see why, note first that A H A is Hermitian, so all its eigenvalues are real by Fact 3. If λ o is an eigenvalue of A H A and x a corresponding eigenvector, then ≤ k Ax k 2 = x H A H Ax = λ o k x k 2 , which implies that λ o ≥ 0 because x 6 = 0. Recall that C n is an innerproduct space with inner product h v,w i = w H v for all v,w ∈ C n . In what follows, when I use the word “orthonormal” I’ll be referring to orthonormality with respect to that inner product. Furthermore, I’ll state all remaining results in terms of complex matrices and vectors. 1 2 Fact 5: If Q ∈ C n × n is Hermitian, then there exists an orthonormal basis { u 1 ,u 2 ,...,u n } for C n consisting solely of eigenvectors for Q . Proving this fact is not straightforward, but you’ve all seen it before. Now for the singularvalue decomposition. Let A ∈ C m × n have rank r . By Fact 2, the Hermitian matrix A H A also has rank r , so its nullspace has dimension n r by Fact 1....
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This note was uploaded on 08/10/2010 for the course ECE 4370 at Cornell.
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