Array sensor Adaptive and Array Signal Processing Homework03 Solved

1. Given the SVD of a matrix A ∈ Cm×n

                              A = UΣVH                    (1)
with the unitary matricesr×r U ∈ Cm×m, V ∈ Cn×n and the positive definite diagonal matrix Σ1 ∈ R , where r = rank(A), the Moore-Penrose pseudo inverse A+ of A is defined as

                                                               A                                                    (2)

(a)    What are the dimensions of the matrices U1, U2, V1, V2, Σ and A+ ?

(b)   Show that U            and V 

(c)    Show that (2) satisfies the four Moore-Penrose conditions for a pseudo inverse

AA+A
=
A
(3)
A+AA+
=
A+
(4)
                                                                         AA+ = AA                                                (5)

                                                                        A+A = A                                                (6)
2.   Look at the matrix A and its pseudo inverse A+

A  ; A 

in terms of four real numbers a,b,c,d ∈ R.

(a)    Write down the three linear equations of the variables a,b,c,d that follow from (3), (5) and (6) in the form:

                                                              D e                                           (7)

(Hint: since we are dealing with real numbers, the (•)H operator becomes a pure transposition)

(b)   Compute the general solution of the underdetermined system (7) in terms of d.

(c)    Now determine the unique value d, that will also satisify the last Moore-Penrose condition (4)

(d)   Write down the pseudo inverse A+ you have obtained in this way.

2

(e)    A SVD of A is given as

 

Compute A+ by the definition in (2) and compare to the previous result.

3.   Consider the matrix

                                                          P                                                 (8)

(a)    Show that P is a projector onto a vector space.

(b)   This vector space S = range(P) is a subspace of C3. What is its dimension?

(c)    What is the dimension of the orthogonal complement S⊥ ⊂ C3 of S in C3?

(d)   Compute the projector P⊥ onto S⊥

(e)    Using this result compute an orthonormal base of S⊥

(f)    Have a look at the following subspace

                                                        S2 = range                                       (9)

Is S2 = S ? (Hint: Compute the projector onto S2 and compare to (8).)

4.   The system Ax = b with A ∈ Cm×n, x ∈ Cn and b ∈ Cm has an exact solution only if b ∈ range(A). If m n (i.e. more equations than unknowns) there is usually no exact solution, since b /∈ range(A) in most cases. The best we can do is modify the righthand have a SVD as in (1) and define a projector Pbas side of the system such that b is replaced by b, its projection onto the range of A. Let A

                                                                    P                                                       (10)

Show that

(a)    the least-squares solution obtained by the pseudo inverse, i.e. xLS = A+b, is really an exact solution to the modified system Ax Pb

(b)   the error   of the righthand sides is orthogonal to bb.