[Solved] Array-sensor Homework03

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  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)

  • What are the dimensions of the matrices U1, U2, V1, V2, Σ and A+ ?
  • Show that U and V
  • 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)

  1. Look at the matrix A and its pseudo inverse A+

A ; A

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

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

De (7)

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

  • Compute the general solution of the underdetermined system (7) in terms of d.
  • Now determine the unique value d, that will also satisify the last Moore-Penrose condition (4)
  • Write down the pseudo inverse A+ you have obtained in this way.

2

  • A SVD of A is given as

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

  1. Consider the matrix

P (8)

  • Show that P is a projector onto a vector space.
  • This vector space S = range(P) is a subspace of C3. What is its dimension?
  • What is the dimension of the orthogonal complement S⊂ C3 of S in C3?
  • Compute the projector Ponto S
  • Using this result compute an orthonormal base of S
  • Have a look at the following subspace

S2 = range (9)

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

  1. 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

  • the least-squares solution obtained by the pseudo inverse, i.e. xLS = A+b, is really an exact solution to the modified system AxPb
  • the error of the righthand sides is orthogonal to bb.

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[Solved] Array-sensor Homework03
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