- Given the SVD of a matrix
**A**∈ C^{m}^{×n}

# A = UΣV^{H } (1)

^{H }

with the unitary matrices^{r}^{×r }**U **∈ C^{m}^{×m}, **V **∈ C^{n}^{×n }and the positive definite diagonal matrix **Σ**_{1 }∈ 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
**U**_{1},**U**_{2},**V**_{1},**V**_{2},**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)

- 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:

**D****e **(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.

- 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 C^{3}. What is its dimension? - What is the dimension of the orthogonal complement S
^{⊥ }⊂ C^{3 }of S in**C**^{3}? - Compute the projector
**P**^{⊥ }onto 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).)

- The system
**Ax**=**b**with**A**∈ C^{m}^{×n},**x**∈ Cand^{n }**b**∈ Chas an exact solution only if^{m }*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**P**bas 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.
**x**=_{LS }**A**^{+}**b**, is really an*exact*solution to the modified system**Ax****Pb** - the error of the righthand sides is orthogonal to
**b**b.

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