Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 335 / 3 / i

Solution

Let

(σ_{j}, u_{j}, v_{j})

A : X \to Y

, so

A v_{j} = σ_{j} u_{j}, A^{*} u_{j} = σ_{j} v_{j}, σ_{j} > 0.

D (A^{†}) = ran A \oplus (ran A)^{⊥}

and acts by

A^{†} y = j \sum \frac{⟨ y , u _{j} ⟩}{σ _{j}} v_{j},

with the orthogonal component of

y

sent to zero. Equivalently, its domain consists of the data satisfying the Picard criterion. It obeys

A A^{†} y = P_{\overline{ran A}} y, A^{†} A x = P_{(k e r A)^{⊥}} x .

A x = y

is exactly solvable, every solution is

x^{†} + z

with

z \in ker A

, and

x^{†} = A^{†} y

is the unique minimum-norm least-squares solution. It recovers the component of the original

x

orthogonal to the null space; no data can determine the null-space component.

There are no discussions about this article yet.

There are currently no matching articles.