Nonlinear Equations - UFRJ

Chapter 8
Condition number
theory
8.1 Linear equations
The following classical theorem in linear algebra is known
as the singular value decomposition (svd for short).
Theorem 8.1. Let A : R n ↦→ R m (resp. C n → C m ) be linear. Then,
there are σ 1 ≥ · · · ≥ σ r > 0, r ≤ m, n, such that
A = UΣV ∗
with U ∈ O(m) (resp. U(m)), V ∈ O(n) (resp. U(n)) and Σ ij = σ i
for i = j ≤ r and 0 otherwise.
It is due to Sylvester (real n × n matrices) and to Eckart and
Young [37] in the general case, now exercise 8.1 below.
Gregorio Malajovich, Nonlinear equations.
28 o Colóquio Brasileiro de Matemática, IMPA, Rio de Janeiro, 2011.
Copyright c○ Gregorio Malajovich, 2011.
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