Introduction to Computational Linear Algebra

42 Introduction to Computational Linear Algebra

2.3 Rectangular Matrices: Ranks and Singular Values

Rectangular matrices do not have eigenvalues, but do have singular values.

For that purpose, we start by introducing the concept of matrix rank. Specif-

ically, given a matrix A ∈ C

m×n

, one may consider A as a mapping from C

n

to C

m

, as:

x ∈ C

n

: y = Ax ∈ C

m

.

Hence, we can start by stating some basic concepts from Linear Algebra.

1. Range(A) = {y = Ax ∈ C

m

|x ∈ C

n

}, implying that C

m

= Range(A) ⊕

(Range(A))

⊥

.

2. Null(A) = {x ∈ C

n

|Ax = 0}, implying Null(A) 6= {0} that

C

n

= Null(A) ⊕ (Null(A))

⊥

One has the following property:

Proposition 2.5 Let A

∗

be the adjoint