In
[1, p. 34 exercise 2.9] we are asked to prove that the rank of the complex

matrix
where the

are linearly independent complex

vectors and the

‘s are real and positive, is equal to

if

. Furthermore we are asked what the rank equals to if

>

.
Solution:
Let

where
![{\bf U} = \left[ {\begin{array}{*{20}c} {{\bf u}_1 } & \cdots & {{\bf u}_m } \\ \end{array}} \right]](https://lysario.de/wp-content/cache/tex_65418a74374f858d31b0defab985a91c.png)
and

. If

is the linear transform associated to

then

can be written as the composite of the linear transformations
[2, propsition 6.3, p. 41] 
and

which are respectively associated to the matrices

and

,

.
For

the

linear independent vectors span the subspace
[2, proposition 4.3, page 112] thus the column rank of the matrix

is

. The matrix

has the same rank as the matrix

. The proof of this statement can be obtained by the observation that given

linear independent vectors, then the complex conjugates of those vectors are also linear independent. Thus

. Furthermore because the column rank equals the the row rank of a matrix
[2, Theorem 4.4, page 218] we have

.
The linear transformation

is surjective as its

.

is bijective as the diagonal matrix associated with it is invertible
[2, proposition 2.3, page 209]. The composite

is thus surjective (onto) to

and thus the dimension of the image obtained by

(per definition the rank of the matrix) is the same as the one that would be obtained by

by linear combinations of a basis in

. Thus

. QED.
For

>

there can be no

linear independent

vectors. The maximum number of linear independent

vectors is simply

. This assertion can be proven by the equality of the row and column rank of a matrix.
For in this case the column rank of

would be

while the row rank of

would not be larger than

. The rank of the matrix cannot be larger than

in this case.
[1] Steven M. Kay: “Modern Spectral Estimation – Theory and Applications”, Prentice Hall, ISBN: 0-13-598582-X. [2] Lawrence J. Corwin and Robert H. Szczarba: “Calculus in Vector Spaces”, Marcel Dekker, Inc, 2nd edition, ISBN: 0824792793.
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