numpy-discussion

[Numpy-discussion] Rank deficient matrices

[Nils W. <nw...@me...>]

Hi,

Let us assume that

r = rank(C) < N               (1)

where C is a symmetric NxN matrix. This implies that the number
of non-zero eigenvalues of C is r. Because C is a symmetric
matrix there exists an orthogonal matrix U whose columns are
the eigenvectors of C such that

U^\top C U = [ d , O
               O , O].         (2) 

In the above equation d \in rxr is a diagonal matrix consisting
of only the non-zero eigenvalues of C.
For convenience, partition U as

U = [U_1 | U_2]                (3)

where the columns of U_1 (Nxr) are the eigenvectors corresponding to the
non-zero block
d and the columns of U_2 are the eigenvectors corresponding to the rest
(N-r) number
of zero eigenvalues.

Defining a rectangular transformation matrix

R = U_1                        (4)

it is easy to show that

R^\top C R = d.                 (5)

Therefore, the matrix R in equation (4) transforms the originally rank
deficient
matrix C to a full-rank (diagonal) matrix of rank r.

I am looking for an efficient Numpy implementation of this
transformation.

Thanks in advance 

                   Nils Wagner

RE: [Numpy-discussion] Rank deficient matrices

[Paul F. D. <pa...@pf...>]

Use LinearAlgebra.singular_value_decomposition

-----Original Message-----
From: num...@li...
[mailto:num...@li...] On Behalf Of Nils
Wagner
Sent: Tuesday, October 30, 2001 12:24 AM
To: num...@li...
Subject: [Numpy-discussion] Rank deficient matrices


Hi,

Let us assume that

r = rank(C) < N               (1)

where C is a symmetric NxN matrix. This implies that the number of
non-zero eigenvalues of C is r. Because C is a symmetric matrix there
exists an orthogonal matrix U whose columns are the eigenvectors of C
such that

U^\top C U = [ d , O
               O , O].         (2) 

In the above equation d \in rxr is a diagonal matrix consisting of only
the non-zero eigenvalues of C. For convenience, partition U as

U = [U_1 | U_2]                (3)

where the columns of U_1 (Nxr) are the eigenvectors corresponding to the
non-zero block d and the columns of U_2 are the eigenvectors
corresponding to the rest
(N-r) number
of zero eigenvalues.

Defining a rectangular transformation matrix

R = U_1                        (4)

it is easy to show that

R^\top C R = d.                 (5)

Therefore, the matrix R in equation (4) transforms the originally rank
deficient matrix C to a full-rank (diagonal) matrix of rank r.

I am looking for an efficient Numpy implementation of this
transformation.

Thanks in advance 

                   Nils Wagner

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