# Appendix A: Factorization of the data matrix

Because equation 7 is a singular value decomposition (SVD) of the signal matrix , the left matrix in equation 7 is a unitary matrix:

 (21)

Combining equations 4, 8, and 21, we can derive:

 (22)

where and are introduced matrices and are diagonal and positive definite.

In order to make the right matrix orthonormal, we make two assumptions:

• The noise is close to white noise in the sense that .
• The signal is orthogonal to the noise in the sense that .

We let denote the right matrix of the last equation in 22, then

 (23)

where

 (24)

when .

 (25)

Since is an orthogonal matrix, then . Since , then , thus . In the same way, since , thus . Then,

 (26)

 (27)

 (28)

when . Thus, we prove that when and are appropriately chosen, and is orthonormal.

2020-02-21