# Analysis of small and large perturbations of matrix subspaces

#### Abstract

This work develops ways of describing small and large perturbations of matrix subspaces. It is shown that the signal and noise subspaces associated with a data matrix are special types of invariant subspaces. Small perturbations in the data matrix cause changes in the signal and noise subspaces which can be tracked by invariant subspace updating. In order to calculate an invariant subspace update, the matrix Riccati, or matrix quadratic equation (MQE) must be solved. Let an orthonormal basis for the signal subspace be given by $X\sb1$ and an orthonormal basis for the noise subspace be given by $X\sb2$. Let $X = \lbrack X\sb1\ X\sb2\rbrack$ be unitary. An update to $X\sb1$ due to a small perturbation can be described as $X\sb1 + X\sb2P$ where P is a solution to the MQE. An efficient, $O(n\sp2)$ algorithm calculates a solution, P, to the MQE which is of small norm. Large perturbations, on the other hand, are defined as either exchanges of basis vectors between the signal and noise subspaces, or the movement of a basis vector from the signal to the noise subspace, or vice versa.^ A new method for the solving the DOA tracking problem is presented. It is based on invariant subspace updating. This method is $O(n\sp2)$, where n is the number of array sensors. One advantage gained by using the invariant subspace updating method is the ability to track rapidly changing DOAs. In addition, it is possible to track sources as they appear or disappear in the incoming data. This amounts to the movement of basis vectors between the signal and noise subspaces, and is termed a large perturbation. Detection of source increases and decreases is accomplished by examination of the current data snapshot, x, and the set of estimated steering vectors, B. Numerical accuracy is maintained by using the QR factorization to obtain an $X = \lbrack X\sb1\ X\sb2\rbrack$ that is unitary to working precision. Simulations showing two types of nonstationarity are presented. ^

#### Subject Area

Engineering, Electronics and Electrical

#### Recommended Citation

Craig Scott Macinnes,
"Analysis of small and large perturbations of matrix subspaces"
(1995).
*Dissertations and Master's Theses (Campus Access).*
Paper AAI9601857.

http://digitalcommons.uri.edu/dissertations/AAI9601857