# Sample matrix inversion

Sample matrix inversion (or direct matrix inversion) is an algorithm that estimates weights of an array (adaptive filter) by replacing the correlation matrix $\displaystyle{ R }$ with its estimate. Using $\displaystyle{ K }$ $\displaystyle{ N }$-dimensional samples $\displaystyle{ X_1, X_2,\dots,X_K }$, an unbiased estimate of $\displaystyle{ R_{X} }$, the $\displaystyle{ N \times N }$ correlation matrix of the array signals, may be obtained by means of a simple averaging scheme:

$\displaystyle{ \hat{R}_{X} = \frac{1}{K} \sum\limits_{k=1}^K X_k X^H_k, }$

where $\displaystyle{ H }$ is the conjugate transpose. The expression of the theoretically optimal weights requires the inverse of $\displaystyle{ R_{X} }$, and the inverse of the estimates matrix is then used for finding estimated optimal weights.