-
Let
. Then is invertible if there exists such that The matrix
is unique. -
(Friedberg e2.4.2) Let
be invertible matrices. Then -
(Friedberg e2.4.3) Let
be invertible, then -
(Friedberg e2.4.4) If
is invertible and , then . -
(Friedberg e2.4.8) A one sided inverse for square matrices is a two sided inverse. If
such that then -
(Friedberg 3.4) Let
. If and are invertible, and matrices respectively, then
Computation
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Let
and . The augmented matrix is defined as the matrix We have that for
-
We can find
by finding the elementary matrices which reduce the LHS of the augmented matrix to the identity.
Theorems
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Murphy Thm. 4.3.2 Let
Where
and are invertible. We have Obtain the second formula, by interchanging the two diagonal entries of the first and then interchanging all
with and all with . We also have the Schur Complements
The Schur Complements arise when performing Gaussian Elimination on the block matrix
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Murphy Cor. 4.3.1. Let
Where
and are invertible. We have The first two are called the Matrix Inversion Lemma or the Sherman-Morrison-Woodbury Formula. The last is called the Matrix Determinant Lemma.
The Sherman-Morrison formula is an important special case given as
Where we assume
. -
The Bunch-Nielsen Sorensen formula is as follows.
Let
denote the eigenvalues of and denote the eigenvalues of the updated matrix . In the special case when
is diagonal, the eigenvalues of is given as Where
is a number that makes normalized.