Matrix determinant lemma

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In mathematics, in particular linear algebra, the matrix determinant lemma^[1][2] computes the determinant of the sum of an invertible matrix A and the dyadic product, u v^T, of a column vector u and a row vector v^T.

Contents 1 Statement 2 Application 3 Generalization 4 See also 5 References

[edit] Statement

Suppose A is an invertible square matrix and u, v are column vectors. Then the matrix determinant lemma states that

Here, uv^T is the dyadic product of two vectors u and v.

[edit] Application

If the determinant and inverse of A are already known, the formula provides a numerically cheap way to compute the determinant of A corrected by the matrix uv^T. The computation is relatively cheap because the determinant of A+uv^T does not have to be computed from scratch (which in general is expensive). Using unit vectors for u and/or v, individual columns, rows or elements^[3] of A may be manipulated and a correspondingly updated determinant computed relatively cheaply in this way.

When the matrix determinant lemma is used in conjunction with the Sherman-Morrison formula, both the inverse and determinant may be conveniently updated together.

[edit] Generalization

Suppose A is an invertible n-by-n matrix and U, V are n-by-m matrices. Then

In the special case this is Sylvester's theorem for determinants.

Given additionally an invertible m-by-m matrix W, the relationship can also be expressed as

[edit] See also

• The Sherman-Morrison formula, which shows how to update the inverse, A^-1, to obtain (A+uv^T)^-1.
• The Woodbury formula, which shows how to update the inverse, A^-1, to obtain (A+UV^T)^-1.

[edit] References

1. ^ Harville, D. A. (1997). Matrix Algebra From a Statistician’s Perspective. Springer-Verlag. 
2. ^ Brookes, M. (2005). The Matrix Reference Manual [online].
3. ^ (1992) Numerical Recipes in C: The Art of Scientific Computing. Cambridge University Press, p.73. ISBN 0-521-43108-5.