Sherman-Morrison formula - Revision history

Vipul: Created page with "==Definition== Suppose $n$ is a positive integer, $A$ is an invertible $n \times n$ matrix, and $\vec{u},\vec{v}$ are $n$
2014-05-09T16:06:18Z

Created page with "==Definition== Suppose <math>n</math> is a positive integer, <math>A</math> is an invertible <math>n \times n</math> matrix, and <math>\vec{u},\vec{v}</math> are <math>n</mat..."

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==Definition==

Suppose <math>n</math> is a positive integer, <math>A</math> is an invertible <math>n \times n</math> matrix, and <math>\vec{u},\vec{v}</math> are <math>n</math>-dimensional column vectors. The [[Hadamard product]] <math>\vec{u}\vec{v}^T</math> is therefore a <math>n \times n</math> matrix of rank one. We have the following:

* The matrix <math>A + \vec{u}\vec{v}^T</math> is invertible if and only if the real number <math>1 + \vec{v}^TA^{-1}\vec{u}</math> is nonzero.
* If the condition above holds, we have the formula:

<math>(A + \vec{u}\vec{v}^T)^{-1} = A^{-1} - \frac{A^{-1}\vec{u}\vec{v}^TA^{-1}}{1 + \vec{v}^TA^{-1}\vec{u}}</math>