Math 2.0

Linear and multilinear algebra; matrix theory Determinants Theorem

Matrix determinant lemma

Builds on DeterminantInvertible matrix

Statement

Let KK be a commutative ring, let AA be an invertible n×nn \times n matrix over KK, and let u,v∈Knu, v \in K^n be column vectors. Then det⁡(A+uvT)=(1+vTA−1u)det⁡A,\det(A + u v^T) = (1 + v^T A^{-1} u) \det A, where det⁡\det is the determinant.

Remarks

The matrix uvTu v^T has rank at most 11, so the lemma describes the effect of a rank one update on the determinant. It follows from Sylvester's determinant theorem with the matrices A−1uA^{-1} u and vTv^T. A variant without inverses is det⁡(A+uvT)=det⁡A+vTadj⁡(A)u\det(A + u v^T) = \det A + v^T \operatorname{adj}(A) u.

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Identity matrix, object, no evidence levelIdentity matrixobjectMatrix, definition, no evidence levelMatrixdefinitionVector space, definition, no evidence levelVector spacedefinitionMatrix multiplication, definition, no evidence levelMatrix multiplicationdefinitionSquare matrix, definition, no evidence levelSquare matrixdefinitionLinear map, definition, no evidence levelLinear mapdefinitionInvertible matrix, definition, no evidence levelInvertible matrixdefinitionMultilinear map, definition, no evidence levelMultilinear mapdefinitionAlternating multilinear map, definition, no evidence levelAlternating multiline…definitionDeterminant, definition, no evidence levelDeterminantdefinitionMatrix determinant lemma, statement, no evidence levelMatrix determinant le…statement
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  1. Identity matrix object
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  8. Multilinear map definition
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  10. Determinant definition
  11. Matrix determinant lemma statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Matrix_determinant_lemma, Statement
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MSC15A15
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