Math 2.0

Linear and multilinear algebra; matrix theory Theorem

Sylvester's determinant theorem

Builds on DeterminantIdentity matrix

Statement

Let KK be a commutative ring, let AA be an m×nm \times n matrix and BB an n×mn \times m matrix over KK. Then det⁡(Im+AB)=det⁡(In+BA),\det(I_m + AB) = \det(I_n + BA), where ImI_m and InI_n are identity matrices and det⁡\det is the determinant.

Remarks

The two sides are determinants of different sizes, mm and nn. For a column uu and a row vTv^T, it gives det⁡(Im+uvT)=1+vTu\det(I_m + u v^T) = 1 + v^T u. The statement is also called the Weinstein–Aronszajn identity.

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Identity matrix, object, no evidence levelIdentity matrixobjectMatrix, definition, no evidence levelMatrixdefinitionVector space, definition, no evidence levelVector spacedefinitionSquare matrix, definition, no evidence levelSquare matrixdefinitionLinear map, definition, no evidence levelLinear mapdefinitionMultilinear map, definition, no evidence levelMultilinear mapdefinitionAlternating multilinear map, definition, no evidence levelAlternating multiline…definitionDeterminant, definition, no evidence levelDeterminantdefinitionSylvester's determinant theorem, statement, no evidence levelSylvester's determina…statement
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  1. Identity matrix object
  2. Matrix definition
  3. Vector space definition
  4. Square matrix definition
  5. Linear map definition
  6. Multilinear map definition
  7. Alternating multilinear map definition
  8. Determinant definition
  9. Sylvester's determinant theorem statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Weinstein–Aronszajn_identity, lead section
About this page
Tag0G24VQ1088
URIm2://0G24VQ1088
Aliasesweinstein-aronszajn-identity
Kindstatement, theorem
WritingW1 stated
MSC15A15
StatusThis copy of the site has no build data. Run m2 status 0G24VQ1088.
UsesDeterminant, Identity matrix
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