Math 2.0

Linear and multilinear algebra; matrix theory Determinants Theorem

Determinant via the Schur complement

Builds on DeterminantBlock matrixInvertible matrix

Statement

Let KK be a commutative ring and let AA be an invertible p×pp \times p matrix over KK. Let BB, CC and DD be matrices over KK of sizes p×qp \times q, q×pq \times p and q×qq \times q. Then det⁡(ABCD)=det⁡A⋅det⁡(D−CA−1B),\det \begin{pmatrix} A & B \\ C & D \end{pmatrix} = \det A \cdot \det(D - C A^{-1} B), where det⁡\det is the determinant. The matrix D−CA−1BD - C A^{-1} B is the Schur complement of AA in the block matrix.

Remarks

The proof factors the block matrix as (I0CA−1I)(AB0D−CA−1B)\begin{pmatrix} I & 0 \\ C A^{-1} & I \end{pmatrix} \begin{pmatrix} A & B \\ 0 & D - C A^{-1} B \end{pmatrix}. Then it uses the rule for block triangular matrices. For p=q=1p = q = 1 and a≠0a \ne 0, it gives det⁡(abcd)=a(d−cb/a)=ad−bc\det \begin{pmatrix} a & b \\ c & d \end{pmatrix} = a(d - cb/a) = ad - bc.

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Identity matrix, object, no evidence levelIdentity matrixobjectMatrix, definition, no evidence levelMatrixdefinitionVector space, definition, no evidence levelVector spacedefinitionBlock matrix, definition, no evidence levelBlock matrixdefinitionMatrix 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 levelDeterminantdefinitionDeterminant via the Schur complement, statement, no evidence levelDeterminant via the S…statement
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  1. Identity matrix object
  2. Matrix definition
  3. Vector space definition
  4. Block matrix definition
  5. Matrix multiplication definition
  6. Square matrix definition
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  8. Invertible matrix definition
  9. Multilinear map definition
  10. Alternating multilinear map definition
  11. Determinant definition
  12. Determinant via the Schur complement statement

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Further reading

  1. Web https://en.wikipedia.org/wiki/Schur_complement, Properties
About this page
Tag0KA7A4ENVJ
URIm2://0KA7A4ENVJ
Kindstatement, theorem
WritingW0 stub
MSC15A15
StatusThis copy of the site has no build data. Run m2 status 0KA7A4ENVJ.
UsesDeterminant, Block matrix, Invertible matrix

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