Math 2.0

Linear and multilinear algebra; matrix theory Proposition

Change of basis for a bilinear form

Builds on Matrix of a bilinear formChange of basis matrix

Statement

Let FF be a field and let VV be a finite-dimensional vector space over FF, with two bases ℬ\mathcal{B} and ℬ′\mathcal{B}'. Let PP be the change of basis matrix whose columns are the coordinate vectors of the vectors of ℬ′\mathcal{B}' in ℬ\mathcal{B}. Let BB be a bilinear form on VV, with matrix MM in ℬ\mathcal{B} and M′M' in ℬ′\mathcal{B}'. Then M′=PTMP.M' = P^T M P.

Remarks

Two square matrices MM and M′M' are congruent when M′=PTMPM' = P^T M P for an invertible matrix PP. So two matrices represent the same bilinear form in two bases if and only if they are congruent. Congruence keeps the rank and the symmetry of a matrix, but in general not its eigenvalues. Compare this with the rule P−1APP^{-1} A P for the matrix of a linear map.

Reading path

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Vector space, definition, no evidence levelVector spacedefinitionBilinear form, definition, no evidence levelBilinear formdefinitionLinear combination, definition, no evidence levelLinear combinationdefinitionLinear subspace, definition, no evidence levelLinear subspacedefinitionLinear independence, definition, no evidence levelLinear independencedefinitionLinear span, definition, no evidence levelLinear spandefinitionSpanning set, definition, no evidence levelSpanning setdefinitionBasis, definition, no evidence levelBasisdefinitionCoordinate vector, definition, no evidence levelCoordinate vectordefinitionMatrix of a bilinear form, definition, no evidence levelMatrix of a bilinear …definitionChange of basis matrix, definition, no evidence levelChange of basis matrixdefinitionChange of basis for a bilinear form, statement, no evidence levelChange of basis for a…statement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
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  1. Vector space definition
  2. Bilinear form definition
  3. Linear combination definition
  4. Linear subspace definition
  5. Linear independence definition
  6. Linear span definition
  7. Spanning set definition
  8. Basis definition
  9. Coordinate vector definition
  10. Matrix of a bilinear form definition
  11. Change of basis matrix definition
  12. Change of basis for a bilinear form statement

Where this leads

2 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Matrix_congruence, lead section
About this page
Tag0SPYAGE7T0
URIm2://0SPYAGE7T0
Kindstatement, proposition
WritingW0 stub
MSC15A63, 15A21
StatusThis copy of the site has no build data. Run m2 status 0SPYAGE7T0.
UsesMatrix of a bilinear form, Change of basis matrix

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