Math 2.0

Linear and multilinear algebra; matrix theory Proposition

Matrix of the adjoint is the conjugate transpose

Builds on Adjoint of a linear mapConjugate transposeMatrix of a linear mapOrthonormal basis

Statement

Let VV and WW be finite-dimensional inner product spaces over ℝ\R or ℂ\C, with orthonormal bases BB and CC. Let T ⁣:V→WT \colon V \to W be a linear map. Then the matrix of the adjoint is [T∗]B,C=([T]C,B)∗,[T^*]_{B,C} = \left([T]_{C,B}\right)^*, where ∗^* on the right is the conjugate transpose. Here [ ⋅ ][\,\cdot\,] is the matrix of a linear map.

Remarks

Over ℝ\R, the matrix of T∗T^* is the transpose of the matrix of TT. The statement needs orthonormal bases; in other bases, the Gram matrices of the bases appear in the formula. The proof compares entries: both are ⟨T(bj),ci⟩\langle T(b_j), c_i \rangle, up to conjugation.

Reading path

To understand this page from the ground up, read these 20 pages in order. The foundations come first.

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Matrix, definition, no evidence levelMatrixdefinitionVector space, definition, no evidence levelVector spacedefinitionTranspose, definition, no evidence levelTransposedefinitionInner product, definition, no evidence levelInner productdefinitionLinear combination, definition, no evidence levelLinear combinationdefinitionLinear map, definition, no evidence levelLinear mapdefinitionLinear subspace, definition, no evidence levelLinear subspacedefinitionConjugate transpose, definition, no evidence levelConjugate transposedefinitionInner product space, definition, no evidence levelInner product spacedefinitionOrthogonal vectors, definition, no evidence levelOrthogonal vectorsdefinitionLinear independence, definition, no evidence levelLinear independencedefinitionLinear span, definition, no evidence levelLinear spandefinitionAdjoint of a linear map, definition, no evidence levelAdjoint of a linear m…definitionOrthonormal family, definition, no evidence levelOrthonormal familydefinitionSpanning set, definition, no evidence levelSpanning setdefinitionBasis, definition, no evidence levelBasisdefinitionCoordinate vector, definition, no evidence levelCoordinate vectordefinitionOrthonormal basis, definition, no evidence levelOrthonormal basisdefinitionMatrix of a linear map, definition, no evidence levelMatrix of a linear mapdefinitionMatrix of the adjoint is the conjugate transpose, statement, no evidence levelMatrix of the adjoint…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.
  • no evidence level
  1. Matrix definition
  2. Vector space definition
  3. Transpose definition
  4. Inner product definition
  5. Linear combination definition
  6. Linear map definition
  7. Linear subspace definition
  8. Conjugate transpose definition
  9. Inner product space definition
  10. Orthogonal vectors definition
  11. Linear independence definition
  12. Linear span definition
  13. Adjoint of a linear map definition
  14. Orthonormal family definition
  15. Spanning set definition
  16. Basis definition
  17. Coordinate vector definition
  18. Orthonormal basis definition
  19. Matrix of a linear map definition
  20. Matrix of the adjoint is the conjugate transpose statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Conjugate_transpose, Properties
About this page
Tag0DBTE6HVBD
URIm2://0DBTE6HVBD
Kindstatement, proposition
WritingW0 stub
MSC15A04, 47A05
StatusThis copy of the site has no build data. Run m2 status 0DBTE6HVBD.
UsesAdjoint of a linear map, Conjugate transpose, Matrix of a linear map, Orthonormal basis

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