Math 2.0

statement proposition

Matrix of the adjoint is the conjugate transpose

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.

Explanations

L1 human

In orthonormal coordinates, taking the adjoint of a map means taking the conjugate transpose of its matrix. This makes the abstract adjoint easy to compute. In other bases, the rule is more complicated.

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To read this node, read these 20 nodes in this order. The foundations come first, and the list ends with this node.

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 node uses the nodes above it. The colour of the bar on a box shows the evidence level, and the box also gives it as text. The list below has the same nodes.
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  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

Sources

  1. Web https://en.wikipedia.org/wiki/Conjugate_transpose, Properties

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