Math 2.0

Linear and multilinear algebra; matrix theory Definition

Trace of a linear map

Builds on Trace of a matrixEndomorphism

Definition

Let FF be a field, let VV be a finite-dimensional vector space over FF, and let T ⁣:V→VT \colon V \to V be an endomorphism. The trace of TT is tr⁡T=tr⁡[T]B,\tr T = \tr [T]_B, the trace of the matrix of TT in a basis BB of VV.

Remarks

The value does not depend on BB: by the change of basis formula, another basis gives P−1[T]BPP^{-1} [T]_B P, and tr⁡(P−1AP)=tr⁡A\tr(P^{-1} A P) = \tr A. The trace is a linear map End⁡F(V)→F\End_F(V) \to F, and tr⁡(ST)=tr⁡(TS)\tr(ST) = \tr(TS). The trace of the identity map of VV is dim⁡V⋅1\dim V \cdot 1. A projection on a real or complex space has trace equal to its rank.

Reading path

To understand this page from the ground up, read these 7 pages in order. The foundations come first.

Show the path as a graph
Matrix, definition, no evidence levelMatrixdefinitionVector space, definition, no evidence levelVector spacedefinitionSquare matrix, definition, no evidence levelSquare matrixdefinitionLinear map, definition, no evidence levelLinear mapdefinitionTrace of a matrix, definition, no evidence levelTrace of a matrixdefinitionEndomorphism, definition, no evidence levelEndomorphismdefinitionTrace of a linear map, definition, no evidence levelTrace of a linear mapdefinition
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. Square matrix definition
  4. Linear map definition
  5. Trace of a matrix definition
  6. Endomorphism definition
  7. Trace of a linear map definition

Where this leads

7 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Trace_(linear_algebra), Trace of a linear operator
About this page
Tag0RPDTWY2JS
URIm2://0RPDTWY2JS
Kinddefinition
WritingW0 stub
MSC15A04, 15A15
DefinesTrace of a linear map
UsesTrace of a matrix, Endomorphism

Markdown · JSON · Edit on GitHub

What this page needs for W1

  • an inventory row: the row of the inventory that the node accounts for (0 of 1)

The writing levels say how complete the text of a page is. They do not measure its truth.

What links here

Used by