Math 2.0

Linear and multilinear algebra; matrix theory Proposition

Trace of a transpose

Builds on Trace of a matrixTranspose

Statement

Let RR be a commutative ring and let AA be an n×nn \times n matrix over RR. Then tr⁡(AT)=tr⁡A\tr(A^T) = \tr A, where ATA^T is the transpose of AA and tr⁡\tr is the trace.

Remarks

The transpose does not move the entries on the main diagonal, so the sum of the diagonal entries stays the same. For a complex matrix, tr⁡(A∗)=tr⁡A‾\tr(A^*) = \overline{\tr A}.

Reading path

To understand this page from the ground up, read these 5 pages in order. The foundations come first.

Show the path as a graph
Matrix, definition, no evidence levelMatrixdefinitionSquare matrix, definition, no evidence levelSquare matrixdefinitionTranspose, definition, no evidence levelTransposedefinitionTrace of a matrix, definition, no evidence levelTrace of a matrixdefinitionTrace of a transpose, statement, no evidence levelTrace of a transposestatement
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. Square matrix definition
  3. Transpose definition
  4. Trace of a matrix definition
  5. Trace of a transpose statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Trace_(linear_algebra), Basic properties
About this page
Tag09E1RCZ7YT
URIm2://09E1RCZ7YT
Kindstatement, proposition
WritingW0 stub
MSC15A15
StatusThis copy of the site has no build data. Run m2 status 09E1RCZ7YT.
UsesTrace of a matrix, Transpose

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