Math 2.0

statement proposition

Trace of a transpose

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}.

Explanations

L1 human

Transposing flips a matrix over its diagonal. The diagonal itself does not move. So the trace, which only sees the diagonal, does not change.

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To read this node, read these 5 nodes in this order. The foundations come first, and the list ends with this node.

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
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  1. Matrix definition
  2. Square matrix definition
  3. Transpose definition
  4. Trace of a matrix definition
  5. Trace of a transpose statement

Sources

  1. Web https://en.wikipedia.org/wiki/Trace_(linear_algebra), Basic properties

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