Math 2.0

Linear and multilinear algebra; matrix theory Matrix analysis Theorem

Determinant of a matrix exponential

Builds on Matrix exponentialDeterminantTrace of a matrix

Statement

Let 𝕂\mathbb{K} be ℝ\R or ℂ\C, and let AA be an n×nn \times n matrix over 𝕂\mathbb{K}. Then det⁡(eA)=etr⁡A,\det(e^A) = e^{\tr A}, where eAe^A is the matrix exponential, det⁡\det is the determinant and tr⁡\tr is the trace.

Remarks

If AA is triangular over ℂ\C with diagonal entries λi\lambda_i, then eAe^A is triangular with diagonal entries eλie^{\lambda_i}, and both sides equal eλ1+⋯+λne^{\lambda_1 + \cdots + \lambda_n}. The general case follows by triangularisation over ℂ\C. It shows again that eAe^A is invertible, because etr⁡A≠0e^{\tr A} \ne 0. If tr⁡A=0\tr A = 0, then eAe^A lies in the special linear group.

Reading path

To understand this page from the ground up, read these 11 pages in order. The foundations come first.

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Identity matrix, object, no evidence levelIdentity matrixobjectMatrix, definition, no evidence levelMatrixdefinitionVector space, definition, no evidence levelVector spacedefinitionSquare matrix, definition, no evidence levelSquare matrixdefinitionLinear map, definition, no evidence levelLinear mapdefinitionMatrix exponential, definition, no evidence levelMatrix exponentialdefinitionTrace of a matrix, definition, no evidence levelTrace of a matrixdefinitionMultilinear map, definition, no evidence levelMultilinear mapdefinitionAlternating multilinear map, definition, no evidence levelAlternating multiline…definitionDeterminant, definition, no evidence levelDeterminantdefinitionDeterminant of a matrix exponential, statement, no evidence levelDeterminant of a matr…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.
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  1. Identity matrix object
  2. Matrix definition
  3. Vector space definition
  4. Square matrix definition
  5. Linear map definition
  6. Matrix exponential definition
  7. Trace of a matrix definition
  8. Multilinear map definition
  9. Alternating multilinear map definition
  10. Determinant definition
  11. Determinant of a matrix exponential statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Matrix_exponential, The determinant of the matrix exponential
About this page
Tag0A1QQN333A
URIm2://0A1QQN333A
Kindstatement, theorem
WritingW0 stub
MSC15A16, 15A15
StatusThis copy of the site has no build data. Run m2 status 0A1QQN333A.
UsesMatrix exponential, Determinant, Trace of a matrix

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