Math 2.0

Linear and multilinear algebra; matrix theory Theorem

Principal axis theorem

Builds on Quadratic formSpectral theorem for Hermitian matrices

Statement

Let AA be a real symmetric n×nn \times n matrix, and let Q(x)=xTAxQ(x) = x^T A x for x∈ℝnx \in \R^n. Then there are an orthogonal matrix PP and real numbers λ1,…,λn\lambda_1, \dots, \lambda_n such that, for x=Pyx = P y, Q(x)=λ1y12+λ2y22+⋯+λnyn2.Q(x) = \lambda_1 y_1^2 + \lambda_2 y_2^2 + \cdots + \lambda_n y_n^2. The numbers λi\lambda_i are the eigenvalues of AA, and the columns of PP are orthonormal eigenvectors of AA.

Remarks

It follows from the spectral theorem A=PDPTA = P D P^T with DD diagonal. The lines spanned by the columns of PP are the principal axes of QQ. For example, Q(x)=5x12+4x1x2+2x22Q(x) = 5x_1^2 + 4x_1 x_2 + 2x_2^2 has the eigenvalues 66 and 11, so it becomes 6y12+y226y_1^2 + y_2^2. So the curve Q(x)=1Q(x) = 1 is an ellipse, with semi-axes 1/61/\sqrt 6 and 11.

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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 matrixdefinitionTranspose, definition, no evidence levelTransposedefinitionBilinear form, definition, no evidence levelBilinear formdefinitionInner 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 transposedefinitionQuadratic form, definition, no evidence levelQuadratic formdefinitionInner product space, definition, no evidence levelInner product spacedefinitionOrthogonal vectors, definition, no evidence levelOrthogonal vectorsdefinitionLinear independence, definition, no evidence levelLinear independencedefinitionEndomorphism, definition, no evidence levelEndomorphismdefinitionLinear span, definition, no evidence levelLinear spandefinitionHermitian matrix, definition, no evidence levelHermitian matrixdefinitionUnitary matrix, definition, no evidence levelUnitary matrixdefinitionOrthonormal family, definition, no evidence levelOrthonormal familydefinitionAdjoint of a linear map, definition, no evidence levelAdjoint of a linear m…definitionEigenvalue, definition, no evidence levelEigenvaluedefinitionSpanning set, definition, no evidence levelSpanning setdefinitionEigenvector, definition, no evidence levelEigenvectordefinitionSelf-adjoint operator, definition, no evidence levelSelf-adjoint operatordefinitionBasis, definition, no evidence levelBasisdefinitionOrthonormal basis, definition, no evidence levelOrthonormal basisdefinitionSpectral theorem, statement, no evidence levelSpectral theoremstatementSpectral theorem for Hermitian matrices, statement, no evidence levelSpectral theorem for …statementPrincipal axis theorem, statement, no evidence levelPrincipal axis theoremstatement
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. Transpose definition
  6. Bilinear form definition
  7. Inner product definition
  8. Linear combination definition
  9. Linear map definition
  10. Linear subspace definition
  11. Conjugate transpose definition
  12. Quadratic form definition
  13. Inner product space definition
  14. Orthogonal vectors definition
  15. Linear independence definition
  16. Endomorphism definition
  17. Linear span definition
  18. Hermitian matrix definition
  19. Unitary matrix definition
  20. Orthonormal family definition
  21. Adjoint of a linear map definition
  22. Eigenvalue definition
  23. Spanning set definition
  24. Eigenvector definition
  25. Self-adjoint operator definition
  26. Basis definition
  27. Orthonormal basis definition
  28. Spectral theorem statement
  29. Spectral theorem for Hermitian matrices statement
  30. Principal axis theorem statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Principal_axis_theorem, Formal statement
About this page
Tag0PR5P9Z4KM
URIm2://0PR5P9Z4KM
Kindstatement, theorem
WritingW1 stated
MSC15A63, 15A18
StatusThis copy of the site has no build data. Run m2 status 0PR5P9Z4KM.
UsesQuadratic form, Spectral theorem for Hermitian matrices
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