Math 2.0

statement theorem

Spectral theorem

Statement

Let VV be a finite-dimensional inner product space over R\R or C\C, and let T ⁣:V→VT \colon V \to V be a self-adjoint linear map. Then VV has an orthonormal basis that consists of eigenvectors of TT. The corresponding eigenvalues are real.

Remarks

In this basis, the matrix of TT is a real diagonal matrix. Equivalently, VV is the orthogonal direct sum of the eigenspaces of TT. Over C\C, the same conclusion holds, with complex eigenvalues, for each normal operator. The matrix form is the spectral theorem for Hermitian matrices.

Explanations

L1 human

A self-adjoint map is a pure stretch along perpendicular axes. The axes are the eigenvectors, and the stretch factors are the real eigenvalues. Every symmetric real matrix is diagonal in a suitable orthonormal basis.

What this node needs for W2
  • edge about out: the concepts that the statement is about (0 of 1)
  • edge proves in: at least one proof node (0 of 1)
  • edge instance_of or illustrates in: one example that shows the statement at work (0 of 1)
  • explanation L2 (0 of 1)

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What this needs

To read this node, read these 19 nodes in this order. The foundations come first, and the list ends with this node.

Vector space, definition, no evidence levelVector spacedefinitionInner product, definition, no evidence levelInner productdefinitionLinear combination, definition, no evidence levelLinear combinationdefinitionLinear map, definition, no evidence levelLinear mapdefinitionLinear subspace, definition, no evidence levelLinear subspacedefinitionInner 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 spandefinitionAdjoint of a linear map, definition, no evidence levelAdjoint of a linear m…definitionOrthonormal family, definition, no evidence levelOrthonormal familydefinitionEigenvalue, definition, no evidence levelEigenvaluedefinitionSpanning set, definition, no evidence levelSpanning setdefinitionSelf-adjoint operator, definition, no evidence levelSelf-adjoint operatordefinitionEigenvector, definition, no evidence levelEigenvectordefinitionBasis, definition, no evidence levelBasisdefinitionOrthonormal basis, definition, no evidence levelOrthonormal basisdefinitionSpectral theorem, statement, no evidence levelSpectral theoremstatement
Each node uses the nodes above it. The colour of the bar on a box shows the evidence level, and the box also gives it as text. The list below has the same nodes.
  • no evidence level
  1. Vector space definition
  2. Inner product definition
  3. Linear combination definition
  4. Linear map definition
  5. Linear subspace definition
  6. Inner product space definition
  7. Orthogonal vectors definition
  8. Linear independence definition
  9. Endomorphism definition
  10. Linear span definition
  11. Adjoint of a linear map definition
  12. Orthonormal family definition
  13. Eigenvalue definition
  14. Spanning set definition
  15. Self-adjoint operator definition
  16. Eigenvector definition
  17. Basis definition
  18. Orthonormal basis definition
  19. Spectral theorem statement

What depends on this

2 nodes use this node, directly or through other nodes.

Sources

  1. Web https://en.wikipedia.org/wiki/Spectral_theorem, Hermitian maps and Hermitian matrices

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