Theorems · Theorem · linear algebra
spectrum_diagonal
∀ {R : Type u_1} {n : Type u_2} [inst : DecidableEq n] [inst_1 : Fintype n] [inst_2 : Field R] (d : n → R),
spectrum R (Matrix.diagonal d) = Set.range dThe spectrum of the diagonal operator is the range of the diagonal viewed as a function.
- Defined in
- Mathlib.LinearAlgebra.Eigenspace.Matrix
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 126 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- DecidableEqFintypeField
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites13
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setstatement and proof · cited by 53,352
- Fintypestatement and proof · cited by 7,736
- Fieldstatement and proof · cited by 7,404
- Set.rangestatement and proof · cited by 4,705
- Matrixstatement · cited by 4,303
- Set.extproof · cited by 2,266
- spectrumstatement · cited by 510
- Matrix.diagonalstatement and proof · cited by 314
- Pi.basisFunproof · cited by 78
- Matrix.toLinAlgEquivproof · cited by 13
- AlgEquiv.spectrum_eqproof · cited by 9
- Module.End.hasEigenvalue_iff_mem_spectrumproof · cited by 4
Cited by2
Results whose statement or proof uses this declaration.
- Matrix.IsHermitian.spectrum_eq_image_rangeproof · cited by 1
- Matrix.IsHermitian.spectrum_real_eq_range_eigenvaluesproof · cited by 1