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  • Eigenvalues and eigenvectors

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  • Particular types of matrices
  • Definition of eigenvalue of a matrix
  • Definition of eigenvector of a matrix
  • Eigenspaces
  • Every matrix has an eigenvalue over the complex numbers.
  • Eigenvalues and operations on matrices
  • Eigenvectors with distinct eigenvalues are linearly independent.
  • Multiplicity
  • Characteristic and minimal polynomials
  • The dimension of a eigenspace is less than or equal to the (algebraic) multiplicity of the eigenvalue.
  • Definition of eigenvalue/characteristic value of a linear transformation
  • Definition of eigenvector/characteristic vector of a linear transformation
  • Definition of characteristic polynomial of a linear transformation
  • Definition of minimal polynomial of a linear transformation
  • The Cayley-Hamilton theorem for a linear transformation
  • The minimal polynomial of a linear transformation exists and is unique.
  • Definition of applying a polynomial to a linear transformation
  • A linear transformation on a finite dimentional nontrivial vector space has at least one eigenvalue.
  • Definition of eigenspace of a linear transformation
  • The eigenspace of a linear transformation is a nontrivial subspace.
  • Definition of invariant subspace of a linear transformation.
  • If a space is the direct sum of invariant subspaces
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  1. If a space is the direct sum of invariant subspaces

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701–715 of 715 nodes
  • Triangular matrices#23561
  • Two matrices of the same size are equivalent if and only if they have the same rank.#23607
  • Two vectors are orthogonal if and only if the Pythagorean Theorem holds.#23918
  • Unitary matrices#23587
  • Unitary matrices are invertible.#23589
  • Unitary matrices have orthogonal (orthonormal) rows/columns.#23590
  • Unitary matrices preserve inner products.#23591
  • Unitary matrices preserve orthogonal (orthonormal) bases.#23592
  • Using matrices to solve linear systems#23760
  • Vector space isomorphism is an equivalence relation.#24003
  • Vector spaces#23775
  • Vector spaces with the same dimension are isomprphic.#24005
  • Vector sum/addition interpreted geometrically in R^n (or C^n)#23797
  • Vector sum/addition is commutative and associative#23787
  • Visualise a linear transformation on R^2 by looking at the image of a region#23852
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  • Aspen Summer 2016 - Entanglement Matters#1213
  • Math#23168

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