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  • Eigenvalues and eigenvectors

Siblings22
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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

Children3
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  • Definition of (algebraic) multiplicity of an eigenvalue
  • Definition of geometric multiplicity of an eigenvalue
  • An n-by-n matrix nas n (complex) eigenvalues
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