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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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  • Definition of eigenspace of a matrix
  • An eigenspace of a matrix is a nontrivial subspace.
  • An eigenspace of a matrix is the null space of a related matrix.
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  • Matrix transpose commutes with matrix inverse.#23526
  • Matrix transpose is an involution.#23504
  • Matrix-vector multiplication is a linear transformation.#23856
  • Matrix-vector product is associative#23486
  • Matrix-vector products#23482
  • Multiplication#23487
  • Multiplication by a change of coordinates matrix converts representations for different bases.#23997
  • Multiplication by a Hermitian matrix commutes with the standard inner product on C^n.#23598
  • Multiplication of block/partitioned matrices#23569
  • Multiplicity#23695
  • Multiplying a row by a scalar multiplies the determinant by that scalar.#23739
  • Nilpotent matrices#23580
  • Non-example of a linear transformation#23850
  • Nonsingular matrices and equivalences#23640
  • Normal matrices#23599
  • Norm and length#23907
  • Notation for entry of matrix#23450
  • Notation for the set of m by n matrices#23453
  • Operations on matrices#23460
  • Orthogonality#23914
  • Orthogonality and projection#23893
  • Parametric form of the solution set of a system of linear equations#23419
  • Parametric vector form of the solution set of a system of linear equations#23420
  • Particular types of matrices#23535
  • Projection#23937
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  • Aspen Summer 2016 - Entanglement Matters#1213
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