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  • Matrices
  • Eigenvalues and eigenvectors
  • Determinants

Siblings41
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  • Basic terminology and notation
  • Operations on matrices
  • Particular types of matrices
  • Matrix equivalence
  • Canonical forms of matrices
  • Factorization of matrices
  • Similarity of matrices
  • Nonsingular matrices and equivalences
  • Rank and mullity
  • Eigenvalues and eigenvectors
  • 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.
  • Determinants
  • Definition of trace of a matrix
  • Cofactors
  • Determinants and operations on matrices
  • Determinants axiomatically
  • The determinant of a matrix measures the area/volume of the parallelogram/parallelipiped determined by its columns.
  • The determinant of the matrix of a linear transformation is the factor by which the area/volume changes.
  • Definition of adjugate/classical adjoint of a matrix
  • A matrix is called ill-conditioned if it is nearly singular
  • The condition number of matrix measures how close it is to being singular
  • 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

Children24
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  • Echelon matrices
  • Definition of unit matrix
  • Definition of permutation matrix
  • Elementary matrices
  • Triangular matrices
  • Block matrices
  • Symmetric matrices
  • Nilpotent matrices
  • Definition of orthogonal matrix
  • Unitary matrices
  • Definition of band matrix
  • Definition of Vandermonde matrix
  • Definition of Markov matrix
  • Hermitian matrices
  • Normal matrices
  • The eigenvalues of a triangular matrix are the entries on the main diagonal.
  • A matrix with real entries has eigenvalues occurring in conjugate pairs.
  • Hermitian matrices have real eigenvalues.
  • Distinct eigenvalues of a Hermitian matrix have orthogonal eigenvectors.
  • Definition of positive-definite matrix
  • Formula for the determinant of a 2-by-2 matrix.
  • Formula for the determinant of a 3-by-3 matrix.
  • The determinant of a triangular matrix is the product of the entries on the diagonal.
  • Theorem describing the determinants of elementary matrices.
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