Matrix Algebra

• Introduction
  • Block matrix
• Linear independence, span, rank
  • Spaces associated with a matrix
  • Rank
• Invertible matrices
  • Similar matrices
  • Gram matrices
  • Pseudo inverses
• Trace and determinant
  • Trace
  • Determinants
• Unitary and orthogonal matrices
  • Orthogonal matrix
  • Unitary matrix
  • F unitary matrix
• Eigen values
  • Invariant subspaces
  • Triangular matrices
  • Similar matrices
  • Linear independence of eigen vectors
  • Diagonalization
  • Symmetric matrices
  • Hermitian matrices
  • Miscellaneous properties
  • Diagonally dominant matrices
  • Gershgorin’s theorem
• Singular values
  • The largest singular value
  • SVD and pseudo inverse
  • Full column rank matrices
  • Low rank approximation of a matrix
• Matrix norms
  • Norms like \(\ell_p\) on complex vector space
  • Properties of Frobenius norm
  • Consistency of a matrix norm
  • Subordinate matrix norm
  • Operator norm
  • p-norm for matrices
  • The 2-norm
  • Unitary invariant norms
  • More properties of operator norms
  • Row column norms
  • Block diagonally dominant matrices and generalized Gershgorin disc theorem
• Matrix Calculus
  • Trace
  • Frobenius Norm