Matrices and Linear Transformations, Volume 1

a₁ a₁x AE Fn b₁ b₂ characteristic polynomial characteristic value characteristic vectors commute Consider converges defined Definition denoted diagonal matrix dim CS(A E L(V E₁ elementary matrices elementary row operations elements equation Example Exercise exists field F follows from Theorem form a basis function hence Hermitian Hermitian matrix integers inverse Jordan canonical form Jordan canonical matrix linear operator linear transformation linearly independent lower triangular mA(x minimum polynomial monic multiplication n-invariant N₁ nilpotent nonsingular matrix NS(A one-to-one orthogonal orthonormal basis P-¹AP P₁ p₁(x product of elementary Proof Prove Theorem reader real numbers represents result row equivalent row-reduced echelon form RS(A satisfies scalar Section sequence Show Smith canonical matrix solution space over F subdeterminants subspace symmetric system AX Theorem 3-2 triangular matrix unique unitary unitary matrix V₁ vector space Vn(F Vn(R X₁ zero αι λι