A Survey of Matrix Theory and Matrix Inequalities

a₁ b₁ called charac characteristic polynomial characteristic roots characteristic vector column operations commutes complex numbers convex function corresponding defined denote determinantal divisors diag diagonal matrix direct sum doubly stochastic matrices elementary divisors elementary matrices elements equal equivalent example exists a unitary field of values follows Hence hermitian matrices hermitian matrix indecomposable matrix inequality invariant factors linearly independent M₂(F M₂(R main diagonal Marcus Math maximal root minimal polynomial Mn(C Mn(R Moreover multiplicity n-square matrix n-tuples nonnegative hermitian nonnegative matrix nonsingular normal matrix notation obtain On,n orthogonal orthonormal set P₁ permutation matrix positive definite hermitian positive semidefinite Proof Qk,n real numbers result row and column row stochastic row sums S₁ sequences similarity invariants skew-hermitian subdeterminants submatrix subset Suppose symmetric function teristic roots theorem u₁ unitary matrix vector space zero