BASIC LINEAR ALGEBRA AND MATRIX

minding some classical definitions about matrices. Let A = [aij] be a matrix in Cn×m (whose ij-th element is aij) and let A = [aij] be a matrix in Cp×q, then • If n = p and m = q, the ij-th element of the matrix sum C = αA+βB = [cij] is defined by cij = αaij + βbij. • If m = p, then the product C = AB is defined by C = [cij] such that cij = p∑ k=1 aikbkj, i = 1,..., n; j = 1,..., q. • The transpose of the matrix A is the m × n matrix AT = [aji] and the transpose conjugate is defined by A ∗ = ĀT where the bar denotes the complex conjugaison. • The ith row of A is defined by ai ∗ = [ai1, ai2,..., aim] • The jth column is defined by a∗j = a1j a2j anj Matrices are related to linear mapping between vector spaces. Let f be a linear mapping from Rm onto Rn and consider two bases B1 = {v1,..., vm} and B2 = {w1,..., wn} then A = MatB1,B2f where the j-th column of A is expressed as a∗j = f(vj) = a1jw1 +... + anjwn, j = 1,...,m. In matrix computation, we usually use the following particular matrices. The square matrix (m = n) A is said to be • Diagonal iff aij = 0 if i 6 = j and we denote A = diag(a11,..., ann). • Identity is defined by I = diag(1, 1,..., 1). • Hermitian iff A ∗ = A and skew-Hermitian • Symmetric iff AT = A and skew-symmetric iff AT = −A. • Normal iff A∗A = AA∗. • Nonnegative if aij ≥ 0 for all i, j. • Unitary iff A∗A = I (orthonormal in the real case) and A is said to be orthogonal if A is not necessarly square and ATA = I • Upper triangular iff aij = 0 for i> j. 1 • Lower triangular if aij = 0 for i < j. • Tridiagonal if aij = 0 for i, j such that |i − j |> 1 and we denote A = tridiag(ai,i−1, aii, ai,i+1). • Permutation if the column (or rows) of A are a permutation of the identity matrix. • A Toeplitz matrix is a matrix whose each diagonal is constant. If T = [tij], we have tij = ti+1,j+1 = ai−j. • A Hankel matrix H = [hij] is a square matrix such that hi,j = hi−1,j+1