Matrix Calculus

A₁ A₂ approximate arbitrary B₁ Bodewig C₁ calculation characteristic equation coefficients component computed condensation convergence corresponding d₁ definite derived system determinant diagonal elements diagonal matrix diagonal terms dominant eigenvalue e₁ eigen eigencolumns eigenrows eigenvalues eigenvectors elementary divisors elimination equal error follows formula Frobenius Further Gauss holds i-th inserted inverse iteration methods likewise linear combination linearly linearly independent M₁ matrix of order multiplications null matrix number of operations obtained orthogonal transformation P₁ parallel plane premultiplied procedure PROOF punched cards Q₁ quadratic form quadric quotient R₁ reciprocal replaced requires resp right sides row and column S₁ scalar product Seidel sequence Similarly simple product solution solved square matrix starting vector subtracting sum of squares sum-column sum-row symmetrical matrix system Ax t₁ THEOREM triangular matrix triangular system u₁ unknown V₁ vanish Vm+1 x'Ax x₁ y₁ yields zero λ₁ ΣΣ