Similarly, for we compute which shows that the formula given in the claim is an upper bound for . It is a major theorem of Linear Algebra that such matrices have nonnegative eigenvalues and orthogonal eigenvectors. A t AQ is a diagonal matrix which we denote by D . It follows that the claimed formula is an upper bound. Our formulas clearly hold also for non-square matrices.
All partial derivatives exist and are continuous everywhere.
The matrix of derivatives are determined easily by computing the partial derivatives. Proof We need to compute If u is the zero vector, the limit clearly exists and equals zero. We have show that all directional derivatives exist. Theorem Let f be a real-valued function defined on an open set . If the partial derivatives , then f is continuous in E .