Maximizing bilinear forms subject to linear constraints
Let \(A \in\mathbb{R}^{m \times n}\), \(B \in\mathbb{R}^{m \times p}\), \(C \in\mathbb{R}^{k \times m}\), \(G \in\mathbb{R}^{p \times n}\), \(d \in\mathbb{R}^k\), \(h \in\mathbb{R}^p\). The author studies the problem of maximizing the bilinear form \((x^T Ay)/(|x|\cdot |y|)\), or its absolute value subject to linear constraints of the form \(Cx = d\), \(Gy = h\) or of the form \(Cx \in R(D)\), \(Gy \in R(H)\), where \(R(D)\) and \(R(H)\) represent the column spaces of the matrices \(D \in\mathbb{R}^{k \times r}\) and \(H \in\mathbb{R}^{p \times \ell}\). Finally, he determines the angle between two subspaces of \(\mathbb{R}^m\) and the canonical correlations for the pair \((A,B)\) subject to the linear equality constraints \(Cx = d\), \(Gy = h\).
scientific article; zbMATH DE number 698651