Principal values and principal subspaces of two subspaces of vector spaces with inner product
Two subspaces \(\Sigma_1\) and \(\Sigma_2\) of arbitrary dimensions in the Euclidean space \(E_n\) are defined. It is proved that the angle between these two subspaces \(\varphi(\Sigma_1, \Sigma_2)\) is equal to the angle between their orthogonal complements \(\Sigma_1^*\) and \(\Sigma_2^*\). Matrices \(A_1\) and \(A_2\) whose vector rows generate the subspace \(\Sigma_1\) and \(\Sigma_2\), respectively, are used for the definitions of principal values as common eigenvalues of these matrices. The principal subspaces or principal directions for the eigenvalues with multipolarity \(1\) are introduced as the orthogonal eigenspaces resulting from unique decomposition of the subspaces \(\Sigma_1\) and \(\Sigma_2\). As a final result the canonical matrix for two subspaces \(\Sigma_1\) and \(\Sigma_2\) is given together with its relation to the duality principle.
scientific article; zbMATH DE number 1549918