is a nearest matrix from the set of matrices having largest eigenvalue mul-tiplicity at least k. The (locally identical) set of matrices having largest eigenvalue multiplicity exactly k is a manifold, and [Ous00] uses the corre-sponding projection as part of an eigenvalue optimization algorithm. We
Projection matrix. Suppose that is the space of complex vectors and is a subspace of . By the results demonstrated in the lecture on projection matrices (that are valid for oblique projections and, hence, for the special case of orthogonal projections), there exists a projection matrix such that for any .
If the vector space is complex and equipped with an inner product, then there is an orthonormal basis in which an arbitrary non-orthogonal projection matrix P is P = [1 σ1 0 0] ⊕ ⋯ ⊕ [1 σk 0 0] ⊕ Im ⊕ 0s, where σ1 ≥ σ2 ≥ … ≥ σk > 0. The integers k, s, m, and the real numbers σi are uniquely determined.
Elementary Matrices, Homogeneous Equations and Non-homogeneous Equations 7 Invertible matrices, Homogeneous Equations Non-homogeneous Equations 8 Vector spaces 9 Elementary Properties in Vector Spaces. Subspaces 10 Subspaces (continued), Spanning Sets, Linear Independence, Dependence 11 Basis for a vector space 12 Dimension of a vector space