Treffer: Examples of Chebyshev Sets in Matrix Spaces: Examples of Chebyshev sets in matrix spaces

Let A be a quadratic matrix of order \(n\) with complex elements (we denote this by \(A\in \mathbb{C}^{n\times n}\)) and \(U\Sigma V^{\ast }\) \(the\) singular value decomposition of \(A,\) where \(U,\) \(V\) are unitary matrices, \(\Sigma =\text{diag}( \sigma _{1}( A),\sigma _{2}( A) ,\dots,\sigma _{n}( A)) \) and \(\sigma _{1}( A) \geq \sigma _{2}( A) \geq \cdots \geq \sigma _{n}( A) \geq 0\) are the singular values of \(A.\) For \(k=1,2,\dots,n\) let us consider \( S_{1}^{(k)}=\{ S\in \mathbb{C}^{n\times n}:\sum \sigma _{j_{1}}( S) \dots\sigma _{j_{k}}( S) \leq 1\} .\) The authors prove that the set \( S_{1}^{( k) }( 1\leq k\leq n) \) is a Chebyshev set in \(\mathbb{C}^{n\times n}\) with respect to the spectral norm. One obtains the formula for the distance from \(A\in \mathbb{C} ^{n\times n}\) to \( S_{1}^{( k) }\) and proves that the metric projection \( P_{S_{1}^{(k)}}\) is globally Lipschitz-continuous.