Generalized singular values and interlacing inequalities (Q583329)

If a complex matrix B has singular values \(\tau_ 1,\tau_ 2,...,\tau_ b\), let \(\| B\|_ g=(\sum^{b}_{i=1}| \tau_ i|^ p)^{1/p},\quad p\geq 1.\) The g-singular values of an \(m\times n\) complex matrix A, \(m\geq n\), are defined by \(\gamma_ k=\| A^{[k]}\|_ g/\| A^{[k-1]}\|_ g,\quad k=1,2,...,n,\) where \(A^{[k]}\) denotes the kth compound of A. The author proves the following interlocking inequalities: if \(\gamma_ k\), \(k=1,2,...,n\), are the g-singular values of A and if \(\lambda_ j\), \(j=1,2,...,n-1\), are the g-singular values of an \(m\times (n-1)\) matrix obtained by deleting any one column of A, then \(\gamma_ k\geq \lambda_ k\geq \gamma_{k+1},\quad k=1,2,...,n-1.\)