The authors present explicit expressions, computable from the data, for mixed and componentwise condition numbers for both Moore-Penrose inverse and linear least squares problems. In both cases the data matrices have full column (row) rank.
For a matrix \(A \in \mathbb R^{m \times n}\), \(vec(A) \in \mathbb R^{mn}\), is defined by \(vec(A)=[a_1^T, \ldots, a_n^T]^T\), where \(A=[a_1,\ldots,a_n]\) with \(a_i \in \mathbb R^m\), \(i=1, 2, \ldots, n\). If \(m_+(A)\) and \(c_+(A)\) denote the mixed and componentwise condition numbers of the Moore-Penrose inverse of \(A\), respectively, the authors obtain the following expressions: \[ m_+(A)=\frac{\| | M(A)| vec(| A| )\| _{\infty}} {\| vec(A^+) \| _{\infty}}, \qquad c_+(A)=\left \| \frac{| M(A)| vec(| A| )} {vec(A^+)} \right \| _{\infty}, \] where \(A^+\) is the Moore-Penrose inverse of \(A\), and \[ M(A)=[-(A^{+^T} \otimes A^+)+ ((I_m-AA^+)\otimes(A^TA)^{-1})\Pi], \] with \(\Pi\) the vec-permutation matrix.
On the other hand, the authors obtain explicit expressions for mixed and componentwise condition numbers for linear least squares problems, \(m_{ls}(A,b)\) and \(c_{ls}(A,b)\).
Finally, they also obtain upper bounds for the condition numbers \(m_+(A)\), \(c_+(A)\), \(m_{ls}(A,b)\) and \(c_{ls}(A,b)\).