The paper deals with the space of real \(n\times n\) matrices. The material covered is based on the properties of the Frobenius inner product on this space defined by \(\langle A,B\rangle_F=\mathrm{tr}(A^T B)\). The major part of the paper is devoted to the study of the geometrical structure of the subspace of symmetric matrices \(S_n\) and the location of symmetric orthogonal matrices within this space. The results of this study are then used to obtain an easily computable lower bound on the Frobenius condition number \(\kappa_F(A)=\|A\|_F\|A^{-1}\|_F\) of \(A\in S_n\). Furthermore, a brief discussion and comparison of the Frobenius condition number and the one based on the operator norm corresponding to the 2-norm for vectors is provided. Finally, some results for the symmetric case are generalized to non-symmetric matrices.
The principal notion for describing the location of matrices is the angle of two matrices \(A\) and \(B\) defined by \(\cos(A,B)=\langle A,B\rangle_F/(\|A\|_F\|B\|_F)\). For the Frobenius condition number it follows that \(\kappa_F(A)=n/\cos(A,A^{-1})\).
For nonsingular \(A\in S_n\) with the spectral decomposition \(A=QDQ^T\) a new symmetric orthogonal matrix \(Q_A=Q\,\mathrm{sgn}(D)Q^T\) is defined, where each element of \(\mathrm{sgn}(D)\) is the sign of the corresponding element in \(D\).
First it is shown that in the case \(n=2\) the angle between \(A\) and \(A^{-1}\) is twice as large as between \(A\) and \(Q_A\) and can be computed using the angle between \(A\) and \(I\). Hence \(\kappa_F(A)\) is obtained easily. Then, for general \(n\), the estimate \[ \kappa_F(A)\geq\frac{n}{\cos(A,Q_A)} \] is proved. A substantial effort is needed to compute the right-hand side except for special cases, e.g., if \(A\) is positive definite since then \(Q_A=I\). It turns out that any symmetric orthogonal matrix, like, e.g. \(Q_A\), with \(k\) negative eigenvalues belongs to the cone surface \(S(I,c_k)=\{B\in S_n:\cos(B,I)=c_k\}\) with \(c_k=(n-2k)/n\). In general, \(A\) lies between two of these surfaces. A matrix of the form \(A+\alpha I\) is constructed such that it belongs to the surface closest to \(A\). It is then shown that \[ \kappa_F(A)\geq\frac{n}{\cos(A,Q_A)}\geq\frac{n}{\cos(A,A+\alpha I)}. \] Since to find \(\alpha\) just \(\mathrm{tr}(A)\) and \(\mathrm{tr}(A^2)\) need to be evaluated, one obtains a practically computable lower bound for the Frobenius condition number.
The paper is well structured and the exposition is clear.