# Observations on the Computation of Eigenvalue and Eigenvector Jacobians +
## 5 On the Choice of a Normalization Vector  
As mentioned earlier, the choice of \(\mathbf{v}_{0}\) is arbitrary, so long as \(\mathbf{v}_{0}^{H}\mathbf{v}\
eq 0\). While this is true, the choice of \(\mathbf{v}_{0}\) can play an important role in the numerical stability of the system in finite precision computing. In particular, the closer the normalization vector gets to being orthogonal to the eigenvector, the more numerically unstable the system becomes. To demonstrate this phenomena consider Figure 1, which shows the condition number of the term \(\mathbf{A}-\lambda\mathbf{I}-\mathbf{v}\mathbf{v}_{0}^{H}\mathbf{A}/\alpha\) normalized by the number of dimensions, \(n\), as a function of the angle between \(\mathbf{v}_{0}\) and \(\mathbf{v}\). A higher condition number indicates a more poorly conditioned matrix prone to issues in finite precision computing. The samples were generated by producing 10,000 random matrices for each integer value in degrees for the angle between \(\mathbf{v}_{0}\) and \(\mathbf{v}\). Figure 1 shows matrices whose elements are real values which are normally distributed with zero mean and unit variance. In these experiments, the eigenvector, \(\mathbf{v}\), for each matrix is randomly selected. Since \(\mathbf{v}_{0}\) and/or \(\mathbf{v}\) may be complex, the angle between the vectors is computed using  
\[\cos\theta=\frac{\text{Re}\left(\mathbf{v}_{0}^{H}\mathbf{v}\right)}{\left\| \mathbf{v}_{0}\right\|\left\|\mathbf{v}\right\|}. \tag{54}\]  
Figure 1 indicates that the probability of poor matrix conditioning is greatest when \(\mathbf{v}_{0}\) and \(\mathbf{v}\) are nearly orthogonal. Presuming they are are in random directions, as \(n\) increases, it becomes highly probable that \(\mathbf{v}_{0}\) and \(\mathbf{v}\) will be nearly orthogonal. Figure 1: The condition number of the term \(\mathbf{A}-\lambda\mathbf{I}-\mathbf{v}\mathbf{v}_{0}^{H}\mathbf{A}/\alpha\) normalized by the number of dimensions, \(n\), as a function of the angle between \(\mathbf{v}_{0}\) and \(\mathbf{v}\). Here, \(\mathbf{A}\) is a real matrix. The numerical results are annotated to make the statistics more clearly visible according to the legend in the bottom right frame. Results are similar for a complex \(\mathbf{A}\). Therefore, it is clear that choosing the normalization vector \(\mathbf{v}_{0}\equiv\mathbf{v}\) (leading to an angle of 0 deg; or, equivalently, \(\mathbf{v}_{0}\equiv-\mathbf{v}\) leading to an angle of 180 deg) will ensure the best conditioning for the system and will usually lead to the most intuitive solution. The worst conditioning occurs as \(\mathbf{v}_{0}^{H}\mathbf{v}\to 0\) (angle of 90 deg). This is why \(\mathbf{v}_{0}\equiv\mathbf{v}\) is often chosen in practice, unless there is a compelling reason to choose otherwise. Note that selecting \(\mathbf{v}_{0}=\mathbf{v}\) causes the condition number of \(\mathbf{A}-\lambda\mathbf{I}-\mathbf{v}\mathbf{v}_{0}^{H}\mathbf{A}/\alpha\) to share many of the statistical properties of the condition number of \(\mathbf{A}\). This is especially true for large values of the condition number. [PERSON] (2000) developed probability density functions for the condition number of \(\mathbf{A}\) normalized by \(n\) when the elements of \(\mathbf{A}\) are real values with zero mean and unit variance,  
\[p\left(x\right)=\frac{2x+4}{x^{3}}\exp\left(-\frac{2}{x}-\frac{2}{x^{2}}\right). \tag{55}\]  
Likewise, for the case where \(\mathbf{A}\) is composed of complex elements with real and imaginary parts which are each normally distributed with zero mean and unit variance,  
\[p\left(x\right)=\frac{8}{x^{3}}\exp\left(-\frac{4}{x^{2}}\right). \tag{56}\]  
Histograms for when \(\mathbf{v}_{0}=\mathbf{v}\) are overlaid with [PERSON]'s probability density functions in Figure 2. These figures show that the probability of large condition numbers occurring are well described by [PERSON]'s result. Unfortunately, [PERSON]'s density function does not work as well for small condition numbers, but these are of little consequence.