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Observations on the Computation of Eigenvalue and Eigenvector Jacobians^{ †}

^{1}

^{2}

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^{†}

## Abstract

**:**

## 1. Introduction

## 2. Existence of Eigenvalue and Eigenvector Jacobians

#### 2.1. Constraining the Eigenvectors to the Unit Hypersphere

#### 2.2. Constraining the Eigenvectors to a Hyperplane

## 3. Previous Work

## 4. Compact Expressions for Eigenvalue and Eigenvector Jacobians

#### 4.1. Eigenvector Jacobian

#### 4.2. Eigenvalue Jacobian

## 5. On the Choice of a Normalization Vector

## 6. Simplified Cases

#### 6.1. Real Symmetric Parent Matrix

#### 6.2. Real Diagonal Parent Matrix

#### 6.2.1. Simplified Jacobians for a Diagonal Matrix

#### 6.2.2. Perturbation to the Eigenspace of a Diagonal Matrix

#### 6.2.3. Perturbations to the Eigenspace of a Diagonalizable Matrix

## 7. Numerical Validation

## 8. Comparison of Performance

## 9. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**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 $\mathbf{A}$.

**Figure 2.**Histograms of 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, compared to Edelman’s probability density function for the condition number of $\mathbf{A}$. Here, $\mathbf{A}$ is a real matrix.

**Figure 3.**Histograms of percent difference between analytic derivatives computed using Equations (36) and (40) (eigenvalue derivatives top and eigenvector derivatives bottom) and finite forward differencing for 5000 randomly generated matrices of each size. The histograms are of the percent difference for each element of the eigenvalue and eigenvector derivatives (for example, for each n × n matrix there are n

^{2}eigenvalue derivative elements and n × n

^{2}eigenvector derivative elements). Similar histograms are presented in [39] for the method discussed in that paper.

**Figure 4.**A plot of minimum computation time versus matrix size for the method from [39] (original method) and the method proposed in this paper (new method). Note that the method from [39] encounters numerical stability issues around a matrix size of 35 due to Equation (29). This is why there is a cut-off in the data.

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**MDPI and ACS Style**

Liounis, A.J.; Christian, J.A.; Robinson, S.B.
Observations on the Computation of Eigenvalue and Eigenvector Jacobians. *Algorithms* **2019**, *12*, 245.
https://doi.org/10.3390/a12120245

**AMA Style**

Liounis AJ, Christian JA, Robinson SB.
Observations on the Computation of Eigenvalue and Eigenvector Jacobians. *Algorithms*. 2019; 12(12):245.
https://doi.org/10.3390/a12120245

**Chicago/Turabian Style**

Liounis, Andrew J., John A. Christian, and Shane B. Robinson.
2019. "Observations on the Computation of Eigenvalue and Eigenvector Jacobians" *Algorithms* 12, no. 12: 245.
https://doi.org/10.3390/a12120245