Gradient Of X ↦ Tr ⁡ { X T A X − 1 B } X \mapsto \operatorname{tr}\left\{ X^T A X^{-1} B\right\} X ↦ Tr { X T A X − 1 B }

Mar 9, 2025 by ADMIN 122 views

$**Gradient of $X \mapsto \operatorname{tr}\left\{ X^T A X^{-1} B\right\}$: A Matrix Calculus Perspective**$

Introduction

Matrix calculus is a powerful tool for computing derivatives of scalar fields with respect to matrices. In this article, we will explore the gradient of the scalar field $X \mapsto \operatorname{tr}\left\{ X^T A X^{-1} B\right\}$ , where $A$ and $B$ are given matrices. This problem is a classic example of a matrix calculus problem, and its solution will provide valuable insights into the properties of matrix derivatives.

Background

Matrix calculus is a branch of mathematics that deals with the differentiation of scalar fields with respect to matrices. It is a fundamental tool in many fields, including machine learning, optimization, and signal processing. The gradient of a scalar field is a measure of the rate of change of the field with respect to its input variables. In the context of matrix calculus, the gradient is used to compute the derivative of a scalar field with respect to a matrix.

The Problem

Given matrices $A$ and $B$ , we want to compute the gradient of the scalar field $X \mapsto \operatorname{tr}\left\{ X^T A X^{-1} B\right\}$ . This problem can be solved using the chain rule and the product rule of matrix calculus.

Solution

To compute the gradient of the scalar field, we will use the chain rule and the product rule of matrix calculus. The chain rule states that the derivative of a composite function is equal to the derivative of the outer function evaluated at the inner function, multiplied by the derivative of the inner function. The product rule states that the derivative of a product of two functions is equal to the derivative of the first function evaluated at the second function, multiplied by the second function.

Using the chain rule and the product rule, we can write the gradient of the scalar field as:

\nabla_{X} \operatorname{tr}\left\{ X^T A X^{-1} B\right\} = \operatorname{tr}\left\{ A X^{-1} B\right\} + \operatorname{tr}\left\{ X^T A \nabla_{X} X^{-1} B\right\}

To compute the derivative of $X^{-1}$ , we will use the formula:

\nabla_{X} X^{-1} = -X^{-1} \nabla_{X} X X^{-1}

Using this formula, we can write the derivative of $X^{-1}$ as:

\nabla_{X} X^{-1} = -X^{-1} A X^{-1}

Substituting this expression into the gradient of the scalar field, we get:

\nabla_{X} \operatorname{tr}\left\{ X^T A X^{-1} B\right\} = \operatorname{tr}\left\{ A X^{-1} B\right\} - \operatorname{tr}\left\{ X^T A X^{-1} A X^{-1} B\right\}

Simplification

The expression for the gradient of the scalar field can be simplified using the properties of the trace function. Specifically, we can use the fact that the trace of a product of matrices is equal to the sum of the diagonal elements of the product.

Using this property, we can write the gradient of the scalar field as:

\nabla_{X} \operatorname{tr}\left\{ X^T A X^{-1} B\right\} = \operatorname{tr}\left\{ A X^{-1} B\right\} - \operatorname{tr}\left\{ A X^{-1} B\right\}

This expression simplifies to:

\nabla_{X} \operatorname{tr}\left\{ X^T A X^{-1} B\right\} = 0

Conclusion

In this article, we have computed the gradient of the scalar field $X \mapsto \operatorname{tr}\left\{ X^T A X^{-1} B\right\}$ using the chain rule and the product rule of matrix calculus. The resulting expression for the gradient of the scalar field is:

\nabla_{X} \operatorname{tr}\left\{ X^T A X^{-1} B\right\} = 0

This result provides valuable insights into the properties of matrix derivatives and has important implications for many fields, including machine learning, optimization, and signal processing.

References

[1] Amari, S. (1985). Differential-Geometric Methods in Statistics. Springer-Verlag.
[2] Magnus, J. R., & Neudecker, H. (1988). Matrix Differential Calculus with Applications in Statistics and Econometrics. John Wiley & Sons.
[3] Minka, T. P. (2000). A Comparison of Numerical Methods for Maximum Likelihood Estimation of Gaussian Mixtures. Technical Report, Microsoft Research.

Appendix

A.1 Derivative of $X^{-1}$