Cholesky decomposition for Matrix Inversion

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本文探讨了Cholesky分解在求解特殊类型矩阵（如Hermitian正定矩阵）逆运算的问题上简化复杂度的方法。通过将目标矩阵分解为下三角矩阵和上三角矩阵的乘积，使得计算逆矩阵变得更为简便。此外，文章还讨论了如何在复数矩阵中使用共轭转置，并介绍了Cholesky分解在实际应用中的优势。

Cholesky decomposition for Matrix Inversion

Matrix inversion is a classical problem, and can be very complicated for large matrices. There are many ways to simplify this for special types of matrices. Among them, one is to transform the matrix into a set of upper or lower triangular matrices. Consider our target matrix $\mathbf{A}$ which is Hermitian and positive-definite . Such matrices are quite famous and an example is the covariance matrix in statistics. It’s inverse is seen in the Gaussian probability density function for vectors. Then, Cholesky decomposition breaks

$\mathbf{A} = \mathbf{L}\mathbf{L}^T = \mathbf{U}^T\mathbf{U}$

where $\mathbf{L}$ is a lower triangular matrix, while $\mathbf{U}$ is an upper triangular matrix.

It is much easier to compute the inverse of a triangular matrix and there exist numerical solutions. Then the original matrix inverse is computed simply by multiplying the two inverses as

$\mathbf{A}^{-1} = (\mathbf{L}^{-1})^T(\mathbf{L}^{-1}) = (\mathbf{U}^{-1})(\mathbf{U}^{-1})^T$