Overview - 2023.2 English

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2023.2 English

Cholesky_Inverse, matrix inversion with the usage of Cholesky decomposition.

Cholesky decomposition is a decomposition of a Hermitian, positive-definite matrix into the product of a lower triangular matrix and its conjugate transpose, in the form of \(A = LL^*\). \(A\) is a Hermitian positive-definite matrix, \(L\) is a lower triangular matrix with real and positive diagonal entries, and \(L^*\) denotes the conjugate transpose of \(L\). Cholesky decomposition is useful for efficient numerical solutions.

matrix \(A^{-1}\) could be computed in the form of \(A^{-1} = (L^{-1})^*L^{-1}\).

\[ \begin{align}\begin{aligned}AA^{-1} = I\\A = L*L^*\\A^{-1} = (L^{-1})^*L^{-1}\end{aligned}\end{align} \]

As matrix \(L\) is a triangular matrix, \(L^{-1}\) is easy to compute.