What are some applications for inverting matrices

linear algebra

I would like to know some of the real-world (preferably not in the mathematics field) applications for inverting matrices, specifically doing it fast. It would be great if you could provide online resources to help me answer this question.

Thanks.

Best Answer

Matrix inversion is useful whenever you are working with matrices, and the actual applications of this are actually pretty great these days (e.g. fields like machine learning and statistics readily use the language of linear algebra, matrices, etc.). Just like we want to be able to solve equations like $xy = 10x$ when $x$ is a nonzero real number (clearly $y = 10$), inverting provides the theory to solve things like $AB = 10A$ when $A$ and $B$ are matrices.

For an example of an application, least squares regression uses matrix inversion. If you have some data arranged in a matrix $X$ that predicts response values $\mathbf{y}$, the "best linear estimate" (in a least squares sense) in the form $\widehat{\mathbf{y}} = X \widehat{\beta}$ has parameters calculated by $\widehat{\beta} = (X^T X)^{-1} X^T \mathbf{y}$. (The first part $(X^T X)^{-1} X^T \mathbf{y}$ is known as the "hat" or "projection matrix".)

So I think this is a very "real-world" application of matrix inversion. Doing it fast is of practical importance, since very large datasets correspond to very large matrices, and computing matrix inverses quickly is a challenging problem. (For large matrices and when speed is an issue, it seems that there are different approaches that try and avoid the inversion step as much as possible when speed and efficiency is of importance -- things like QR/SVD/Cholesky factorisation, etc. -- that you might search up if you are interested.)

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