The Matrix

© 2001 Daniel du Prie Our world is everywhere, but that is not the place our bodies live. (Barlow, 1996) You have always lived in a dream world. This is the world where this exists today. Welcome to the desert - it is true. From Plato's "Charmides" to the Wachowski brothers "The Matrix" (1999), there is a tradition of Western literature that regards the city as a utopia, a distant pier, or both.

The transpose of the matrix product takes a simple form. (AB) '= B'A'. The inverse of A is expressed as A ^ -1 and is defined as a matrix like A ^ -1 A = I (I is the identity matrix). However, A ^ -1 is mainly used as a theoretical tool, so please do not actually use it in most software applications. Because A ^ -1 can only be represented with limited precision in digital computers, the algorithm is often able to obtain a more accurate estimate of x using the value of b. Many mathematical objects can be better understood by dividing them into components, or finding their common characteristics, rather than a way to express them. For example, an integer can be broken down into prime factors. The number 12 depends on whether it is decimal or binary, but 12 = 2 × 2 × 3 always applies.

A well-known matrix decomposition method is singular value decomposition (SVD). At higher levels, SVD is an algorithm that decomposes matrix A into the best low rank (ie, smaller / simpler) approximation of the original matrix A. Mathematically, we break up A into two unitary matrices and diagonal matrices. Where A is the input data matrix (user evaluation), U is the left singular vector (user's "feature" matrix) and Sum is a singular value pair. Angle matrix (basically the weight / intensity of each concept) V ^ T is the right singular vector (the "feature" matrix of the movie). U and V ^ T are column orthogonal and represent different things. U indicates the degree to which the user "likes" each function, and V ^ T indicates to what extent each function is related to each movie.