The Matrix

In the near future, humans have become slaves of artificial machines with artificial intelligence. To control humans, machines put them in Matrix which is a very sophisticated computer simulation program that determines the real world we know. A group of freedom fighters left the war of the dark empire and fought for the liberation of mankind. They are guided by a strange man named Morpheus. A young man named Neo was adopted by the organization as a voter who finally released mankind.

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.

Let A be an m × n matrix. We define U as an m × m matrix, D as an m × n matrix, and V as an n × n matrix. Each of these matrices is defined as having a special structure. Both matrices U and V are defined as orthogonal matrices. The matrix D is defined as a diagonal matrix. Please note that D is not necessarily a square. The diagonal element of D is called the singular value of matrix A. The U column is called the left singular vector. Right singular vector