The Matrix

Morpheus: I think I feel like Alice now. Oh? Rolling a hole in a rabbit?

Morpheus: I saw your eyes. I am looking forward to waking up, so you have someone who accepts what he sees. Ironically, this is not far from the truth. Do you believe in fate, neo?

Morpheus: I know * complete * what you mean. Why are you here? You are here because you know something. You can not explain what you know, but you feel it. You already feel what is wrong with your life, the world. You do not know what it is, but it's there, just like you split in your mind, I am crazy for you. This feeling will bring you to my side. Do you know what I am saying?

Morpheus: The matrix is ​​everywhere. That is right next to us. Even now, in this room. I can see it when I turn on the TV or outside the window. When you go to work, you can feel ... when you go to church ... when you pay tax. The world is attracted to your eyes, you are blinded from the truth.

Morpheus: You are a slave, Neo. Just like anyone else, you are born and bound. Once in prison, you can neither taste nor see nor touch it. For your soul prison

However, I feel that I escaped the corporate slavery procession on the first day. Ultimately I realized there was another matrix outside the matrix. Like Russian nesting dolls, each matrix is ​​the creation of a matrix outside it. eternally. Every day there is another escape, but that is only when I have enough courage

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 elements along the diagonal of D are called singular values ​​of matrix A. The U column is called the left singular vector. Right singular vector

A well-known matrix decomposition method is singular value decomposition (SVD). At higher levels, SVD is an algorithm that decomposes Matrix A into the lowest level (ie, smaller / simpler) 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 a right singular vector ("feature" matrix of the movie). U and V ^ T are column orthogonal and represent different things. U indicates how much the user likes each function, and V ^ T indicates how much each function is associated with each movie.

X represents the utility matrix and U is the left singular matrix representing the relationship between the user and the underlying element. S is a diagonal matrix describing the strength of each potential factor and V transpose is a regular matrix representing the similarity between the project and the underlying factor. Well, you may want to know the root cause here. Explaining the attributes or concepts of a user or project is a broad concept. For example, in the case of music, the underlying element can refer to the type to which the music belongs. SVD reduces the dimension of the utility matrix by extracting its underlying factors. Essentially, we map each user and each item to a potential space of dimension r. Therefore, users and projects can be compared directly, which helps to understand the relationship between users and projects. The figure below shows this idea