Linear Algebra-basic concepts of Matrices and stochastic Matrices

Prepare to write an article about Singular Value DecompositionArticleIt suddenly found that it needed a lot of linear algebra knowledge. Therefore, we will first introduce the basic concepts and operations of linear algebra to help readers understand Singular Value Decomposition.

1. Basic Concepts

A matrix is a number composed of several rows and columns, which is equivalent to a table.

In particular, if the number of rows and columns in the matrix is the same, the matrix is a square matrix. In the matrix, the element AIJ in column A represents the element in column J of row I. The line from the element in the upper left corner to the lower right corner is the main diagonal line. The line between the elements on the primary diagonal line is called a Matrix Trace.

Only one row of matrix is called a row matrix, and only one column of matrix is called a column matrix. The matrix with all elements 0 is an empty matrix.

If a matrix has no element on the diagonal line, such a matrix is called a diagonal line.

If all elements in a diagonal matrix are equal, it is called a scalar matrix. In particular, if all elements are 1, it is called a matrix of units.

Exchange the rows and columns of a matrix. The obtained matrix is called the transpose matrix of the original matrix.

2. rank of the matrix

The element above the main diagonal line of a matrix is 0, which is called the lower diagonal matrix. Otherwise, if the element below is 0, it is called the upper diagonal matrix.

The rank of the matrix is equal to a smaller number of Linearly Independent rows or columns in the matrix. Therefore, convert the Matrix to an equivalent lower diagonal matrix, and then see how many non-zero columns are there or convert them to the upper diagonal matrix to see how many non-zero rows are there.

Example

3 random matrix

If all the elements in the matrix are non-negative values, you can divide each element in each row by the sum of all elements in this row to normalize them.