Eigenvalue Decomposition

EigendecompositionIt is a method of representing the product of matrices by decomposing the matrix into eigenvalues and eigenvectors, but only diagonalizable matrices can be eigendecomposed.

Matrix multiplication corresponds to a transformation, which is to transform any vector into a new vector of another direction and length. In this process, the original vector will rotate and scale. If the matrix only scales some vectors without rotating them, then they are called the eigenvectors of the matrix, and the scaling ratio is the eigenvalue.

Eigenvalue decomposition decomposes the matrix A into the following form:

Among them, the matrix Q is composed of the eigenvectors of the matrix A,Σ is a diagonal matrix. Each diagonal element is an eigenvalue. The eigenvalues are arranged from large to small. The eigenvectors corresponding to these eigenvalues describe the direction of matrix change. In other words, the information of matrix A can be represented by eigenvalues and eigenvectors.

References

【1】Concept summary of eigenvalue decomposition, singular value decomposition and PCA