An Introduction to Principal Component Analysis and Online Singular Value Decomposition

2.1 Computing Eigenvectors from the Characteristic Polynomial The characteristic polynomial can be used to develop an algorithm for computing the eigenvectors and eigenvalues. This algorithm, which is known as Power Iteration, takes advantage of the property that vectors which are transformed by the matrix A will be scaled in the direction of the largest eigenvector. Initially, the vector is chosen to be some random non-zero vector. This vector is iteratively multiplied by the matrix, which causes it to gradually align with the largest eigenvector. This is repeated until the vector has converged. The eigenvector and eigenvalue are stored, and the eigenvector’s contribution is subtracted from the matrix. The process is then repeated to compute the next largest eigenvector. Here is the pseudocode for determining an eigenvector: