• Standard Power Method:

Matrix iteration using infinity-norm. Need to normalize the vector everytime before next loop, and the (first) entry agrees with the infinity-norm is the approximation of eigenvalue with the largest magnitude (spectral radius of the matrix). Notice that this includes the sign.

• Inverse Power Method:

Matrix iteration using infinity-norm. This matrix is not our original matrix A, instead, it’s the inverse of A-qI, where we hope to find the eigenvalue closest to q. This method converges faster than the standard one, and also can be treated as a modified method from the standard one by making p be the approximation of the spectral radius, e.g. using Rayleigh quotient. The final result you get should be 1/(lambda-q), and the expensive inverse operation is unnecessary: we have multiple choices of iteration techniques (e.g. standard Gaussian elimination, Gaussian elimination with partial/scaled partial/complete pivoting strategies, Jacobi, Gauss-Seidel, SOR, etc.) to solve a linear system in each step.

• Symmetric Power Method:

Matrix iteration using 2-norm. Special methods for special cases, symmetric power method converges faster than the standard one. It uses 2-norm instead of infinity-norm, and making full use of Rayleigh quotient, i.e. normalize the vector everytime before next loop, then use that unit vector to construct a quadratic form, which is the approximation of eigenvalue with the largest magnitude (spectral radius of the matrix). The sign of this eigenvalue mainly depends on the definite of our matrix.

• Deflation Method:

This technique helps us to find out other eigenvalues inductively, i.e. given k known dominant eigenvalues, we can compute the k+1 th dominant eigenvalue. A – (sum of) lambda x^t v, where x is the vector s.t. x^t v be an unit vector. However, this method may get roundoff error blows up, so usually coupled with IPM.

• QR Factorization

This method is focusing on a class of more specialized matrices: tridiagonal matrices. Equipped with Householder method, we can also use QR factorization method to arbitrary symmetric matrices. Q represents an orthogonal matrix, while R represents an upper triangular matrix. Like we can do A1=Q1R1, then our A2:=R1Q1 and do factorization again for A2=Q2R2, etc. The sequence of {An} produced from this process is a sequence of similar matrices, with the larger n, the smaller magnitude of off-diagonal entries, and An approaches to the diagonalized A with diagonal entries approach to eigenvalues. The exact way to factorize A is using rotation matrices multiple times, and those matrices are orthogonal so their inverse are direct transpose.

* for MATH 105A Review, Ch 9