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.
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.
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.
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.
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