Matrix Multiplication

The obvious matrix multiplication algorithm for an n₁×n₂ matrix by an n₂×n₃ matrix has O(n₁ n₂ n₃)-time complexity, i.e., n³ if n₁ = n₂ = n₃ = n. Strassen (1969) showed how to reduce this from n³ to O(n^log₂(7)) by recursively performing seven multiplications of n/2×n/2 matrices, e.g., [code]. Note that log₂(7) ≈ 2.8 < 3. The improvement is not dramatic, but it sparked a cottage industry looking for further improvements -- search for 'fast matrix multiplication' in the [bib].

Matrix Inversion

The inverse of a square k×k matrix, m, is a k×k matrix, m^-1, s.t.

m m^-1 = m^-1 m = id,

where id is the k×k identity matrix.

(Note, a singular matrix does not have an inverse.)

The matrix inversion problem can be solved by Gauss-Jordon elimination.

The "naive" Gauss-Jordon elimination algorithm is numerically unstable. Gauss-Jordon elimination with pivoting is more stable.

There seems to be some debate in the literature over whether (i) partial pivoting (row pivoting) is adequate or (ii) full pivoting (row and column pivoting) is needed in general.

Random example generated by JavaScript ...

(try "reload" for a new example)

Eigen Values and Eigen Vectors

See maths/Eigen-values and algorithm.