- Gauss-Jordan Elimination
- Symmetric Positive Deﬁnite(SPD) Matrices
- The key of Gauss-Jordan Elimination is to transfer matrix A to the identity matrix:
- Notice ,. Every iteration, we scale to 1.
- Via Gauss-Jordan, taking advantage of zeroes in the appended identity matrix, requires approximately floating point operations.
- Solving Ax = b should be accomplished by first computing its LU factorization (possibly with partial pivoting) and then solving with the triangular matrices.
Deﬁnition: Let . Matrix A is said to be symmetric positive definite(SPD) if
- A is symmetric; and
- for all nonzero vector .
Consider the quadratic polynomial
- The graph of this function is a parabola that is “concaved up” if . In that case, it attains a minimum at a unique value .
- Now consider the vector function given by where are all given. If A is a SPD matrix, then this equation is minimized for a unique vector x. If , plotting this function when A is SPD yields a paraboloid that is concaved up:
- Let be a SPD matrix. Then there exists a lower trianglar matrix such that . If the diagonal elements of L are chosen to be positive, this factorization is unique.
- Notice that and which are legal if . It turns out that if A is SPD, then
- in the first iteration and hence and are legal; and
- is again a SPD matrix.