What is LDV Decomposition?
LDV decomposes a matrix A s.t. where L is a lower unitriangular matrix, D is a diagonal matrix w/ non-zero entries & V is an upper unitriangular matrix. A is only regular if and only if it can be decomposed. from our LU factorization, see here Elementary Matrices, LU Factorization, and Backwards Substitution. The LDV decomposition of a regular symmetric matrix S looks like: .
How do we compute LDV factorization?
We use gaussian elimination on A to get the matrix U, while solving for U, we form L. Then, with U, we take just its diagonals to form D, and we normalize U (divide each row by its pivot) to get V.
What is row echelon form?
A matrix in row echelon form has each row start with a nonzero entry (a pivot) to the right of the pivot of the row above it. Allow entries below the pivot must be zero and all zero rows should be on the bottom.
What is the rank of a matrix?
The rank of a matrix is the number of pivots in echelon form. The rank tells us the number of unique equations this system has. A matrix is nonsingular if its size is equal to its rank. A system of linear equations can have no solutions, infinitely many solutions, or one unique solution.