LU Decomposition Calculator

LU decomposition expresses a given square matrix A as the product of two specific triangular matrices such that A = L * U. Here, L represents a lower triangular matrix featuring zeros above the main diagonal and ones along the diagonal (in standard Doolittle algorithm), while U represents an upper triangular matrix containing zeros below the main diagonal. For a 3x3 matrix, the factorization expands as:

A = [a11, a12, a13; a21, a22, a23; a31, a32, a33] = [1, 0, 0; l21, 1, 0; l31, l32, 1] * [u11, u12, u13; 0, u22, u23; 0, 0, u33].

By matching elements through row operations or Crout/Doolittle methods, you isolate the unknown entries of L and U sequentially.

Let us find the LU decomposition of a 2x2 matrix A where:

A = [[4, 3], [6, 3]]

Step 1: Set up the matrices L and U for a 2x2 system. Let L have 1s on the diagonal: L = [[1, 0], [l21, 1]] and U = [[u11, u12], [0, u22]].

Step 2: Multiply L and U to match matrix A. From the top-left entry, 1 * u11 = 4, so u11 = 4. For the top-right entry, 1 * u12 = 3, so u12 = 3.

Step 3: Solve for the lower matrix row. For the bottom-left entry, l21 * u11 = 6, which means l21 * 4 = 6, yielding l21 = 1.5.

Step 4: Solve for the final diagonal entry of U. For the bottom-right entry, (l21 * u12) + (1 * u22) = 3. Substituting values gives (1.5 * 3) + u22 = 3, leading to 4.5 + u22 = 3, so u22 = -1.5.

Result: L = [[1, 0], [1.5, 1]] and U = [[4, 3], [0, -1.5]].

Always verify that your input matrix is square before attempting factorization, as non-square matrices require singular value decomposition or QR factorization instead. Keep an eye out for potential zero pivots during manual calculations; if a zero appears on the diagonal, matrix pivoting or permutation matrices (PLU decomposition) may be required to complete the process accurately.