1. What is LU Decomposition?

LU decomposition factors a square matrix A into the product of a lower triangular matrix L and an upper triangular matrix U. The L matrix has 1s on its diagonal, while U has non-zero values on its diagonal.

2. How Does the Calculator Work?

The calculator performs the following steps:

1. Parses the input matrix
2. Initializes L and U matrices
3. Computes the upper triangular matrix U
4. Computes the lower triangular matrix L
5. Verifies the decomposition is valid

The decomposition uses Doolittle's algorithm which sets the diagonal elements of L to 1.

3. Importance of LU Decomposition

Applications: LU decomposition is fundamental for solving systems of linear equations, computing determinants, and finding matrix inverses. It's more efficient than Gaussian elimination when solving multiple equations with the same coefficient matrix.

4. Using the Calculator

Tips: Enter your square matrix using commas to separate columns and semicolons to separate rows. For example, a 3x3 matrix would be entered as: "1,2,3;4,5,6;7,8,9".

5. Frequently Asked Questions (FAQ)

Q1: What matrices can be LU decomposed?
A: Any square matrix that can be reduced to row echelon form without row exchanges. Some matrices may require pivoting.

Q2: How is LU different from QR decomposition?
A: LU decomposes into lower and upper triangular matrices, while QR decomposes into an orthogonal matrix and upper triangular matrix.

Q3: What if my matrix can't be LU decomposed?
A: Try LU decomposition with partial pivoting (PLU) which rearranges rows to avoid zero pivots.

Q4: Is LU decomposition unique?
A: Yes, for invertible matrices where L has 1s on the diagonal, the LU decomposition is unique.

Q5: How is LU used in solving equations?
A: After decomposition, Ax=b becomes LUx=b. First solve Ly=b for y, then Ux=y for x.