Compute A = LU (L lower triangular, U upper triangular)
Enter 2x2 Matrix
Enter 3x3 Matrix
Result
Lower Triangular Matrix L
Upper Triangular Matrix U
Gaussian Elimination Steps
LU Decomposition Definition
A = L × U
L: Lower triangular (L[i][j] = 0 for i < j)
U: Upper triangular (U[i][j] = 0 for i > j)
L[i][i] = 1 (diagonal is all ones)
LU decomposition factorizes a matrix into a lower triangular matrix L and an upper triangular matrix U. It is widely used for efficient solution of linear systems.
⚠LU decomposition requires all leading principal minors to be non-zero. If pivoting is needed, consider LUP decomposition with permutation matrix.
What is LU Decomposition?
LU decomposition factorizes a matrix A into A = LU where L is a lower triangular matrix with 1s on the diagonal and U is an upper triangular matrix. This decomposition is fundamental for solving systems of linear equations efficiently.
Lower Triangular L
L[i][j] = 0 for i < j, L[i][i] = 1
Upper Triangular U
U[i][j] = 0 for i > j, contains pivots
Efficiency
O(n³) decomposition, O(n²) per solve
Gaussian Elimination
L stores multipliers, U is row-echelon form
LU Algorithm:
For each column k from 1 to n:
1. For rows i = k+1 to n: multiplier = A[i][k] / A[k][k]
2. L[i][k] = multiplier
3. Row[i] = Row[i] - multiplier × Row[k]
4. U is the transformed matrix
Applications
Linear SystemsDeterminantsMatrix InversionRepeated SolvesNumerical Methods
Frequently Asked Questions
What is LU decomposition?▼
A = LU where L is lower triangular with 1s on diagonal, U is upper triangular. Used for solving linear systems efficiently.
How to compute LU?▼
Gaussian elimination: L stores multipliers, U is the row-echelon form.
Properties of LU?▼
L has 1s on diagonal. U has pivots on diagonal. Exists if all leading principal minors are non-zero.
Applications of LU?▼
Solving linear systems, computing determinants, matrix inversion, repeated solves with same matrix.
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