Cholesky decomposition factorizes a positive definite matrix A into A = LLᵀ where L is a lower triangular matrix with positive diagonal entries. It is particularly efficient for symmetric matrices.
⚠Cholesky decomposition only exists for positive definite matrices. All leading principal minors must be positive.
What is Cholesky Decomposition?
Cholesky decomposition factorizes a positive definite matrix A into A = LLᵀ where L is a lower triangular matrix with positive diagonal entries. It is the square root of a matrix and is widely used in numerical computations.
Lower Triangular L
L[i][j] = 0 for i < j, positive diagonal entries.
Positive Definite
Requires xᵀAx > 0 for all x ≠ 0.
Uniqueness
Unique factorization for positive definite matrices.
Efficiency
O(n³/3) operations, faster than LU decomposition.
Cholesky Algorithm:
For i from 1 to n:
L[i][i] = sqrt(A[i][i] - Σk<i L[i][k]²)
For j from i+1 to n:
L[j][i] = (A[j][i] - Σk<i L[j][k]L[i][k]) / L[i][i]
Applications
Monte CarloOptimizationKalman FilteringLinear SystemsStatistics
Frequently Asked Questions
What is Cholesky decomposition?▼
A = LLᵀ where L is lower triangular with positive diagonal entries. Only exists for positive definite matrices.
How to compute Cholesky?▼
Recursive formula: L[i][i] = sqrt(A[i][i] - sum of L[i][j]^2), L[i][j] = (A[i][j] - sum)/L[j][j].
Properties of Cholesky?▼
L is lower triangular with positive diagonal. Uniquely defined for positive definite matrices.
Applications of Cholesky?▼
Solving linear systems, Monte Carlo simulation, optimization, Kalman filtering, numerical stability.
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