AMU 2026 — Mathematics PYQ

1. What is Cholesky Method?

The Cholesky method (or Cholesky decomposition) is a mathematical technique used in numerical analysis to decompose a square matrix $A$ into the product of a lower triangular matrix and its transpose. The equation is represented as:

$$A=L{L}^T$$

Where:

• $A$ is the original square matrix.

• $L$ is a unique lower triangular matrix with positive diagonal entries.

• $L^T$ is the transpose of the lower triangular matrix.

2. Essential Requirements for Cholesky Decomposition

For a real matrix $A$ to be factored uniquely using the Cholesky method, it must satisfy two strict criteria:

• Symmetry: The matrix must be equal to its own transpose.

  $$A=A^T$$

  If a matrix is not symmetric, the formulation $A=L{L}^T$ cannot hold because the product of any lower triangular matrix $L$ with its transpose $L^T$ automatically yields a symmetric matrix.

• Positive Definiteness: A symmetric matrix $A$ is positive definite if for every non-zero vector $x$, the quadratic form is strictly positive:

  x^T A x > 0

  This condition ensures that during the computation process, you do not have to perform a division by zero or take the square root of a negative number when evaluating diagonal elements (i.e., $L_{ii}=\sqrt{A_{ii}-{\sum }_{k=1}^{i-1}{L}_{ik}^2}$).

Conclusion

Since the matrix $A$ must be both symmetric and positive definite for the Cholesky method to be validly applied, the correct choice encompassing both requirements is (c).