6.49.1  Cholesky decomposition: cholesky

If M is a square symmetric positive definite matrix, the Cholesky decomposition is M=P^TP, where P is a lower triangular matrix. The cholesky command finds the matrix P.

• cholesky takes one argument:
  M, a square symmetric positive definite matrix.
• cholesky(M) returns a symbolic or numeric matrix P given by the Cholesky decomposition.

Examples.

• Input:
  cholesky([[1,1],[1,5]])
  Output:

  ⎡ ⎢ ⎣

  1 0 1 2

  ⎤ ⎥ ⎦

• Input:
  cholesky([[3,1],[1,4]])
  Output:

  ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

  √ 3 0 √ 3 3 √ 33 3

  ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

• Input:
  cholesky([[1,1],[1,4]])
  Output:

  ⎡ ⎢ ⎢ ⎢ ⎣

  1 0 1 √ 3

  ⎤ ⎥ ⎥ ⎥ ⎦

Warning: If the matrix argument A is not a symmetric matrix, cholesky(A) does not return an error, instead cholesky(A) will use the symmetric matrix B of the the quadratic form q corresponding to the (non symmetric) bilinear form of the matrix A.

Example.
Input:

or:

Output: