Understanding an Example of LDL Decomposition

Take the original matrix as an example:
A = | 4 2 |
     | 2 5 |

The following steps can be used to compute the LDL decomposition.

1. Initialize a unit lower-triangular matrix L and a diagonal matrix D.

L = | 1 0 |
     |  l 1 |

D = | d1 0 |
     | 0 d2 |

2. By the definition of the LDL decomposition we have A = L * D * L^T. We need suitable values of l and d that make this equation hold. First compute the first elements of L and D:

4 = 1 * d1 * 1 + 0 * 0 * l
So, d1 = 4.

2 = 1 * 0 * 1 + 0 * l * 1

Here we may choose any nonnegative value for l; for example, l = 0.

3. Next, compute the second elements of L and D. First compute the first element of the second row of L, namely l:

2 = 0 * d1 * 1 + l * d2
So, l = 2 / d1 = 0.5.
Then compute the second diagonal element of D, d2:

5 = 1 * d1 * l + l^2 * d2
Substituting the known values of d1 and l, we obtain:
5 = 4 * 0.5 + (0.5)^2 * d2
Solving this equation gives d2 = 3.75.

4. Finally, we obtain the LDL decomposition:

L = | 1 0.5 |
     | 0   1 |

D = | 4    0 |
     | 0 3.75 |

The product L * D * L^T equals the original matrix A.