11.5. Cholesky Decomposition Homework¶
Consider the following two matrices.
\[\begin{split}\mathbf{A} = \begin{bmatrix}
51 & -4.5 & 56 & -20.5 \\
-4.5 & 34.5 & -8 & 1.75 \\
56 & -8 & 64 & -16 \\
-20.5 & 1.75 & -16 & 43.5
\end{bmatrix} \qquad
\mathbf{B} = \begin{bmatrix}
17 & -2 & 4 & 11 \\
-2 & 78 & 8 & 3 \\
4 & 8 & 28 & 12 \\
11 & 3 & 12 & 9
\end{bmatrix}\end{split}\]
A = np.array([[51, -4.5, 56,-20.5],
[-4.5, 34.5, -8, 1.75],
[56, -8, 64, -16],
[-20.5, 1.75, -16, 43.5]]); print(A)
B = np.array([[17, -2, 4, 11],
[-2, 78, 8, 3],
[4, 8, 28, 12],
[11, 3, 12, 9]]).astype(float); print(B)
Does each matrix appear to be positive definite from the first two positive definite tests listed in the study guide?
Attempt to find the Cholesky decomposition of each matrix. Is the matrix positive definite? If it is, what is the Cholesky decomposition? Verify that the product of the Cholesky factors matches the values in the original matrix.
