Algorithm: Cholesky Decomposition Given a symmetric positive definite n x n matrix A, this algorithm overwrites part of it with its Cholesky factor: kta for k = 1 to n-1 akk = √Vakk for i = k+1 to n Qik dik = akk end for for j = k+1 to n for i = j to n Qij = Qij - Qik * ajk end end end ann = √Vann
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The algorithm takes a symmetric positive definite n X n matrix A as input. Show more…
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Suppose you are given an n × n symmetric positive definite matrix A and its Cholesky factor G such that GGT = A. Describe how you can efficiently solve the following problems. (Note that you should not need to compute the explicit inverse of A or any other matrix.) Assume c and b are known vectors given to you. (a) Compute α = c^T A^(-1) b. (b) Solve the matrix equation Bx = b where B = A + Ã. The matrix à satisfies the condition that all of its columns are zero vectors, except the kth column where Ã_i,k = i, for 1 ≤ i ≤ n. (Hint: Consider the Sherman-Morrison formula.)
Madhur L.
Let $A=\left[a_{i j}\right]$ be an $n \times n$ matrix. Determine whether $A$ is symmetric. (a) $a_{i j}=i^{2}+j^{2}$ (b) $a_{i j}=i^{2}-j^{2}$ (c) $a_{i j}=2 i+2 j$ (d) $a_{i j}=2 i^{2}+2 j^{3}$
Systems of Linear Equations and Matrices
Diagonal, Triangular, and Symmetric Matrices
A positive definite matrix A can be factored into A = T'T, (2.79) where T is a nonsingular upper triangular matrix. One way to obtain T is the Cholesky decomposition, which can be carried out in the following steps. Let A = (aij) and T = (tij) be n x n. Then the elements of T are found as follows: t11 = sqrt(a11), t1j = a1j / t11 2 <= j <= n tii = sqrt(aii - sum_{k=1}^{i-1} t_{ki}^2) 2 <= i <= n tij = (aij - sum_{k=1}^{i-1} t_{ki} t_{kj}) / tii 2 <= i < j <= n tij = 0 1 <= j < i <= n For example, let A = (3 0 -3; 0 6 3; -3 3 6). Then by the Cholesky method, we obtain T = (sqrt(3) 0 -sqrt(3); 0 sqrt(6) sqrt(1.5); 0 0 sqrt(1.5)), T'T = (sqrt(3) 0 0; 0 sqrt(6) 0; -sqrt(3) sqrt(1.5) sqrt(1.5))(sqrt(3) 0 -sqrt(3); 0 sqrt(6) sqrt(1.5); 0 0 sqrt(1.5)) = (3 0 -3; 0 6 3; -3 3 6) = A.
Adi S.
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