MATLAB Mathematics - SERC - Index of

1 Matrices and Linear Algebra
The elements of A are binomial coefficients. Each element is the sum of its
north and west neighbors. The Cholesky factorization is
R = chol(A)
R =
1 1 1 1 1 1
0 1 2 3 4 5
0 0 1 3 6 10
0 0 0 1 4 10
0 0 0 0 1 5
0 0 0 0 0 1
The elements are again binomial coefficients. The fact that R'*R is equal to A
demonstrates an identity involving sums of products of binomial coefficients.
Note The Cholesky factorization also applies to complex matrices. Any
complex matrix which has a Cholesky factorization satisfies A' = A and is said
to be Hermitian positive definite.
The Cholesky factorization allows the linear system
Ax
=
b
to be replaced by
R′Rx = b
Because the backslash operator recognizes triangular systems, this can be
solved in MATLAB quickly with
x = R\(R'\b)
If A is n-by-n, the computational complexity of chol(A) is O(n 3 ), but the
complexity of the subsequent backslash solutions is only O(n 2 ).
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