A symmetric square matrix A ∈ â„^(n×n) is called positive definite if A_c > 0 for all c ≠0 ∈ â„^n, and called positive semidefinite if A_r ≥ 0 for all r ∈ â„^n. A matrix A is positive definite if and only if all its eigenvalues are strictly positive, and A is positive semidefinite if and only if all its eigenvalues are nonnegative.
Given a positive (semi)definite matrix A, our aim is to find a positive (semi)definite matrix B such that B · B = A. B is called the square root of A, and denoted by B = √A. This matrix B can be computed as B = T · D' · T^(-1) = T · D' · T^T, where D' is a diagonal matrix with the square roots of the eigenvalues of A, and T is the matrix whose columns are the normalized eigenvectors of A.
Let A = ...
Compute the square root of A by using the method given in ...
