An important factorization in mechanics is the polar decomposition. For an × matrix A with a positive determinant, the factorization is A = QP, where Q is an orthogonal matrix and P is a symmetric...

An important factorization in mechanics is the polar decomposition. For an

×

matrix
A
with a positive determinant, the factorization is
A
=
QP, where
Q
is an orthogonal matrix and
P
is a symmetric positive definite matrix.

(a) Assuming the SVD of
A
has been computed, explain how this can be used to compute
Q
and
P. Make sure to explain why the formulas you derive for
Q
and
P
guarantee that they have their required properties.

(b) According to Theorem 4.4, det(Q) = ±1. An orthogonal matrix with det(Q) = −1 corresponds to a reflection, and these are considered to be unphysical. For this reason, in mechanics one is interested in having det(Q) = 1, which corresponds to what is known as a proper orthogonal matrix and physically they correspond to rotations. Explain how to modify, if necessary, your algorithm or formulas in part (a) so that
Q
is a proper orthogonal matrix.