If A is symmetric and positive definite, there is an orthogonal matrix Q for which A = QΛQT. Here Λ is the matrix of eigenvalues. Singular Value Decomposition lets us write any matrix A as a product UΣVT where U and V are orthogonal and Σ is a diagonal matrix whose non-zero entries are square roots of the eigenvalues of ATA. The columns of U and V give bases for the four fundamental subspaces.
Lecture 29: Singular Value Decomposition
Problem Solving: Determinants and VolumeComputing the Singular Value Decomposition
Work the problems on your own and check your answers when you're done.