MTH 451: Numerical Linear Algebra
Sample Final
1. Prove one of the following statements
(a) The eigenvalues of a hermetian positive definite matrix are all
positive real numbers.
(b) If Ais hermetian and QT Q∗is a Schur factorization, then Tis
diagonal.
(c) If Ais an m×mhermetian matrix, then x∗Ax is real, ∀x∈Cm.
(d) Show that Ahas real eigenvalues and orthogonal eigenvectors if
and only if Ais hermitian.
2. Suppose QT Q∗is a Schur factorization of A, show that the eigenvalues
of Aare the diagonal elements of T.
3. Compute the LU factorization of the matrix
A=
1 1/2 1/3
1/2 1/3 1/4
1/3 1/4 1/5
4. Compute the reduced QR factorization of the matrix
A=
3−6
4−8
0 1
5. Compute the eigenvalue decomposition of the matrix
A=2 1
1 2
6. Compute the SVD of the matrix
A=0−11
5 0
7. Compute the SVD of the matrix
A=2 1
−1−2
1
