91Ó°ÊÓ

Find all numbers \(d\) such that \(A=\left[\begin{array}{rr}1 & -2 \\ -2 & d\end{array}\right]\) is positive-definite.

(a) Assume that \(A\) is an \(n \times n\) matrix with entries \(\left|a_{i j}\right| \leq 1\) for \(1 \leq i, j \leq n\). Prove that the matrix \(U\) in its \(\mathrm{PA}=\mathrm{LU}\) factorization satisfies \(\left|u_{i j}\right| \leq 2^{n-1}\) for all \(1 \leq i, j \leq n\). See Exercise 9 (b). (b) Formulate and prove an analogous fact for an arbitrary \(n \times n\) matrix \(A\).

Find all numbers \(d\) such that \(A=\left[\begin{array}{rrr}1 & -1 & 0 \\ -1 & 2 & 1 \\ 0 & 1 & d\end{array}\right]\) is positive-definite.

Prove that a principal submatrix of a symmetric positive-definite matrix is symmetric positive-definite. (Hint: Consider an appropriate \(X\) and use Property 2 .)

Solve the problems by carrying out the Conjugate Gradient Method by hand. (a) \(\left[\begin{array}{ll}1 & 2 \\ 2 & 5\end{array}\right]\left[\begin{array}{l}u \\\ v\end{array}\right]=\left[\begin{array}{l}1 \\ 1\end{array}\right]\) (b) \(\left[\begin{array}{ll}1 & 2 \\ 2 & 5\end{array}\right]\left[\begin{array}{l}u \\\ v\end{array}\right]=\left[\begin{array}{l}1 \\ 3\end{array}\right]\)

Solve the problems by carrying out the Conjugate Gradient Method by hand. (a) \(\left[\begin{array}{rr}1 & -1 \\ -1 & 2\end{array}\right]\left[\begin{array}{l}u \\\ v\end{array}\right]=\left[\begin{array}{l}0 \\ 1\end{array}\right]\) (b) \(\left[\begin{array}{ll}4 & 1 \\ 1 & 4\end{array}\right]\left[\begin{array}{l}u \\\ v\end{array}\right]=\left[\begin{array}{r}-3 \\ 3\end{array}\right]\)