Numerical Analysis I, Fall 2009 (http://www.math.nthu.edu.tw/˜ wangwc/)

Quiz 05

Dec 29, 2009.

1. Write down the matrix representation of orthonormal projection onto span{(3, 0, −4)^{T} } relative to standard inner product in ℝ^{3}.

2. Let B be an N^{3} ×N^{3} nonsingular matrix with B_{ij} = 0 if |i−j| > N and N >> 1. Give the operation count (multiplication and division only) of solving Bx = f by Gaussian elimination. Express your answer as O(N^{k}) and find out k. Explain.

3. Find LU factorization of A = ( \begin{matrix}1 & 1 & −1 \\ 1 & 2 & −2 \\ −2 & 1 & 1 \end{matrix} ) . Verify your answer.

4. True or False?

If L is a non-singular lower-triangular matrix, then so is L^{−1}. Explain.

5. True or False?

If T is a non-singular tridiagonal matrix, then so is T^{−1}. Explain.

Numerical Analysis I, Fall 2009 (http://www.math.nthu.edu.tw/˜ wangwc/)

Quiz 05

Dec 29, 2009.

1. Write down the matrix representation of orthonormal projection onto span{(3, 0, −4)^{T} } relative to standard inner product in ℝ^{3}.

2. Let B be an N^{3} ×N^{3} nonsingular matrix with B_{ij} = 0 if |i−j| > N and N >> 1. Give the operation count (multiplication and division only) of solving Bx = f by Gaussian elimination. Express your answer as O(N^{k}) and find out k. Explain.

3. Find LU factorization of A = ( \begin{matrix}1 & 1 & −1 \\ 1 & 2 & −2 \\ −2 & 1 & 1 \end{matrix} ) . Verify your answer.

4. True or False?

If L is a non-singular lower-triangular matrix, then so is L^{−1}. Explain.

5. True or False?

If T is a non-singular tridiagonal matrix, then so is T^{−1}. Explain.