MATH 661.03

1.令A∈C
m×m是厄米。
（a）证明A的所有特征值都是真实的。
（b）证明如果x和y是对应于不同特征值的特征向量，

（c）令k·k表示C上的任何范数
m，以及C上的诱导矩阵范数

m×m，ρ（A）≤kAk，其中ρ（A）最大

2.假设A∈C
m×m具有SVD A =UΣV

（a）让
S =
1个

2
</ s> </ s> </ s>

</ s> </ s> </ s>

Σ0

（b）使用（a）部分的结果，否则，找到特征值分解
2m×2m厄米矩阵的A =XΛX-1
</ s> </ s> </ s>
0 A ∗
A 0
</ s> </ s> </ s>

3.设一个∈C，并考虑矩阵
A =

1 0
1
1 0

。
（a）证明正交投影仪P在范围（A）上不依赖于

（b）使用您喜欢的任何方法，计算出简化的QR因式分解A = QˆRˆ

2。
4. Consider the following MATLAB code for computing a reduced QR factorization
of a nonsingular matrix A:
function [A,R] = mgs(A);
% Computes the QR factorisation of A, using A to store Q.
m = size(A,1);
n = size(A,2);
R = zeros(n,n);
for i = 1:n
R(i,i) = norm(A(:,i));
A(:,i) = A(:,i)./R(i,i);
for j = 1:n
R(i,j) = A(:,j)’*A(:,i);
A(:,j) = A(:,j) – R(i,j)*A(:,i);
end
end
(a) There are two mistakes in this code. Identify and correct them.
(b) Calculate the exact number of floating-point operations performed by the corrected code as a function of m and n.
5. Show that (1 + O(machine))−1 = 1 + O(machine).
6. Here is a stripped-down version of one of MATLAB’s built-in m-files.
[U,S,V] = svd(A);
S = diag(S);
tol = max(size(A))*S(1)*eps;
r = sum(S > tol);
S = diag(ones(r,1)./S(1:r));
X = V(:,1:r)*S*U(:,1:r);
What does this program compute?
7. Let A be a nonsingular square matrix, and let A = QR and A∗A = U
∗U be QR and
Cholesky factorizations, respectively, with the usual normalizations rjj , ujj > 0. Is
it true or false that R = U? Explain your reasoning.
8. What happens if you apply the unshifted (pure) QR algorithm to an orthogonal matrix? How does your answer relate to the convergence theorem for this method?
(C.f. page 14 of the ’Eigenvalues’ lecture notes.)
9. Show that the Ritz values at step n of the Lanczos iteration (the eigenvalues of Tn)
are the stationary values of r(x) = (x
TAx)/(x
T x) if x is restricted to Kn.
10. Suppose A is a dense symmetric positive definite 1000×1000 matrix, with κ(A) =
100. Estimate roughly how many flops are required to solve Ax = b to ten-digit
accuracy by (a) Cholesky factorization, and (b) conjugate gradients.