14.16

(a)注意到

(b)

(c)假设$A$的满奇异值分解为

14.26

(a)将卷积写成矩阵形式，记

(b)

c = [  0.0455; -0.2273; -0.0455;  0.2727;  0.4545;  0.4545;  0.2727; -0.0455; -0.2273;  0.0455;];
k = 1;
n = length(c);
A = zeros(n, n);
B = zeros(n - 1, n);
for i = 1: n
for j = 1:i
A(i, j) = c(i + 1 - j)
end
end

for i = 1: (n - 1)
for j = 1 : (n - i)
B(i, j + i) = c(n + 1 - j)
end
end

C = [A; B];
D = [zeros(2 * k + 1, n - k - 1), eye(2 * k + 1), zeros(2 * k + 1, n - k - 1)];
%E = inv(C' * C) * (C' * D' * D * C);
E = (C' * C) \ (C' * D' * D * C);
Eig = eig(E);
res = max(Eig)
0.9375

(a)回顾定义

15.8

(a)

(b)

expA = expm(3 * A);
[U, S, V] = svd(expA);
% (a)
x0_1 = V(:, 1);

% (b)
x0_2 = V(:, 5);

15.10

(a)注意到，如果

(b)

A = [2 4 5 4 5;    0 5 7 7 1;    7 8 0 6 7;     7 0 4 9 4;    9 1 1 8 7];
Vmax = 3;
[U, S, V] = svd(A);
k = Vmax / S(1, 1);
s1 = k * V(:, 1)
s2 = - k * V(:, 1)
s1 =
-0.0606
-0.0373
-0.0312
-0.0746
-0.0549
s2 =
0.0606
0.0373
0.0312
0.0746
0.0549

15.11

(a)回顾结论，我们有

A = [1, 0, 0, 0; 1, 1, 0, 0; 0, 1, 1, 0; 1, 0, 0, 0];
B = [0, 1; 0, 1; 1, 0; 0, 0];
x0 = [1; 0; -1; 1];
T = 1;
C = B;
tmp = B;
x = A * x0;

while true
if rank(C) == rank([C, x])
break
end
x = A * x;
tmp = A * tmp;
C = [C, tmp];
T = T + 1;
end

% (a)
u = - pinv(C) * x;
J1 = norm(u) ^ 2

(b)利用第7,8讲的内容求解该问题。

% (b)
C = [];
tmp = B;
x10 = x0;
for i = 1:10
C = [C, tmp];
tmp = A * tmp;
x10 = A * x10;
end

P = C;
v = - x10;
[m, n] = size(P);
N = 100;
Lambda = logspace(1, -1, N);
res = zeros(1, N);
for i = 1: N
lambda = Lambda(i);
u = inv(eye(n) + lambda * P' * P) * lambda * P' * v;
res(i) = norm(P * u - v) - 0.1;
if i > 1 && res(i) * res(i - 1) < 0
u_res = u;
%break;
end
end

J9 = norm(u_res) ^ 2;
plot(res);