Solution:

%code

%Define matrix A

A = [1, 2, 3; 3, 3, 4; 5, 6, 7];

%Compute SVD of A

[U, S, V] = svd(A);

%Rank-1 approx

A1 = U(:,1) * S(1,1) * V(:,1)';

RMSE = sqrt(mean((A(:) - A1(:)).^2));

%Display A1 rounded to 4 decimal places

disp(round(A1, 4));

1.7039 2.0313 2.4935

2.7243 3.2477 3.9867

4.9087 5.8517 7.1832

%Display RMSE

disp(RMSE);

0.3257

Problem 2

Use the svd() function in MATLAB to compute , the rank-2 approximation of . Clearly state what is, rounded to 4 decimal places. Also, compute the root-mean square error (RMSE) between and . Which approximation is better, or ? Explain.

Solution:

%code

A = [2, 4, 7; 3, 3, 5; 1, 6, 6];

% Compute SVD of A

[U, S, V] = svd(A);

% Rank-2 approximation

A2 = U(:,1:2) * S(1:2,1:2) * V(:,1:2)';

% Round A2 to 4 decimal places

A2 = round(A2, 4);

% Display A2

disp(A2);

2.4486 4.3608 6.5949

2.6356 2.7069 5.3291

0.8347 5.8671 6.1493

% Compute RMSE between A and A2

RMSE = sqrt(mean((A(:) - A2(:)).^2));

disp(RMSE);

0.3144

Problem 3

For the matrix A, the singular value decomposition is where . Use MATLAB to compute the dot product .

Also, use MATLAB to compute the cross product and dot product . Clearly state the values for each of these computations. Do these values make sense? Explain.