Numerical Linear Algebra and MATLAB
Numerical Linear Algebra is a core topic in scientific computing, focusing on the application of linear algebra techniques to real-world numerical problems. MATLAB offers a robust environment for performing these operations efficiently.
Solving Systems of Linear Equations
MATLAB provides efficient methods for solving systems of linear equations. Let’s consider the system of equations:
3x - 2y = 5
x + 4y = 6
% Solving a System of Linear Equations
A = [3 -2; 1 4];
b = [5; 6];
x = A \ b; % Solve Ax = b
disp('Solution x:');
disp(x);
Output:
Solution:
1.0000
1.2500
LU Decomposition
LU decomposition factors a square matrix A into a lower triangular matrix L and an upper triangular matrix U, such that A = L * U. Let’s consider:
Matrix A:
4 3
6 3
% LU Decomposition
A = [4 3; 6 3];
[L, U] = lu(A); % Perform LU decomposition
disp('Lower Triangular Matrix L:');
disp(L);
disp('Upper Triangular Matrix U:');
disp(U);
Output:
Lower Triangular Matrix L:
1.0000 0.0000
1.5000 1.0000
Upper Triangular Matrix U:
4.0000 3.0000
0.0000 -1.5000
Eigenvalues and Eigenvectors
Eigenvalues and eigenvectors are critical for stability analysis, vibration analysis, and solving systems of differential equations. Consider the matrix:
Matrix A:
2 1
1 2
% Finding Eigenvalues and Eigenvectors
A = [2 1; 1 2];
[V, D] = eig(A); % Compute eigenvalues and eigenvectors
disp('Eigenvectors:');
disp(V);
disp('Eigenvalues (Diagonal Matrix):');
disp(D);
Output:
Eigenvectors:
-0.7071 -0.7071
0.7071 -0.7071
Eigenvalues (Diagonal Matrix):
1.0000 0
0 3.0000
QR Decomposition
QR decomposition factors a matrix A into an orthogonal matrix Q and an upper triangular matrix R, such that A = Q * R. Let’s consider the matrix:
Matrix A:
12 -51 4
6 167 -68
-4 24 -41
% QR Decomposition
A = [12 -51 4; 6 167 -68; -4 24 -41];
[Q, R] = qr(A); % Perform QR decomposition
disp('Orthogonal Matrix Q:');
disp(Q);
disp('Upper Triangular Matrix R:');
disp(R);
Output:
Orthogonal Matrix Q:
-0.8571 0.3943 0.3314
-0.4286 -0.9029 -0.0343
0.2857 -0.1714 0.9429
Upper Triangular Matrix R:
-14.0000 21.0000 -14.0000
0.0000 -175.0000 70.0000
0.0000 0.0000 -35.0000
Singular Value Decomposition (SVD)
SVD is a robust tool for solving ill-conditioned systems, data compression, and image processing. Let’s consider:
Matrix A:
1 0
0 2
3 0
% Singular Value Decomposition
A = [1 0; 0 2; 3 0];
[U, S, V] = svd(A); % Perform SVD
disp('Matrix U:');
disp(U);
disp('Diagonal Matrix S:');
disp(S);
disp('Matrix V:');
disp(V);
Output:
Matrix U:
-0.3162 0 0.9487
0 -1.0000 0
-0.9487 0 -0.3162
Diagonal Matrix S:
3.1623 0
0 2.0000
0 0
Matrix V:
-1.0000 0
0 1.0000
Cholesky Decomposition
Cholesky decomposition is used for solving systems of linear equations and for numerical optimization. It decomposes a positive definite matrix A into a product of a lower triangular matrix L and its transpose: A = L * L'.
Consider the matrix:
Matrix A: 4 12 -16 12 37 -43 -16 -43 98
% Cholesky Decomposition
A = [4 12 -16; 12 37 -43; -16 -43 98];
L = chol(A, 'lower'); % Perform Cholesky decomposition
disp('Lower Triangular Matrix L:');
disp(L);
disp('Reconstructed Matrix A from L:');
disp(L * L');
Output:
Lower Triangular Matrix L: 2.0000 0 0 6.0000 1.0000 0 -8.0000 -5.0000 3.0000 Reconstructed Matrix A from L: 4.0000 12.0000 -16.0000 12.0000 37.0000 -43.0000 -16.0000 -43.0000 98.0000
Determinant Calculation
The determinant of a square matrix is a scalar value used to determine invertibility and other properties. Consider:
Matrix A: 1 2 3 0 4 5 1 0 6
% Determinant of a Matrix
A = [1 2 3; 0 4 5; 1 0 6];
detA = det(A); % Compute determinant
disp('Determinant of A:');
disp(detA);
Output:
Determinant of A: 22
Moore-Penrose Pseudoinverse
The pseudoinverse is used to find solutions to overdetermined or underdetermined systems. Consider the matrix:
Matrix A: 1 0 2 0 1 3
% Pseudoinverse of a Matrix
A = [1 0 2; 0 1 3];
pinvA = pinv(A); % Compute pseudoinverse
disp('Pseudoinverse of A:');
disp(pinvA);
Output:
Pseudoinverse of A:
0.2000 0
0 0.2000
0.1333 0.2000
Norm of a Matrix
The norm of a matrix measures its size or length in a specific sense. Consider:
Matrix A: 1 -2 3 4
% Norm of a Matrix
A = [1 -2; 3 4];
normF = norm(A, 'fro'); % Frobenius norm
norm1 = norm(A, 1); % 1-norm
normInf = norm(A, Inf); % Infinity norm
disp('Frobenius Norm:');
disp(normF);
disp('1-Norm:');
disp(norm1);
disp('Infinity Norm:');
disp(normInf);
Output:
Frobenius Norm: 5.4772 1-Norm: 6 Infinity Norm: 7
Matrix Rank
The rank of a matrix is the dimension of its row or column space. For:
Matrix A: 1 2 3 4 5 6 7 8 9
% Rank of a Matrix
A = [1 2 3; 4 5 6; 7 8 9];
r = rank(A); % Compute rank
disp('Rank of A:');
disp(r);
Output:
Rank of A: 2
Least Squares Solution
For overdetermined systems, the least squares method minimizes the error. Consider:
System: 2x + y = 3 x + 2y = 4 x + y = 1
% Least Squares Solution
A = [2 1; 1 2; 1 1];
b = [3; 4; 1];
x_ls = A \ b; % Solve using least squares
disp('Least Squares Solution:');
disp(x_ls);
Output:
Least Squares Solution: 0.3846 1.3846
Useful MATLAB Functions
eig(A) computes the eigenvalues and eigenvectors of A.svd(A) computes the Singular Value Decomposition of A.lu(A) performs LU decomposition of A.qr(A) performs QR decomposition of A.A \ b solves the system of linear equations Ax = b.Practice Problems
Test Yourself
1. Perform LU decomposition on a 3×3 matrix and verify the result.
2. Find the eigenvalues and eigenvectors of a symmetric matrix.
3. Solve the system of equations using QR decomposition.
4. Compute the SVD of a 3×2 matrix and explain the significance of the singular values.
5. Perform Cholesky decomposition on a symmetric positive definite matrix and verify the result.
6. Compute the determinant of a 4×4 matrix and check if it is invertible.
7. Find the pseudoinverse of a rectangular matrix and verify the property A * pinv(A) * A = A.
8. Compute the rank and norms (Frobenius, 1-norm, infinity norm) of a random 3×3 matrix.
9. Solve an overdetermined system using the least squares method.