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Linear Algebra in MATLAB

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Linear algebra is a fundamental aspect of MATLAB, given its name "MATrix LABoratory." This section will cover essential linear algebra concepts and operations that you can perform using MATLAB. By the end of this module, you will be able to handle matrices and vectors efficiently and apply various linear algebra techniques to solve problems.

Key Concepts

  1. Matrix Decomposition
    • LU Decomposition
    • QR Decomposition
    • Singular Value Decomposition (SVD)
  2. Eigenvalues and Eigenvectors
  3. Solving Linear Systems
  4. Matrix Inversion
  5. Determinants

  1. Matrix Decomposition

LU Decomposition

LU decomposition factors a matrix as the product of a lower triangular matrix (L) and an upper triangular matrix (U).

A = [4 3; 6 3];
[L, U] = lu(A);
disp('Lower Triangular Matrix L:');
disp(L);
disp('Upper Triangular Matrix U:');
disp(U);

Explanation:

  • lu(A) decomposes matrix A into L and U.
  • L is a lower triangular matrix.
  • U is an upper triangular matrix.

QR Decomposition

QR decomposition factors a matrix into an orthogonal matrix (Q) and an upper triangular matrix (R).

A = [1 2; 3 4; 5 6];
[Q, R] = qr(A);
disp('Orthogonal Matrix Q:');
disp(Q);
disp('Upper Triangular Matrix R:');
disp(R);

Explanation:

  • qr(A) decomposes matrix A into Q and R.
  • Q is an orthogonal matrix.
  • R is an upper triangular matrix.

Singular Value Decomposition (SVD)

SVD decomposes a matrix into three matrices: U, Σ (Sigma), and V* (V transpose).

A = [1 2; 3 4; 5 6];
[U, S, V] = svd(A);
disp('Matrix U:');
disp(U);
disp('Diagonal Matrix S:');
disp(S);
disp('Matrix V:');
disp(V);

Explanation:

  • svd(A) decomposes matrix A into U, S, and V.
  • U and V are orthogonal matrices.
  • S is a diagonal matrix containing singular values.

  1. Eigenvalues and Eigenvectors

Eigenvalues and eigenvectors are fundamental in many applications, such as stability analysis and vibration analysis.

A = [1 2; 2 1];
[eigenVectors, eigenValues] = eig(A);
disp('Eigenvalues:');
disp(diag(eigenValues));
disp('Eigenvectors:');
disp(eigenVectors);

Explanation:

  • eig(A) returns the eigenvalues and eigenvectors of matrix A.
  • eigenValues is a diagonal matrix with eigenvalues on the diagonal.
  • eigenVectors contains the corresponding eigenvectors.

  1. Solving Linear Systems

To solve a system of linear equations \(Ax = b\), you can use the backslash operator \.

A = [3 2; 1 2];
b = [5; 5];
x = A \ b;
disp('Solution x:');
disp(x);

Explanation:

  • A \ b solves the linear system \(Ax = b\).
  • x is the solution vector.

  1. Matrix Inversion

Matrix inversion is used to find the inverse of a matrix, which is useful in solving linear systems and other applications.

A = [1 2; 3 4];
A_inv = inv(A);
disp('Inverse of Matrix A:');
disp(A_inv);

Explanation:

  • inv(A) computes the inverse of matrix A.

  1. Determinants

The determinant of a matrix is a scalar value that can be computed using the det function.

A = [1 2; 3 4];
det_A = det(A);
disp('Determinant of Matrix A:');
disp(det_A);

Explanation:

  • det(A) computes the determinant of matrix A.

Practical Exercises

Exercise 1: LU Decomposition

Problem: Given the matrix \(A = \begin{bmatrix} 2 & 1 & 1 \ 4 & -6 & 0 \ -2 & 7 & 2 \end{bmatrix}\), perform LU decomposition and verify the result by reconstructing the original matrix.

Solution:

A = [2 1 1; 4 -6 0; -2 7 2];
[L, U] = lu(A);
disp('Lower Triangular Matrix L:');
disp(L);
disp('Upper Triangular Matrix U:');
disp(U);
A_reconstructed = L * U;
disp('Reconstructed Matrix A:');
disp(A_reconstructed);

Exercise 2: Solving Linear Systems

Problem: Solve the system of linear equations given by \(Ax = b\), where \(A = \begin{bmatrix} 1 & 2 & 3 \ 4 & 5 & 6 \ 7 & 8 & 10 \end{bmatrix}\) and \(b = \begin{bmatrix} 6 \ 15 \ 25 \end{bmatrix}\).

Solution:

A = [1 2 3; 4 5 6; 7 8 10];
b = [6; 15; 25];
x = A \ b;
disp('Solution x:');
disp(x);

Common Mistakes and Tips

  • Mistake: Forgetting that not all matrices are invertible. Always check if a matrix is singular before attempting to invert it.

    • Tip: Use the rank function to check the rank of a matrix. If the rank is less than the number of rows or columns, the matrix is singular.

  • Mistake: Misinterpreting the order of matrix multiplication.

    • Tip: Remember that matrix multiplication is not commutative. \(AB \neq BA\) in general.

Conclusion

In this section, you have learned about various linear algebra operations in MATLAB, including matrix decomposition, eigenvalues and eigenvectors, solving linear systems, matrix inversion, and determinants. These tools are essential for many applications in engineering, physics, computer science, and more. Practice these concepts with the provided exercises to reinforce your understanding and prepare for more advanced topics.

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