Computing Real or Complex Eigenvalues in MATLAB

MATLAB provides several commands for computing the eigenvalues of a matrix, including `eig` for full matrices and `eigs` for sparse matrices. In this article, we will discuss how to compute both real and complex eigenvalues using these commands.

Computing Real Eigenvalues

When dealing with real matrices, MATLAB's `eig` command can be used to compute all the eigenvalues of a matrix. For example, if we have a full matrix A, we can compute its eigenvalues using the following code:

```

'eigenvalues, eigenvectors' = eig(A);

```

This will return both the eigenvalues and eigenvectors of the matrix A.

Computing Complex Eigenvalues

When dealing with complex matrices, MATLAB's `eigs` command can be used to compute the 6 largest magnitude eigenvalues of a matrix. For example, if we have a sparse matrix A, we can compute its six largest magnitude eigenvalues using the following code:

```

'eigenvalues, eigenvectors' = eigs(A);

```

This will return both the eigenvalues and eigenvectors of the matrix A.

Properties of Eigenvalues

Eigenvalues are scalar values that represent how a linear transformation changes a vector. They have several important properties:

* Real or Complex: Eigenvalues can be either real or complex.

* Algebraic Multiplicity: The algebraic multiplicity of an eigenvalue is the number of times it appears as a root of the characteristic polynomial.

* Geometric Multiplicity: The geometric multiplicity of an eigenvalue is the dimension of its eigenspace.

Rotation-Scaling Matrices

A rotation-scaling matrix is a special type of matrix that can be used to represent linear transformations in which both rotation and scaling are involved. Rotation-scaling matrices have the form:

```

' a -b '

' ba b '

```

where `a` and `b` are real numbers.

Rotation-Scaling Theorem

The rotation-scaling theorem states that two rotation-scaling matrices A and B are similar if and only if their eigenvalues are equal.

Complex Eigenvalue Decomposition

When dealing with complex matrices, the eigenvalues can be complex. In this case, the eigenvectors of the matrix are also complex. The eigenvectors can be used to decompose the matrix into its real and imaginary parts.

In conclusion, MATLAB provides several commands for computing both real and complex eigenvalues of a matrix. Understanding the properties of eigenvalues, including their algebraic and geometric multiplicities, is important for analyzing linear transformations. Additionally, rotation-scaling matrices and the rotation-scaling theorem are useful tools for understanding the behavior of linear transformations.

References:

* MATLAB documentation: `eig` and `eigs` commands

* Linear Algebra by Gilbert Strang

* Complex Variables by James Ward Brown

Real or complex eigenvalues matlab