Welcome to this lesson on eigenvalues and eigenvectors, critical concepts in the field of linear algebra with wide-reaching applications. Eigenvalues and eigenvectors play a pivotal role in various areas, such as solving differential equations, face recognition in computer vision, and even vibration analysis in physics.

An eigenvector of a matrix is a non-zero vector that changes at most by a scalar factor when that matrix multiplies it. The corresponding eigenvalue is the factor by which the eigenvector is scaled. They provide deeper insights into the properties of a matrix and are fundamental in transforming coordinate systems.

Before diving into eigenvalues and eigenvectors, let’s recall some key concepts. In our previous lesson, we worked with matrices, which are rectangular arrays of numbers, and vectors, which are arrays that can represent points or directions in space.

For eigenvalues and eigenvectors, we focus on square matrices. These are matrices with the same number of rows and columns, which allow us to carry out transformations that are vital in the calculations of eigenvalues and eigenvectors.

Let's recall the math. The relationship between a matrix $A$, an eigenvector $\mathbf{v}$, and its corresponding eigenvalue $\lambda$ is given by the equation:

$A\mathbf{v} = \lambda\mathbf{v}$

This equation states that when the matrix $A$ multiplies the eigenvector $\mathbf{v}$, the result is the same as scaling $\mathbf{v}$ by the eigenvalue $\lambda$.

To find the eigenvalues, we solve the characteristic equation:

$\det(A - \lambda I) = 0$

where $I$ is the identity matrix of the same size as $A$, and $\det$ denotes the determinant of a matrix. The solutions to this equation are the eigenvalues of the matrix $A$.

Let's walk through the process of calculating eigenvalues and eigenvectors using a specific example.

Begin by defining a square matrix using NumPy. This matrix will serve as the basis for our eigenvalue and eigenvector calculations.

Python
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1A = np.array([[2, 0, 1], 
2              [0, 1, 0], 
3              [1, 0, 2]])

Here, A is our 3x3 square matrix.

Next, use the eig function provided by SciPy to calculate the eigenvalues and eigenvectors of the matrix.

Python
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1eigenvalues, eigenvectors = eig(A)
2print("Eigenvalues:", eigenvalues)
3print("Eigenvectors:\n", eigenvectors)

The eig function returns two objects:

• eigenvalues: An array of eigenvalues corresponding to the eigenvectors.
• eigenvectors: A 2D array where each column represents an eigenvector.

When you run this code, you might see an output like this:

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1Eigenvalues: [3.+0.j 1.+0.j 1.+0.j]
2Eigenvectors:
3 [[ 0.70710678  0.         -0.70710678]
4 [ 0.          1.          0.        ]
5 [ 0.70710678  0.          0.70710678]]

This output shows the eigenvalues and corresponding eigenvectors of matrix A. Each eigenvalue represents a factor by which its associated eigenvector is scaled.

Note: Eigenvalues can be complex because they are the roots of the characteristic polynomial of a matrix, which may have complex solutions, especially for non-symmetric matrices. Complex eigenvalues often appear in conjugate pairs and indicate oscillatory or rotational behavior in systems.

Understanding the results is crucial. Each eigenvalue shows how its associated eigenvector is stretched or compressed. In the output above:

• The eigenvalue 3 corresponds to the eigenvector [0.707, 0, 0.707], indicating how this direction in space is affected by the transformation the matrix represents.

These concepts are used in various domains. For instance, in computer graphics, they assist in understanding shape distortions. In physics, eigenvalues can represent natural frequencies in systems.

In this lesson, we explored the computation of eigenvalues and eigenvectors using SciPy. We reviewed previous concepts, detailed the steps to perform the calculations in Python, and interpreted the results. These insights pave the way for comprehending more complex systems and transformations.

You should now try applying these techniques in the practice exercises. Experimenting with different matrices will reinforce your understanding and give you confidence as you continue your studies in linear algebra. Congratulations on reaching the end of this unit! Keep up the great work as you advance in your journey.