Course Content

Mathematics for Data Analysis and Modeling

## Mathematics for Data Analysis and Modeling

# Eigenvalues and Eigenvectors

**Eigenvectors and eigenvalues** are concepts related to linear transformations and matrices. An eigenvector `v`

is a non-zero vector that results in a **scaled version of itself** when multiplied by a given matrix. The eigenvalue `λ`

associated with an eigenvector represents the scalar value by which the eigenvector is scaled.

If we have some matrix `A`

and provide linear transformation `A * v`

, where `v`

- eigenvector of matrix `A`

, we will get the vector with the **same direction** but with **different length**:

## Calculating eigenvalues and eigenvectors

To find eigenvectors and corresponding eigenvalues of a matrix, we can use `np.linalg.eig()`

method:

In this example, we create a 3x3 `matrix`

matrix. We then use the `np.linalg.eig()`

method from `NumPy`

to calculate the eigenvalues and eigenvectors. The function returns two arrays: eigenvalues contain the eigenvalues, and eigenvectors contain the corresponding eigenvectors.

## Practical applications

Eigenvalues and vectors are often used to solve various applied problems. One of these problems is the problem of **dimensionality reduction** for which the PCA algorithm is used: this algorithm is based on using eigenvalues of the feature covariance matrix.

Note

Dimensionality reduction is a fundamental problem in data analysis and machine learning, aiming to reduce the number of features or variables in a dataset while preserving as much relevant information as possible.

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