Eigenvalues and EigenvectorsEigenvalues 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.

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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:

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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.

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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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Assume that v = [2, 4, 6] is a eigenvector of matrix A that correspond so eigenvalue λ=2. Calculate the result of matrix multiplication A * v.

Select the correct answer

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Section 2. Chapter 9