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**Eigenvalues and Eigenvectors**

In linear algebra, an **Eigenvector** or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding **Eigenvalue**, often denoted by **lambda**, is the factor by which the **Eigenvector** is scaled.

Geometrically, an **Eigenvector**, corresponding to a **real nonzero Eigenvalue**, points in a direction in which it is stretched by the transformation and the eigenvalue is the factor by which it is stretched. If the **Eigenvalue **is negative, the direction is reversed. Loosely speaking, in a multidimensional vector space, the eigenvector is not rotated.

An

Eigenvectoris a vector whose direction remains unchanged when a linear transformation is applied to it.

Consider the image below in which three vectors are shown.

Eigenvectors (red) do not change direction when a linear transformation (e.g. scaling) is applied to them. Other vectors (yellow) do.

The **Eigenvectors** and **Eigenvalues** of a **covariance (or correlation) matrix represent the “core” of a PCA**: The **Eigenvectors (principal components)** determine the **directions** of the new feature space, and the **Eigenvalues** determine their **magnitude**. In other words, the **Eigenvalues explain the variance of the data along the new feature axes**.

We need to understand the formula for calculating a determinant of a matrix as it will be used for calculating **Eigenvalues **and **Eigenvectors**.

Now lets go thru the steps to calculate the **Eigenvalues **and **Eigenvectors**

To solve for the **Eigenvalue**, **lambda**, and the corresponding **Eigenvector**, **X**, of an **n x n** matrix **A** follow the following steps.

**Step 1**: Multiply a **n x n** **identity** **matrix** with the scalar **lambda**

**Step 2**: Subtract the result from **Step 1** from the matrix **A**

**Step 3**: Find the **determinant** of the difference matrix from **Step 2**

**Step 4**: Solve for the value of **lambda** that satisfy the equation:

**Step 5**: Solve the correspondent vector for each values of **lambda**

I hope this article provides you with a good understanding of some important concepts of **Eigenvectors and Eigenvalues**.

If you have any questions or if you find anything misrepresented please let me know.

Thanks!

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