# Eigenvalue Calculator

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**How to Use the Eigenvalue Calculator**:

**Matrix Size**: Begin by specifying the size of your matrix using the "Matrix Size" input. Adjust the number, and the matrix will resize accordingly.**Input Matrix**: Fill in the matrix with your desired values. Ensure that the matrix is square (i.e., the number of rows is equal to the number of columns).**Calculate**: Once you've filled in the matrix, click on the "Calculate Eigenvalues" button.**Results**: In a moment, you'll see a table displaying the eigenvalues, eigenvectors, and a visualization of the eigenvectors.**Interrupt**: If you feel that the processing is taking too long (especially for larger matrices), you can interrupt the calculation by clicking the "Stop" button.

**Brief Introduction to Eigenvalues and Eigenvectors**:

Eigenvalues and eigenvectors are foundational concepts in linear algebra and have extensive applications across various fields, from quantum mechanics to machine learning.

**Eigenvalue**: Given a square matrix \( A \), an eigenvalue is a scalar \( \lambda \) such that the product of \( A \) and a vector \( v \) is the same as scaling \( v \) by \( \lambda \). In other words, for some vector \( v \):

\[

A \cdot v = \lambda \cdot v

\]

**Eigenvector**: Following the above definition, the vector \( v \) in the equation is termed the eigenvector corresponding to the eigenvalue \( \lambda \).

**Use Cases**:

Eigenvalues and eigenvectors are pivotal in:

**Physics**: They play a critical role in quantum mechanics, where they represent observable quantities and their states.**Computer Science**: In machine learning, they are used in Principal Component Analysis (PCA) for dimensionality reduction.**Engineering**: Used in the study of linear transformations and system stability analysis.**Economics**: Useful in understanding multi-variable linear models and their behaviors.

Explore the profound world of matrices and transformations with our Eigenvalue Calculator. Dive deep, compute efficiently, and uncover the hidden patterns within your data!