Unlocking the Matrix: JavaScript's Gateway to Linear Transformations

A matrix is a rectangular array of real numbers arranged in m rows and n columns. For instance, a 3x2 matrix looks like this:

Matrix: 
[
  [1, 2],
  [3, 4],
  [5, 6]
]

The constructor of the Matrix class takes as its parameters elements in row order. We can extract a row from the matrix by specifying its row number, and then a column by specifying its column number:

const matrix = new Matrix([
  [1, 2],
  [3, 4],
  [5, 6]
]);

const row2 = matrix.getRow(1); // [3, 4]
const col1 = matrix.getCol(0); // [1, 3, 5]

Matrices play a central role in linear transformations, which are functions that map one vector space to another. They provide a compact and efficient way to represent these transformations, making it possible to perform complex operations with a few simple matrix multiplications.

To understand how matrices represent linear transformations, consider a simple example. Let's say we have a 2D vector space and a linear transformation that rotates vectors by 90 degrees counterclockwise. This transformation can be represented by the following matrix:

[
  [0, -1],
  [1, 0]
]

Multiplying any vector by this matrix will result in a vector that has been rotated by 90 degrees.

const transformationMatrix = new Matrix([
  [0, -1],
  [1, 0]
]);

const vector = new Vector([3, 4]);
const transformedVector = transformationMatrix.multiplyVector(vector); // [-4, 3]

Eigenvalues and eigenvectors are special values and vectors that provide insights into a matrix's behavior. Eigenvalues are the values along the matrix's diagonal, and eigenvectors are the vectors that are scaled by the eigenvalues when multiplied by the matrix.

In JavaScript, we can use the eigen() method of the Matrix class to compute eigenvalues and eigenvectors:

const eigenvaluesAndEigenvectors = matrix.eigen();
const eigenvalues = eigenvaluesAndEigenvectors.eigenvalues;
const eigenvectors = eigenvaluesAndEigenvectors.eigenvectors;

Eigenvalues and eigenvectors have applications in various fields, such as physics, engineering, and data analysis. They can be used to solve systems of differential equations, analyze vibrations, and perform dimensionality reduction.

In conclusion, matrices are fundamental to linear algebra in JavaScript. They provide a compact representation of linear transformations, enabling us to manipulate vectors and explore linear spaces. Understanding the concepts of matrices, eigenvalues, and eigenvectors opens up a world of possibilities for solving complex problems and gaining insights into data.