Eigenvectors, also known as characteristic vectors, are vectors that do not change their direction when a linear transformation is applied to them, such as rotation or reflection, only their modulus. These vectors are very useful in different areas of mathematics and science, as they often have very interesting and useful properties.
Eigenvalues, also known as characteristic values, are scalars that are used to represent the magnitude of the transformation applied to eigenvectors. That is, an eigenvalue is the factor by which the modulus of an eigenvector is multiplied when a linear transformation is applied to it. Each eigenvector is associated with an eigenvalue, and together they form an eigenvalue-eigenvector pair. Eigenvalues can be used to describe how the magnitude of an eigenvector changes when a linear transformation is applied.
To find the eigenvectors and eigenvalues of a matrix, we must first determine the matrix's eigenvalues. Eigenvalues are the values that make the equation (A - λI) singular (not invertible), where A is the matrix in question and I is the identity matrix. Once we have found the eigenvalues, we can solve the equation (A - λI) * x = 0 to find the eigenvectors associated with each of them.
Eigenvectors and eigenvalues have many practical applications. One of the most well-known is in principal component analysis (PCA), which is a data analysis technique used to reduce the dimensionality of a dataset and highlight the most important features. In PCA, eigenvectors are used to construct the principal components, which are vectors used to represent the maximum variability in the dataset.
Another important application of eigenvectors and eigenvalues is in the calculation of rotation and reflection matrices in space. These matrices can be used to rotate and reflect objects in space, which is very useful in applications such as engineering and data visualization.
In addition to these applications, eigenvectors and eigenvalues have other very interesting properties. For example, each matrix has a finite number of eigenvectors and eigenvalues, and each eigenvector is associated with a unique eigenvalue. This means that each matrix has a finite number of eigenvalue-eigenvector pairs.
Eigenvectors also have the property of being orthogonal to each other. This means that if we have two eigenvectors x and y, then x * y = 0. This property is very useful in the calculation of rotation and reflection matrices, as it means that the eigenvectors have
