Gentle Introduction to Eigendecomposition, Eigenvalues, and Eigenvectors for Machine Learning - Machine Learning Mastery

Eigendecomposition can also be used to calculate the principal components of a matrix in the Principal Component Analysis method or PCA that can be used to reduce the dimensionality of data in machine learning. Eigenvectors are unit vectors, which means that their length or magnitude is equal to 1.0. They are often referred as right vectors, which simply means a column vector (as opposed to a row vector or a left vector). A right-vector is a vector as we understand them. Eigenvalues are coefficients applied to eigenvectors that give the vectors their length or magnitude. For example, a negative eigenvalue may reverse the direction of the eigenvector as part of scaling it. A matrix that has only positive eigenvalues is referred to as a positive definite matrix, whereas if the eigenvalues are all negative, it is referred to as a negative definite matrix. Decomposing a matrix in terms of its eigenvalues and its eigenvectors gives valuable insights into the properties of the matrix. Certain matrix calculations, like computing the power of the matrix, become much easier when we use the eigendecomposition of the matrix.