Eigenvectors and eigenvalues are (among other things) connected to something called diagonalization. In math, this is incredibly useful because it lets you desc

Eigenvectors and eigenvalues are (among other things) connected to something called diagonalization. In math, this is incredibly useful because it lets you describe a matrix as being put together from simpler, easy-to-understand matrixes. Eigenvectors and eigenvalues are also a major focus of quantum mechanics. For example, when you measure the energy level of an electron in a quantum mechanical system, the measurement itself is a linear operator on the underlying wavefunction (linear operator = think "like a matrix"). The eigenvalues are the different energy levels of the electron, and the eigenvectors are the wavefunctions which are states of the electron with the corresponding energy level. You can experimentally verify that the eigenvalues correspond to spectral lines (the rainbow you see when you look at the substance through diffraction grating). There are a ton of other things going on with eigenvalues and eigenvectors, they're used all over the place. If you want to understand a Markov process, for example, it can be described in terms of linear equations, and if it has a steady-state, it's an eigenvector.