6.4 Summary

In this chapter, we’ve veered into the theory side of math once again. This time, it was about eigenvalues and eigenvectors of a matrix, that is, scalars λ and vectors x for which

hold.

Just like most mathematical objects, this might seem daunting at first, but geometrically, this means that in the direction x, the linear transformation A is the same as a stretching by λ. In practice, we can find eigenvectors by solving the so-called characteristic equation

for λ.

What are eigenvalues used for? There are tons of applications, but one stands out: according to Theorem 38, if you can build a basis from the eigenvectors of the matrix A ∈ℝ^n×n, then you can find a T ∈ℝ^n×n such that T⁻¹AT is diagonal. This process is extremely useful. For one, multiplication with diagonal matrices is fast and simple, and we prefer to do it whenever we can. For another, diagonalization reveals a ton about the internal...