Why Do We Need Eigenvalues and Eigenvectors?

Mathematics and Such

[ Prerequisites : basic linear algebra, matrices and determinants. ]

Eigenvalues and eigenvectors are often confusing to students the first time they encounter them. This article attempts to demystify the concepts by giving some motivations and applications. It’s okay if you’ve never heard of eigenvectors / eigenvalues, since we will take things one step at a time here.

Note: all vectors are assumed to be column vectors here, and a matrix acts on a vector via left multiplication.

Consider the matrix $latex M = begin{pmatrix} 3 & 1 1 & 3end{pmatrix}$. We know mathematically what it does: it takes a (column) vector v = (xy) to w = (3x+yx+3y). But what’s the geometry behind it? If we take the circle C with radius 1 and centre at origin, then the matrix maps C to the ellipse C’

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