Eigenvectors and Eigenvalues

When a matrix transforms the plane, most vectors get knocked clean off their original line — they rotate as well as stretch, ending up pointing somewhere new. But a few special directions refuse to budge: the transformation only stretches or shrinks them, leaving them pointing exactly the same way (or exactly opposite). Those rare, unmoved directions are the eigenvectors of the matrix, and the stretch factor is the eigenvalue.

It sounds like a narrow, technical curiosity — yet this "stays-on-its-own-line" behaviour turns out to unlock an astonishing range of real problems. The original Google search engine ranked the entire web by finding one eigenvector. Engineers predict whether a bridge will shake itself apart in the wind by finding the eigenvalues of its vibration equations. Both are the same idea underneath: find the directions a transformation treats specially, and the rest of the system's behaviour falls out for free.

For an eigenvector \vec{v}, applying A is the same as multiplying by a single number \lambda — its eigenvalue:

A\vec{v} = \lambda\vec{v}.

Hunt for the unmoved directions

Spin the input vector \vec{v} (faint) and watch its image A\vec{v} (bold) for the matrix A = \begin{bmatrix} 2 & 1 \\ 1 & 2 \end{bmatrix}. Most of the time the two point different ways. But at a couple of special angles they snap onto the same line — there \vec{v} is an eigenvector, and the readout shows its eigenvalue \lambda (how many times longer A\vec{v} is). Watch for the two faint dashed lines: they mark the eigenvector directions, and we'll use this very matrix in the worked example below.

Worked example 1 — verify an eigenvector by hand

Is \vec{v} = \begin{bmatrix} 1 \\ 1 \end{bmatrix} an eigenvector of A = \begin{bmatrix} 2 & 1 \\ 1 & 2 \end{bmatrix} — the same matrix from the diagram above? Just multiply and see what comes out:

A\vec{v} = \begin{bmatrix} 2 & 1 \\ 1 & 2 \end{bmatrix}\begin{bmatrix} 1 \\ 1 \end{bmatrix} = \begin{bmatrix} 2(1) + 1(1) \\ 1(1) + 2(1) \end{bmatrix} = \begin{bmatrix} 3 \\ 3 \end{bmatrix}.

Is (3,3) a scalar multiple of (1,1)? Yes: (3,3) = 3\cdot(1,1). So \vec{v} is an eigenvector of A, with eigenvalue \lambda = 3 — this is exactly the direction where the diagram's two arrows line up. The whole check is mechanical: compute A\vec{v}, then ask "is this just a number times the original \vec{v}?"

Worked example 2 — the easy case: a diagonal matrix

For a diagonal matrix, the eigenvectors are almost embarrassingly obvious. Take D = \begin{bmatrix} 3 & 0 \\ 0 & 7 \end{bmatrix}. Multiply it by \hat{\imath} = \begin{bmatrix}1\\0\end{bmatrix}:

D\hat{\imath} = \begin{bmatrix} 3 & 0 \\ 0 & 7 \end{bmatrix}\begin{bmatrix} 1 \\ 0 \end{bmatrix} = \begin{bmatrix} 3 \\ 0 \end{bmatrix} = 3\,\hat{\imath}.

So \hat{\imath} is an eigenvector with eigenvalue 3 — exactly the diagonal entry sitting in its own column. The same happens for \hat{\jmath}: D\hat{\jmath} = 7\hat{\jmath}, eigenvalue 7. In general, a diagonal matrix's eigenvectors are simply the standard basis vectors, and its eigenvalues are just its diagonal entries, read straight off with no work at all. This trivial case is exactly why diagonalization is so desirable later on: if you can find a basis in which a matrix becomes diagonal, its eigen-structure becomes this easy to read.

Worked example 3 — a reflection, geometrically

Eigenvectors don't always need algebra — sometimes geometry hands them to you directly. Consider the matrix that reflects the plane across the line y = x: R = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}.

Any point on the mirror line, like (1,1), reflects to itself: R\begin{bmatrix}1\\1\end{bmatrix} = \begin{bmatrix}1\\1\end{bmatrix}, an eigenvector with eigenvalue +1 — reflecting it does nothing at all. But a point perpendicular to the mirror, like (1,-1), flips to exactly the opposite side: R\begin{bmatrix}1\\-1\end{bmatrix} = \begin{bmatrix}-1\\1\end{bmatrix} = -1\begin{bmatrix}1\\-1\end{bmatrix}, an eigenvector with eigenvalue -1. Every reflection has exactly this shape: +1 along the mirror, -1 perpendicular to it — you can often name a reflection's eigenvectors just by looking at the picture, no matrix arithmetic required.

An eigenvalue of exactly 0 is not a mistake. If A\vec{v} = 0\cdot\vec{v} = \vec{0}, that simply means A squashes the direction \vec{v} flat onto the zero vector — a real, important eigenvector with eigenvalue 0, not a broken calculation. It happens whenever A collapses some direction entirely, for instance a matrix that projects everything onto a single line: any vector perpendicular to that line gets flattened to nothing, eigenvalue 0.

The other classic trap: eigenvectors are only defined up to scaling. If \vec{v} is an eigenvector of A with eigenvalue \lambda, then so is 2\vec{v}, -\vec{v}, or c\vec{v} for any nonzero c — always with the very same eigenvalue. Check it for Worked Example 1: A(2,2) = (6,6) = 3(2,2), still eigenvalue 3. An eigenvector names a direction, not a specific length — don't be surprised when two textbooks list "different" eigenvectors for the same matrix that are really just scaled copies of each other.

Why anyone cares

Eigenvectors are the "natural axes" of a transformation — the directions along which it does nothing but scale. Find them, as in the three worked examples above, and a tangled matrix becomes simple: it's just stretching along its own private set of axes. That insight powers diagonalization, the stability of physical systems, the ranking behind Google's PageRank, and the principal directions of a dataset. First, though, we need a general way to compute eigenvectors and eigenvalues for a matrix where the answer isn't as obvious as it was above — that's next.

In the late 1990s, Larry Page and Sergey Brin needed a way to rank billions of web pages by importance. Their idea, PageRank, builds a giant matrix where each entry records whether one page links to another, and treats a link as a small "vote" of importance passed along. A page is important if important pages link to it — a circular definition that sounds impossible to pin down.

The resolution is exactly the idea on this page: the ranking of every page on the web is the eigenvector of that link matrix corresponding to eigenvalue 1. Importance flowing around the web settles into the one distribution of "scores" that the link matrix leaves completely unchanged — an eigenvector, by definition. Instead of solving billions of equations directly, early search engines found this eigenvector by repeatedly multiplying an importance guess by the matrix, watching it settle towards the true ranking, page after page after page.

Every bridge, building, and aircraft wing has its own natural rhythms — frequencies at which it loves to vibrate, the same way a wine glass rings at one particular pitch when you flick it. Engineers find those rhythms by writing down the structure's equations of motion as a matrix and computing its eigenvalues: each eigenvalue corresponds to one natural resonant frequency, and its eigenvector describes the exact shape the structure takes as it vibrates at that frequency (a "mode shape").

In 1940, the Tacoma Narrows Bridge in Washington State began twisting itself violently in moderate wind, and collapsed within hours — famous footage shows the deck rippling like a piece of cloth. The wind was feeding energy into the bridge at a rate that matched one of the structure's own eigenvalue-derived resonant frequencies, pumping the vibration bigger and bigger until the bridge tore itself apart. Modern bridges are deliberately engineered so their eigenvalues stay well clear of the frequencies that wind, traffic, or earthquakes are likely to produce — a direct, life-or-death application of finding a matrix's eigenvectors and eigenvalues.

See it explained