Eigenvectors: the arrows a matrix only stretches

Most arrows turn, a few don't

Take A = 2112 and watch the plane move. The two coloured landmarks, 10 and 01, both turn on their way to the columns 21 and 12. The two faint lines are different. The arrow x₁ = 11 stays on its line and grows. The arrow x₂ = 1−1 stays exactly where it is.

The equation Ax = λx

"A keeps the direction of x" means Ax is a multiple of x. Call the multiple λ (the Greek letter lambda):

The zero vector always works (A0 = 0), so it doesn't count. To test an arrow, multiply and compare:

What the number λ tells you

The eigenvalue says what happens along that line:

• λ = 3: the arrow gets 3 times longer.
• λ = ½: it shrinks to half. For 300½, A01 = 0½ = ½·01.
• λ = −1: it flips to point the other way, with the same length.
• λ = 0: it is squashed to the zero vector. For 1000, A01 = 00 = 0·01.

A mirror shows two of these at once. The mirror across the line y = x is 0110. Arrows on the mirror line, like x₁ = 11, stay put, so λ = 1. Arrows straight across the mirror, like x₂ = 1−1, flip, so λ = −1:

A whole line, not one arrow

If x is an eigenvector, so is every multiple of x, with the same λ. For our first A:

So we really look for eigenvector lines. We name one arrow on each line, like 11, and remember that any multiple works too.

Why the other arrows turn

Any arrow is a mix of the two eigenvectors. Each part is stretched by its own λ, and then the parts are added back. For 10:

The 11 part grew 3 times, the other part didn't. So the sum tips toward 11. That is why 10 turned in the first picture.

Not every matrix has them

A quarter turn moves every arrow by 90°. No arrow stays on its own line, so there is no eigenvector made of ordinary (real) numbers. On slide 2 you will learn the test det(A − λI) = 0 (I is the identity matrix). For this turn it gives λ² + 1 = 0, which no real number solves.