Eigenvalues and eigenvectors: directions preserved by a transformation

Where we are on the map

The matrix triples vector (1, 1) in the same direction and leaves vector (1, −1) unchanged

Begin with a familiar image

Stretching an image on a rubber sheet turns most arrows. Along a few lines, however, arrows stay on the same line and change only length or orientation.

Precise meaning

A nonzero vector v is an eigenvector of A when Av=λv. The scalar λ is its eigenvalue: |λ| gives scale, a negative sign reverses direction, and λ=0 collapses it to zero. Find λ from det(A−λI)=0, then solve (A−λI)v=0. The zero vector is excluded.

The lesson’s support signal

meaning → model → calculation → check → explain in your own words

Av=λv → det(A−λI)=0 → find λ → find nonzero v → multiply to verify

Worked example

For A=[[2,1],[1,2]], A(1,1)=(3,3)=3(1,1), so λ=3. Also A(1,−1)=(1,−1)=1(1,−1), so λ=1.

Example with a fading prompt

Change the numbers in the example and repeat the solution without copying its steps. Estimate first, calculate exactly, and check against the original condition.

Recall without a prompt

1. Define the main idea in one sentence.
2. Rebuild the support signal from memory.
3. Solve the example with different numbers and explain why every step is valid.

Find and correct the mistake

Treating v=0 as an eigenvector is wrong. It supplies no direction and satisfies A0=λ0 for every λ, so the definition explicitly excludes it.

Transfer to a new setting

Apply the same relationship to something you can measure yourself. Name the quantities, units, and the boundary within which the model is valid. Check the answer by a second representation or an inverse operation.

Exercise

Required. For A=[[3,1],[1,3]], find the characteristic equation, eigenvalues, and one eigenvector for each. Verify every pair by direct multiplication.

With your own data. Create one everyday example with the same structure and show it in words, a formula, and a check.

Optional. Seven days later, change the numbers and repeat without the support map.

Open perspective

The checkpoint joins matrices, bases, and eigenvector directions through calculation, geometry, and meaning.

Course contents