Eigenvalues & Eigenvectors

AHL AI 1.15

The eigenvectors of a matrix A are the vector(s) v such that

Av=λv

for some constant(s) λ which we call eigenvalues.

AHL AI 1.15

The eigenvalues λ of a matrix A satisfy

det(A−λI)=0

For example, if A=(−1−234) then

det(−1−λ−234−λ)=0⟹(−1−λ)(4−λ)+6=0

which simplifies to

λ2−3λ+2=0⇒λ=−1,−2

We call λ2−3λ+2 the characteristic polynomial of A.

AHL AI 1.15

If we know an eigenvalue of a matrix A, we can find the corresponding eigenvector using its definition:

A=(acbd)(xy)=λ(xy).

When the matrix A is known, we can solve this system of simultaneous equations to find the eigenvector (xy).

AHL AI 1.15

If a matrix A has two distinct, real eigenvalues, then we can write it in the form

A=PDP−1

where P=(v1v2) is formed with the eigenvectors of A as its columns, and D=(λ100λ2) is a diagonal matrix.

Exercises