Eigenvalues and Eigenvectors
The defining equation
For a square matrix A, an eigenvector is a nonzero vector v that A only stretches or shrinks, never rotates off its own line: multiplying by A is the same as multiplying by a plain number λ, the eigenvalue:
Av=λv⟺(A−λI)v=0Since v=0, this only has a solution when (A−λI) is singular, giving the characteristic equation that pins down every eigenvalue:
det(A−λI)=0Worked example
Find the eigenvalues and eigenvectors of A=[2112].
Solution: Set up the characteristic equation:
det[2−λ112−λ]=(2−λ)2−1=0(2−λ)2=1⇒2−λ=±1⇒λ=1 or λ=3For λ=1: solve (A−I)v=0:
[1111][v1v2]=0⇒v1+v2=0⇒v=[1−1]For λ=3: solve (A−3I)v=0:
[−111−1][v1v2]=0⇒v1=v2⇒v=[11]Checking the answer two ways
By direct substitution: A[1−1]=[2−11−2]=[1−1]=1⋅[1−1]. Confirmed. A[11]=[33]=3⋅[11]. Confirmed.
By trace and determinant: the sum of the eigenvalues always equals the trace of A, and their product always equals detA:
λ1+λ2=1+3=4=trace(A)=2+2,λ1λ2=1×3=3=detA=(2)(2)−(1)(1)Both checks pass independently of the substitution check above, a fast way to catch an arithmetic slip in the characteristic equation itself.
Continue learning
The same partial derivatives used to build a Jacobian matrix are covered separately: eigenvalues of a Jacobian are exactly how stability of a system is analyzed near an equilibrium point.