Complex eigenvalues

Complex eigenvalues

Lessons

In this section, we will be finding complex eigenvalues and eigenvectors of 2×22 \times 2 matrices. We will not go any higher than a 2×22 \times 2.

Given a complex vector complex vector v, the real part and imaginary part would be:
real part and imaginary part of complex vector v

Let xx Ax Ax be a transformation that is composed of scaling and rotation. Also, let transformation: scaling and rotation of matrix. Then we can find the angle (argument) φ\varphi in the complex plane by doing the following steps:

1. Find the complex eigenvalues
2. Find the scale factor λ=r=a2+b2| \lambda |=r=\sqrt{a^2+b^2}
3. Divide the scale factor of the matrix
4. Compare with the rotational matrix rotational matrix and create two equations
5. Solve for the argument φ\varphi

Let AA be a real 2×22 \times 2 matrix with a complex eigenvalue λ=abi\lambda =a-bi and associated eigenvector vv in C2\Bbb{C}^2. Then A=PCP1A=PCP^{-1} where

matrix p and matrix c
  • 1.
    Complex Eigenvalues Overview:
    a)
    Review of Complex numbers
    • The square root of -1
    • Finding complex roots using the quadratic equation
    • Re(v)(v) and Im(v)(v) of Complex vectors

    b)
    Complex Eigenvalues and Eigenvectors
    • The Characteristic Equation
    • Using the quadratic formula to find the complex roots
    • The corresponding eigenvectors

    c)
    Finding the Argument
    • Find the Eigenvalues a±bia \pm bi
    • Find the scale factor λ=r=a2+b2| \lambda |=r=\sqrt{a^2+b^2}
    • Divide the scale factor of the matrix
    • Compare with the rotational matrix
    • Solve for the argument φ\varphi

    d)
    The Formula A=PCP1A=PCP^{-1}
    • Get the Eigenvalue λ=abi\lambda =a-bi and Eigenvector vv
    • Find Re(v)(v) and Im(v)(v)
    • Combine them to get the invertible matrix PP
    • Use aa and bb from the eigenvalue abia-bi to get matrix CC


  • 2.
    Finding the Complex Eigenvalues/Eigenvectors
    Find the complex eigenvalues of A and their corresponding eigenvectors.
    Find the complex eigenvalues of matrix

  • 3.
    Finding the Argument using Eigenvalues
    The transformation xx Ax Ax is the composition of a rotation and scaling. Find the eigenvalues of AA. Then give the angle φ\varphi of the rotation, where π-\pi < φπ\varphi \leq \pi, and give the scale factor rr if
    Finding the Argument using Eigenvalues 1

  • 4.
    The transformation xx Ax Ax is the composition of a rotation and scaling. Find the eigenvalues of AA. Then give the angle φ\varphi of the rotation, where π-\pi < φπ\varphi \leq \pi, and give the scale factor rr if
    Finding the Argument using Eigenvalues 2

  • 5.
    Finding the Invertible Matrix and Matrix C
    Find an invertible matrix PP and a matrix CC of the form Find an invertible matrix P and a matrix C such that the given matrix has the form A=PCP1A=PCP^{-1} if
    Find an invertible matrix P and a matrix C

  • 6.
    Advanced Proofs Related to the Eigenvector
    Let AA be a real n×nn \times n matrix, and let xx be a vector in Cn\Bbb{C}^n. Show that

    Re(Ax)=A((Ax)=A(Re x)x)

    Im(Ax)=A((Ax)=A(Im x)x)

  • 7.
    Let AA be a real 2×22 \times 2 matrix with a complex eigenvalue λ=abi\lambda =a-bi where b0b \neq 0 and an associated eigenvector vv in C2\Bbb{C}^2. Show that:
    A(A(Re v)=aRev+bv)=aRe\; v+bIm vv
    A(A(Im v)=bRev+av)=-bRe\; v+aIm vv