Complex eigenvalues

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Intros
Lessons
  1. Complex Eigenvalues Overview:
  2. 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
  3. Complex Eigenvalues and Eigenvectors
    • The Characteristic Equation
    • Using the quadratic formula to find the complex roots
    • The corresponding eigenvectors
  4. 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
  5. 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
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Examples
Lessons
  1. Finding the Complex Eigenvalues/Eigenvectors
    Find the complex eigenvalues of A and their corresponding eigenvectors.
    Find the complex eigenvalues of matrix
    1. 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
      1. 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
        1. 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
          1. 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)
            1. 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)=aRe  v+bv)=aRe\; v+bIm vv
              A(A(Im v)=bRe  v+av)=-bRe\; v+aIm vv