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- Complex Eigenvalues Overview:
- Review of Complex numbers
• The square root of -1
• Finding complex roots using the quadratic equation
• Re and Im of Complex vectors
- Complex Eigenvalues and Eigenvectors
• The Characteristic Equation
• Using the quadratic formula to find the complex roots
• The corresponding eigenvectors
- Finding the Argument
• Find the Eigenvalues
• Find the scale factor
• Divide the scale factor of the matrix
• Compare with the rotational matrix
• Solve for the argument
- The Formula
• Get the Eigenvalue and Eigenvector
• Find Re and Im
• Combine them to get the invertible matrix
• Use and from the eigenvalue to get matrix
- Finding the Complex Eigenvalues/Eigenvectors
Find the complex eigenvalues of A and their corresponding eigenvectors.
- Finding the Argument using Eigenvalues
The transformation → is the composition of a rotation and scaling. Find the eigenvalues of . Then give the angle of the rotation, where < , and give the scale factor if
- The transformation → is the composition of a rotation and scaling. Find the eigenvalues of . Then give the angle of the rotation, where < , and give the scale factor if
- Finding the Invertible Matrix and Matrix C
Find an invertible matrix and a matrix of the form such that the given matrix has the form if
- Advanced Proofs Related to the Eigenvector
Let be a real matrix, and let be a vector in . Show that
- Let be a real matrix with a complex eigenvalue where and an associated eigenvector in . Show that: