Eigenvalues and Eigenvectors

GRE Math Subject · Axiom Academy

LESSON Eigenvalues and Eigenvectors Finding and interpreting the special directions and scaling factors of linear transformations Let be an matrix. A scalar is an eigenvalue of if there exists a nonzero vector such that: Such a vector is called an eigenvector corresponding to eigenvalue . Geometric interpretation: The matrix acts on the eigenvector by simply scaling it by . is an eigenvalue if and only if has a nonzero solution. This happens when: The polynomial is the characteristic polynomial . Its roots are exactly the eigenvalues. For each eigenvalue , find eigenvectors by solving: The solution space is the eigenspace . Key fact: Geometric multiplicity algebraic multiplicity If they're equal for all eigenvalues, the matrix is diagonalizable . Useful Facts About Eigenvalues If is an eigenvalue of , then is an eigenvalue of If is invertible and is an eigenvalue, then is an eigenvalue of Eigenvectors for distinct eigenvalues are linearly independent Eigenvalue Problems on the GRE Compute characteristic polynomial for or matrices Find eigenvalues and eigenvectors explicitly Determine if a matrix is diagonalizable (check multiplicities) Use eigenvalues to find trace and determinant Prove properties using eigenvalue properties Pro tip: Eigenvalue questions appear in ~25% of linear algebra problems. Master characteristic polynomials first!

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