3×3 Eigenvalue Computation

Linear Algebra (Matrices) · Axiom Academy

EXAMPLE Eigenvalues of a Matrix Solving : cofactor expansion to the characteristic cubic, then factoring for the three eigenvalues. Find all eigenvalues of the matrix by solving the characteristic equation . Report each eigenvalue with its algebraic multiplicity. Nicely done. You computed every eigenvalue of a matrix straight from the characteristic equation. Characteristic equation: eigenvalues are exactly the roots of , which is a cubic for a matrix. Expand smartly: cofactor-expand along the row or column with the most zeros. The bottom row collapsed the work to a single determinant. Algebraic multiplicity: a repeated factor means a repeated eigenvalue. Here gives with multiplicity 2 . Result: and (multiplicity 2 ) — three eigenvalues counted with multiplicity, as a matrix must have. The algebraic multiplicity you just found sets the stage for diagonalization: next you would compute eigenvectors and compare the geometric multiplicity of against its algebraic multiplicity of 2 .

This is the written version of the interactive lesson above. See the full Linear Algebra (Matrices) course.