Finding Eigenvalues of 2×2 Matrices

Linear Algebra (Matrices) · Axiom Academy

EXAMPLE Finding Eigenvalues of 2×2 Matrices Solve the characteristic equation to find both eigenvalues of a 2×2 matrix. Find the eigenvalues of the matrix . An eigenvalue is a number for which has a nonzero solution , so we solve the characteristic equation . Most vectors get rotated by A , but vectors along two special directions are only scaled . The scale factors are the eigenvalues — for this matrix they come out to 4 and 2. Nicely done — you found both eigenvalues of a 2×2 matrix straight from the characteristic equation. Characteristic equation: eigenvalues are exactly the that make . Shortcut for 2×2: the characteristic polynomial is always — here and , giving . Check it: the roots should sum to the trace ( 4+2=6 ) and multiply to the determinant ( ) — both hold. Same recipe works for any square matrix: build , set it to zero, and solve. The next step is finding the eigenvector for each .

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