MATH 247 Lecture Notes - Lecture 8: Triangular Matrix, Invertible Matrix, Characteristic Polynomial

• School: McGill University
• Department: Mathematics and Statistics
• Course: MATH 247
• Professor: Axel Hundemer

Document Summary

Main topics: characteristic polynomials, similar matrices, eigenvalues, eigenvectors, funda- mental theorem of algebra, matrix polynomials, upper triangular matrices. Motivation: to nd all the eigenvalues of a transformation. Let a be an n n matrix with coe cients in c. then, we know: Is an eigenvalue of a x k n, x 6= 0 | ax = x. We can re-write this by collecting the terms on the lhs: The correct way to factor this out is: Inx ax = 0 ( i a)x = 0. Here, x 6= 0 and ( i a) is an n n matrix. What does this actually mean: we have a homogeneous linear system with a non-trivial solution, so, i a is a non-invertible matrix, by the fundamental property of determinants, det(a) = 0 a is non-invertible. So, det( i a) = 0: also, note that: det( i a) = det. From this, we obtain a polynomial in of degree n and leading term n.