The Characteristic Polynomial

Linear Algebra (Matrices) · Axiom Academy

LESSON Reading the Characteristic Polynomial One polynomial holds it all — its roots are the eigenvalues, and its coefficients spell out the trace and determinant. 1. The Roots Are the Eigenvalues Take the matrix . Expanding gives a degree-2 polynomial, . Plot it, and the story is geometric: wherever the curve crosses the -axis, that value of is an eigenvalue — because there , which is exactly the eigenvalue condition. A root is a value where the curve hits zero …and that is precisely the eigenvalue test 2. Degree n Counts the Eigenvalues For an matrix, always has degree exactly n . By the Fundamental Theorem of Algebra a degree- n polynomial has n roots counted with multiplicity — so an matrix has n eigenvalues. Watch the parabola (two crossings) grow into a cubic (three crossings): bump the size, gain a root. has degree n for an matrix — the size of A sets the degree. Exactly n roots over , counted with multiplicity — hence n eigenvalues. A repeated root, e.g. , is one eigenvalue with algebraic multiplicity 2 . 3. Trace and Determinant Hide in the Coefficients Multiply out the factored form and the coefficients are symmetric functions of the roots . For that means the coefficient is and the constant term is . So the sum of the eigenvalues is the trace and their product is the determinant — read straight off the page, no eigenvector needed. You've read the characteristic polynomial three ways: its roots, its degree, and its coefficients — each one a window into the matrix.

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