Characteristic polynomial

The characteristic polynomial of an n × n square matrix $A$ is the polynomial $p_A(\lambda) = \det(A - \lambda I)$, where $I$ is the identity matrix and $\det$ denotes the determinant. The roots of this polynomial are precisely the eigenvalues of $A$, and the polynomial itself encodes not only the eigenvalues but also their algebraic multiplicities. The degree of the characteristic polynomial is $n$, guaranteeing exactly $n$ roots (counting multiplicities) in the complex numbers by the fundamental theorem of algebra.

The coefficients of the characteristic polynomial have direct geometric meaning. The coefficient of $\lambda^{n-1}$ is $(-1)^{n-1}$ times the trace of $A$ (the sum of eigenvalues); the constant term is $(-1)^n \det(A)$ (the product of eigenvalues). These relationships make the characteristic polynomial a powerful computational tool: rather than solving the eigenvector equation $Av = \lambda v$ directly, one can find eigenvalues by root-finding on a single polynomial.

The characteristic polynomial is also central to the Cayley-Hamilton theorem, which states that every square matrix satisfies its own characteristic equation: $p_A(A) = 0$. This theorem reveals a deep algebraic constraint on matrix powers and functions, and underlies many results in linear systems theory and control.

The characteristic polynomial reduces the geometry of linear transformations to the algebra of polynomial roots — a compression that is as powerful as it is deceptive. The polynomial tells you where the eigenvalues are, but nothing about the eigenvectors; it gives you the spectrum, but not the operator's geometry. To stop at the characteristic polynomial is to read the table of contents and call it the book.