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Mathematics for Physics

241 Characteristic Polynomial

A polynomial built from a linear transformation that is used to find eigenvalues as its roots.

definition [d] (Characteristic Polynomial) From Axler: the polynomial defined by

  • \(z \mapsto \det(zI - T)\)

is called the characteristic polynomial of \(T\).

where

  • \(T\) is an operator on a finite-dimensional vector space.
  • \(I\) is the identity operator.
  • \(z\) is a scalar variable.
  • \(\det\) is the determinant.

definition [d] (Characteristic Polynomial) From Shifrin and Adams: let \(A\) be a square matrix. Then

  • \(p(t) = p_{A}(t) = \det(A - tI)\)

is called the characteristic polynomial of \(A\).

where

  • \(A\) is a square matrix.
  • \(I\) is the identity matrix of the same size.
  • \(t\) is a scalar variable.

Note:

  • Axler’s convention uses \(\det(zI - T)\); Shifrin’s matrix form uses \(\det(A - tI)\). The two differ by a sign factor \((-1)^{n}\) in dimension \(n\).

241.1 Elementary Example

241.1.1 Simple

For a \(2 \times 2\) diagonal matrix, Axler’s characteristic polynomial factors as a product of linear terms.

\[ A = \begin{pmatrix} 2 & 0 \\ 0 & 3 \end{pmatrix} \]

\[ zI - A = \begin{pmatrix} z-2 & 0 \\ 0 & z-3 \end{pmatrix} \]

\[ \det(zI - A) = (z-2)(z-3) \]

where

  • the roots \(z = 2\) and \(z = 3\) are the eigenvalues of \(A\).

241.1.2 General

In three dimensions, a diagonal matrix yields a cubic characteristic polynomial.

\[ A = \operatorname{diag}(2,3,5) \]

\[ \det(zI - A) = (z-2)(z-3)(z-5) \]

where

  • the three roots are the three eigenvalues of \(A\).

241.2 References

  1. Axler, S. Linear Algebra Done Right. — characteristic polynomial \(z \mapsto \det(zI - T)\).
  2. Shifrin, T., & Adams, M. Linear Algebra: A Geometric Approach. W. H. Freeman, 2010. — \(p_{A}(t) = \det(A - tI)\).