APSC 174: Linear Algebra for Engineers

Definition

For a square matrix $A$ of size $n \times n$ , the characteristic polynomial $p_A(\lambda)$ is defined as:

p_A(\lambda) = \det(A - \lambda I)

where $I$ is the identity matrix of the same size as $A$ .

The eigenvalues of a matrix $A$ are the roots of its characteristic polynomial:

p_A(\lambda) = 0

The characteristic polynomial of an $n \times n$ matrix is a polynomial of degree $n$
The coefficient of $\lambda^n$ is $(-1)^n$
The constant term is $\det(A)$
The coefficient of $\lambda^{n-1}$ is $(-1)^{n-1}\text{tr}(A)$ , where $\text{tr}(A)$ is the trace of $A$

Two matrices are similar if and only if they have the same characteristic polynomial.

Consider the matrix:

A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}

The characteristic polynomial is:

p_A(\lambda) = \det\begin{pmatrix} a-\lambda & b \\ c & d-\lambda \end{pmatrix} = (a-\lambda)(d-\lambda) - bc = \lambda^2 - (a+d)\lambda + (ad-bc)

For a general 3x3 matrix:

A = \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix}

The characteristic polynomial is:

p_A(\lambda) = -\lambda^3 + (a+e+i)\lambda^2 - (ae+ai+ei-bd-cg-fh)\lambda + \det(A)

The characteristic polynomial is fundamental in: