Characteristic polynomial

A number \(\lambda\) is an eigenvalue of \(A\) when \((A-\lambda I)v=0\) has a solution \(v\neq0\), which happens exactly when \(A-\lambda I\) is not invertible, that is, when \(\det(A-\lambda I)=0\). Hence the eigenvalues of \(A\) are the roots of \(p_A(\lambda)=\det(A-\lambda I)\).

For a \(2\times2\) matrix with entries \(a,b,c,d\), \(p_A(\lambda)=\lambda^2-(a+d)\lambda+(ad-bc)\): the coefficient of \(\lambda\) is the negative of the sum of the diagonal and the constant term is \(\det A=p_A(0)\). Example: for \(A=\begin{pmatrix}2&1\\1&2\end{pmatrix}\), \(p_A(\lambda)=\lambda^2-4\lambda+3=(\lambda-1)(\lambda-3)\), so the eigenvalues are 1 and 3.

For an \(n\times n\) matrix, \(p_A\) has degree \(n\), so there are at most \(n\) distinct eigenvalues. If the matrix is triangular, \(p_A\) is the product of the factors \((a_{ii}-\lambda)\) and the eigenvalues are the diagonal entries. Some roots may be complex: the rotation \(\begin{pmatrix}0&-1\\1&0\end{pmatrix}\) has \(p_A(\lambda)=\lambda^2+1\), with no real roots. A repeated root, as in \(\begin{pmatrix}2&1\\0&2\end{pmatrix}\) with \(p_A(\lambda)=(2-\lambda)^2\), counts as a single eigenvalue.

Typical mistakes: subtracting \(\lambda\) from the whole matrix instead of only the diagonal (it is \(A-\lambda I\)), slipping a sign when expanding the determinant, and confusing the polynomial with its roots. Some books define \(p_A\) as \(\det(\lambda I-A)\); it differs from this version by the factor \((-1)^n\) and has the same roots.