Linear Algebra 54 | Characteristic Polynomial [dark version]

Eigenvalues can be found by turning a matrix problem into a single polynomial equation: for a square matrix A, the eigenvalues are exactly the zeros of its characteristic polynomial p_A(Λ)=det(A−ΛI). The key reason this works is that an eigenvector x≠0 for an eigenvalue Λ exists precisely when the matrix A−ΛI becomes singular—meaning its determinant drops to zero.

The discussion starts from the eigenvalue equation Ax=Λx. Rearranging gives (A−ΛI)x=0, so x is an eigenvector exactly when it lies in the kernel of A−ΛI. A nonzero vector exists in that kernel only if the kernel is nontrivial, which happens exactly when A−ΛI is not invertible. Non-invertibility is detected by the determinant: A−ΛI is singular if and only if det(A−ΛI)=0. That equivalence links eigenvalues directly to a determinant condition, turning “find Λ such that Ax=Λx has a nonzero solution” into “find Λ such that det(A−ΛI)=0.”

A concrete 2×2 example makes the method tangible. For A=\begin{pmatrix}3&2\\1&4\end{pmatrix}, subtracting Λ along the diagonal produces A−ΛI=\begin{pmatrix}3−Λ&2\\1&4−Λ\end{pmatrix}. Taking its determinant yields a polynomial in Λ (for 2×2 matrices, it reduces to multiplying the diagonal terms and subtracting the product of the off-diagonal terms). Solving det(A−ΛI)=0 gives the characteristic polynomial’s zeros, which in this case are Λ=2 and Λ=5. Those values are then identified as the complete set of eigenvalues of A.

The transcript then generalizes the pattern to n×n matrices. Forming A−ΛI means subtracting Λ only on the diagonal entries, leaving all off-diagonal entries unchanged. Using the Leibniz formula for determinants shows that det(A−ΛI) is always a polynomial in Λ of degree n. The leading term comes from the permutation that selects all diagonal entries, contributing (−1)^nΛ^n. The remaining coefficients are bundled as constants c_{n−1}, c_{n−2}, …, c_0, so the characteristic polynomial has the form p_A(Λ)= (−1)^nΛ^n + c_{n−1}Λ^{n−1}+…+c_0.

Finally, the characteristic polynomial is formally defined: for any square matrix A, p_A is the function Λ↦det(A−ΛI). Its zeros are exactly the eigenvalues of A. The practical payoff is clear: once the characteristic polynomial is computed, finding eigenvalues becomes a root-finding problem. The transcript closes by noting that solving those polynomials may be straightforward in low dimensions, while higher-dimensional cases require more work—reserved for later videos.