Why the Characteristic Polynomial Matters

The characteristic polynomial condenses a matrix's essential properties into a single expression. For a matrix $A$, it is defined as $\text{det}\left(A-\mathrm{\backslash lambda}I\right)$. The roots of this polynomial are the eigenvalues, the scalars that scale eigenvectors. Eigenvalues capture fundamental behaviors in systems of differential equations, vibrations, and Markov processes. Understanding the characteristic polynomial thus reveals whether solutions grow, decay, or oscillate.

Connections to Matrix Invariants

The coefficients of the characteristic polynomial relate directly to invariants of $A$. For a 2×2 matrix, the polynomial is ${\mathrm{\backslash lambda}}^2-\mathrm{tr}\left(A\right)\mathrm{\backslash lambda}+\text{det}\left(A\right)$. Here $\mathrm{tr}$ denotes the trace and $\text{det}$ the determinant. For a 3×3 matrix, the cubic includes $\mathrm{tr}\left(A^2\right)$ as well. These relationships allow one to derive eigenvalue sums and products without directly solving the polynomial.

Historical Perspective

Augustin-Louis Cauchy and others studied matrices via determinants in the nineteenth century, leading to the notion of the characteristic equation. The Cayley–Hamilton theorem, proved in the 1850s, states that every matrix satisfies its own characteristic polynomial. This surprising fact provides a route to compute matrix functions and proves fundamental in control theory and beyond.

Worked Example

Take $A=\left[\begin{array}{ccc}4& 2& 0\\ 1& 3& 0\\ 0& 0& 5\end{array}\right]$. The trace is $4+3+5=12$. The determinant is $4\times 3\times 5=60$. Evaluating the determinant of $A-\mathrm{\backslash lambda}I$ gives ${\mathrm{\backslash lambda}}^3-12{\mathrm{\backslash lambda}}^2+47\mathrm{\backslash lambda}-60$. Solving this cubic yields eigenvalues $5$, $4$, and $3$.

Practical Usage

By inputting the matrix coefficients, this calculator automates these steps, computing the polynomial coefficients and presenting them in a readable format. For larger systems, the characteristic polynomial can guide numerical methods like the QR algorithm. In control engineering, the roots indicate system stability through their location in the complex plane. In quantum mechanics, the characteristic polynomials of operators correspond to measurable observables. The polynomial thus bridges theoretical and applied disciplines.

Further Insights

Because the characteristic polynomial is a determinant, its coefficients are symmetric functions of the eigenvalues. This symmetry means that permuting eigenvalues leaves the polynomial unchanged, reflecting the underlying algebraic multiplicities. Additionally, knowing the characteristic polynomial allows computation of the matrix's minimal polynomial, which in turn reveals the size of Jordan blocks. Working through examples with this calculator provides intuition for these subtle relationships.