What Is the Minimal Polynomial?

For a square matrix $A$, the minimal polynomial is the monic polynomial of least degree $m$ such that $m\left(A\right)=0$. It divides the characteristic polynomial and shares the same roots, yet reflects deeper information about repeated eigenvalues and the structure of $A$.

For a 2×2 matrix, the characteristic polynomial takes the form $p\left(\lambda \right)={\lambda }^2-\mathrm{tr}\left(A\right)\lambda +\det \left(A\right)$. Let the eigenvalues be $\lambda 1$ and $\lambda 2$. If they are distinct, $m\left(\lambda \right)=(\lambda -\lambda 1)(\lambda -\lambda 2)$. If $\lambda 1$ equals $\lambda 2$, the minimal polynomial is either $\lambda -\lambda 1$ or ${\lambda -\lambda 1}^2$ depending on whether $A$ is diagonalizable.

Procedure Used Here

The calculator first computes the trace and determinant of your matrix. From these it derives the eigenvalues using the quadratic formula. If the eigenvalues are distinct, the minimal polynomial coincides with the characteristic polynomial. When they coincide, the script checks whether $A-\lambda 1I$ is the zero matrix. If not, the minimal polynomial is squared to reflect the defective nature of the matrix.

Example

Take $A$ = $\left[\begin{array}{cc}2& 1\\ 0& 2\end{array}\right]$. The eigenvalue calculation yields a repeated root $\lambda =2$. Because $A$ is not already $\mathrm{2I}$, the minimal polynomial is ${\lambda -2}^2$. Applying the polynomial to $A$ indeed produces the zero matrix.

Why It Matters

The minimal polynomial reveals the size of Jordan blocks associated with each eigenvalue. In turn, this information determines whether the matrix is diagonalizable and helps classify linear transformations up to similarity. The polynomial also enters the Cayley–Hamilton theorem, matrix functions, and solution formulas for differential equations.

By exploring various matrices, you can see how repeated eigenvalues alter the minimal polynomial and connect to geometric multiplicity. This understanding is fundamental when analyzing systems that cannot be fully diagonalized.