Find the smallest monic polynomial for a matrix. Review powers, dependencies, exports, and example calculations. Use clean inputs and verify results with confidence today.
| Case | Matrix | Minimal Polynomial | Why It Happens |
|---|---|---|---|
| Repeated eigenvalue, non diagonal | [[2, 1], [0, 2]] | x^2 - 4x + 4 | The Jordan block forces a repeated factor. |
| Distinct eigenvalues | [[1, 0], [0, 2]] | x^2 - 3x + 2 | Each distinct eigenvalue contributes one linear factor. |
| Scalar matrix | [[5, 0], [0, 5]] | x - 5 | A scalar matrix is annihilated by one linear term. |
| 3 x 3 Jordan block | [[3, 1, 0], [0, 3, 1], [0, 0, 3]] | x^3 - 9x^2 + 27x - 27 | The block size creates a cubic repeated factor. |
The calculator looks for the first linear dependence among the matrices I, A, A2, and higher powers.
Core relation: Ak = c0I + c1A + ... + ck-1Ak-1
Once that first dependence appears, the minimal polynomial is built as:
m(x) = xk - ck-1xk-1 - ... - c1x - c0
The calculator then evaluates m(A). A very small residual confirms the matrix is annihilated numerically.
For reference, the page also computes the characteristic polynomial. The minimal polynomial must divide it.
This minimal polynomial calculator helps you study square matrices in a direct way. Enter a 2 x 2 or 3 x 3 matrix. The page searches for the smallest monic polynomial that sends the matrix to zero. It also shows the characteristic polynomial for comparison. That gives you a clearer view of matrix structure.
The minimal polynomial is important in linear algebra. It describes the smallest exact polynomial relation satisfied by a matrix. That relation helps you test diagonalizability, analyze repeated eigenvalues, and understand Jordan blocks. It is often more informative than a determinant alone. It also supports proofs and exam problems involving matrix powers.
The tool builds the identity matrix, the matrix A, and the next powers of A. Then it checks when those matrices become linearly dependent. The first valid dependence creates the minimal polynomial. This approach follows the standard algebraic definition. A residual check is also shown. That check helps confirm the numerical result is consistent with m(A) = 0.
This page gives more than a final expression. It displays the matrix size, the detected degree, the minimal polynomial, and the first dependence relation. It also shows the characteristic polynomial beside the result. That comparison is useful because the minimal polynomial must divide the characteristic polynomial. Students can see both objects in one place and learn the connection faster.
Use this calculator for homework checks, classroom demonstrations, or self study. It is helpful when working with eigenvalues, diagonalization, Cayley Hamilton problems, recurrence relations, and matrix decomposition topics. The layout stays simple, so the algebra remains the focus. The saved session history also makes repeated testing easy when you want to compare several matrices.
The page includes CSV and PDF options. CSV is helpful for keeping a quick result log. PDF is useful for notes, tutoring handouts, and printable practice sheets. The interface stays light and clear, which makes the calculator easy to adapt for broader maths collections.
A minimal polynomial is the smallest monic polynomial that makes m(A) = 0 for a matrix A. It captures the shortest exact algebraic relation satisfied by that matrix.
No. The minimal polynomial must divide the characteristic polynomial, but it can have lower degree. They match only in some matrices.
This page supports 2 x 2 and 3 x 3 square matrices. Those sizes cover many common classroom and exam examples.
A diagonalizable matrix has distinct linear factors in its minimal polynomial. Repeated factors usually indicate a non diagonal Jordan structure.
The residual measures how close the computed matrix m(A) is to the zero matrix. Small residuals confirm the displayed polynomial works numerically.
Yes. You can enter decimal values. The tolerance setting helps the calculator handle numerical rounding while testing matrix dependencies.
It shows the first power of A that can be written using lower powers and the identity matrix. That exact relation generates the minimal polynomial.
CSV downloads the saved calculation history from the current session. PDF uses the browser print dialog so you can save the page as a PDF file.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.