Minimal Polynomial of a Matrix Calculator

Example Data Table

2 1 0
0 2 0
0 0 3

1 0 0
0 2 0
0 0 4

0 1 0
0 0 1
0 0 0

Formula Used

The calculator searches for the first monic polynomial p(x) with p(A) = 0. It forms I, A, A^2, and higher powers. For each degree k, it tests whether A^k belongs to the span of I through A^(k-1).

If A^k = c0I + c1A + ... + c(k-1)A^(k-1), then the minimal polynomial is p(x) = x^k - c(k-1)x^(k-1) - ... - c1x - c0. The first valid relation gives the lowest degree.

How to Use This Calculator

1. Enter a square matrix. Place each row on a new line.
2. Use commas, spaces, or semicolons between values.
3. Set tolerance for near-zero numerical comparisons.
4. Choose decimal precision and a polynomial variable.
5. Select advanced output if you need powers or residuals.
6. Press the calculate button. The result appears above the form.
7. Use CSV or PDF buttons to save the report.

Understanding Matrix Minimal Polynomials

What the Polynomial Means

A matrix minimal polynomial is the shortest monic polynomial that turns a square matrix into the zero matrix. It is a compact algebraic fingerprint. It records repeated eigenvalue behavior, nilpotent parts, and the power relation that first stops being independent.

Why It Matters

This value is useful in linear algebra, control theory, differential systems, recurrence models, and symbolic computation. It can show whether a matrix is diagonalizable. It also helps reduce high powers of a matrix into smaller combinations. That makes long calculations faster and easier to verify.

How This Tool Works

The calculator starts with the identity matrix. Then it multiplies by the given matrix to build powers. At every step, it solves a linear dependency test. If a new power can be written using earlier powers, a monic polynomial is formed. The first successful relation is reported as the minimal polynomial.

Numerical Care

Real calculator entries often contain decimals. Small rounding errors may appear. The tolerance setting controls when a value is treated as zero. A tighter tolerance is stricter. A wider tolerance can help with rounded data, but it may also hide small meaningful entries. Always compare the residual check with your chosen tolerance.

Reading the Output

The main result shows the polynomial, its degree, and the maximum absolute residual in p(A). A small residual means the polynomial fits the matrix well. The characteristic polynomial is also shown for reference. The minimal polynomial must divide it in exact algebra. For diagonal matrices, the answer usually contains one factor for each distinct eigenvalue.

Limits and Interpretation

Exact algebra can factor the answer neatly. Numeric calculation instead reports a decimal relation. This is still helpful for teaching, checking homework, and exploring models. When results look surprising, rerun with integer entries, a smaller tolerance, and residual output. Compare the characteristic polynomial too. If the minimal degree equals the matrix size, no shorter relation was found before that point for the selected tolerance setting and precision.

Best Practice

Use exact integers or simple decimals when possible. Review the example table before entering larger matrices. Turn on matrix powers when you want to audit the dependency. Turn on the residual matrix when you need detailed error inspection.

FAQs