Understanding the Characteristic Polynomial of a Matrix
The characteristic polynomial of a square matrix is a scalar polynomial whose roots are the eigenvalues of the matrix. For an $n \times n$ matrix $A$, it is defined algebraically by evaluating the determinant $p(\lambda) = \det(\lambda I - A)$, where $I$ represents the identity matrix of identical dimensions and $\lambda$ acts as a scalar variable.
Formula and Mathematical Foundations
Expanding the determinant yields a monic polynomial of degree $n$:
$$p(\lambda) = \lambda^n - \text{tr}(A)\lambda^{n-1} + \dots + (-1)^n \det(A)$$
Coefficients can be systematically derived via advanced linear algebra methods such as the Faddeev-LeVerrier algorithm, avoiding direct symbolic determinant expansion for larger dimensions.
How to Use This Calculator
- Select your preferred matrix dimension from the configuration panel.
- Input numeric values separated cleanly by commas or spaces into the text area.
- Choose your calculation algorithm, precision level, and formatting preferences.
- Click the Calculate button to render exact polynomial outputs instantly above.
Frequently Asked Questions (FAQs)
What is an eigenvalue? Eigenvalues are scalar multipliers corresponding to non-zero eigenvectors satisfying $Av = \lambda v$.
Can this handle non-square matrices? No, characteristic polynomials exist exclusively for square matrices.