Calculate the characteristic polynomial of a square matrix to find eigenvalues. Essential for linear algebra, stability analysis, and matrix diagonalization.
Matrix entries must be valid numbers (use decimals or integers)
The characteristic polynomial of a square matrix A is a fundamental concept in linear algebra, defined as p(λ) = det(A - λI), where I is the identity matrix of the same size as A, λ (lambda) is a scalar variable, and det denotes the determinant. This polynomial encodes critical information about the matrix's behavior: its roots are precisely the eigenvalues of the matrix. For a 2×2 matrix, the characteristic polynomial is always quadratic (degree 2), taking the form λ² - (trace A)·λ + det(A), where the trace is the sum of diagonal elements. For a 3×3 matrix, it's a cubic polynomial (degree 3) with the form λ³ - (trace A)·λ² + (sum of principal minors)·λ - det(A). The coefficients of the polynomial reveal structural properties of the matrix: the coefficient of the highest degree term is always 1, the next coefficient is the negative of the trace, and the constant term is ±det(A).
Understanding the characteristic polynomial is essential for matrix diagonalization, stability analysis in differential equations, vibration analysis in mechanical systems, and quantum mechanics (where eigenvalues represent observable physical quantities). The Cayley-Hamilton theorem states that every square matrix satisfies its own characteristic polynomial—a profound result meaning that if you substitute the matrix A for λ in p(λ), you get the zero matrix. In practical applications, finding eigenvalues by solving the characteristic polynomial is computationally expensive for large matrices (solving high-degree polynomials is numerically unstable), so iterative methods like QR decomposition are preferred. However, for 2×2 and 3×3 matrices—common in physics, computer graphics, and engineering problems—the characteristic polynomial provides an exact analytical approach. The multiplicity of an eigenvalue as a root of the characteristic polynomial (algebraic multiplicity) also provides information about the matrix's diagonalizability and the dimension of the corresponding eigenspace.
Choose between a 2×2 or 3×3 matrix using the toggle buttons. The calculator will reset to a default example matrix for that size.
Input the values for each position in your matrix. You can use integers, decimals, or negative numbers. The calculator updates results in real-time as you type.
The characteristic polynomial appears in the results panel in the form det(A - λI). For 2×2 matrices, you'll also see the eigenvalues calculated using the quadratic formula.
The results display includes the trace (sum of diagonal elements) and determinant. These values are the coefficients in the characteristic polynomial and serve as useful verification checks.
Click "Load Example" to see a worked demonstration matrix, or "Reset" to return to the default starting values for your current matrix size.
Analyzing a Vibrating System
The system has two natural frequencies corresponding to eigenvalues 4 and 2. These tell the engineer at which frequencies the system will resonate and oscillate most strongly.
Eigenvalues represent natural frequencies, growth rates, or stability indicators. They're crucial in applications like vibration analysis, population dynamics, quantum mechanics, and machine learning (PCA, spectral clustering).
Set the characteristic polynomial equal to zero: p(λ) = 0 and solve for λ. For 2×2 matrices, use the quadratic formula. For 3×3, use cubic solving techniques or numerical methods like Newton-Raphson.
Yes. Complex eigenvalues occur in non-symmetric matrices. They appear in conjugate pairs for real matrices (e.g., 3+2i and 3-2i) and indicate rotational or oscillatory behavior in the system.
The constant term of the characteristic polynomial equals (-1)^n times the determinant of the matrix, where n is the matrix size. For 2×2: constant = +det(A). For 3×3: constant = -det(A).
No. A matrix and its transpose have identical characteristic polynomials, hence the same eigenvalues. This is because det(A - λI) = det(A^T - λI) for any matrix A.
No. The characteristic polynomial is only defined for square matrices because you need to subtract λI (which must have the same dimensions). For non-square matrices, use singular value decomposition (SVD).
The degree equals the size of the matrix (n×n gives degree n). A 2×2 matrix produces a quadratic (degree 2), a 3×3 produces a cubic (degree 3), and an n×n produces degree n.
The trace (sum of diagonal elements) equals the sum of all eigenvalues (counting multiplicities). This is a useful verification: if eigenvalues are λ₁, λ₂, ..., λₙ, then trace(A) = λ₁ + λ₂ + ... + λₙ.
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