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3.5
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The Eigenvalue Calculator finds the eigenvalues of a 2×2 matrix by solving the characteristic polynomial. Eigenvalues are among the most important concepts in linear algebra, revealing the intrinsic scaling factors of a linear transformation. An eigenvalue $$\lambda$$ of matrix $$A$$ satisfies $$Av = \lambda v$$ for some non-zero vector $$v$$ (the eigenvector) — meaning the transformation simply scales the eigenvector by factor $$\lambda$$ without changing its direction.
For a 2×2 matrix, the characteristic equation is a quadratic polynomial: $$\lambda^2 - \text{tr}(A)\lambda + \det(A) = 0$$, where $$\text{tr}(A) = a_{11} + a_{22}$$ is the trace and $$\det(A) = a_{11}a_{22} - a_{12}a_{21}$$ is the determinant. By the quadratic formula, the eigenvalues are $$\lambda = \frac{\text{tr}(A) \pm \sqrt{\text{tr}(A)^2 - 4\det(A)}}{2}$$. The discriminant $$\Delta = \text{tr}^2 - 4\det$$ determines the nature of the eigenvalues.
Three cases arise depending on the discriminant. When $$\Delta > 0$$, there are two distinct real eigenvalues — the transformation stretches differently along two independent directions. When $$\Delta = 0$$, there is one repeated eigenvalue — the matrix may or may not be diagonalizable (it is if and only if it is already a scalar multiple of the identity). When $$\Delta < 0$$, the eigenvalues are complex conjugates $$\alpha \pm \beta i$$, indicating that the transformation involves rotation and scaling rather than pure stretching.
Eigenvalues encode crucial information about the behavior of the matrix. The sum of the eigenvalues equals the trace: $$\lambda_1 + \lambda_2 = \text{tr}(A)$$. The product of the eigenvalues equals the determinant: $$\lambda_1 \cdot \lambda_2 = \det(A)$$. These relationships provide quick consistency checks and connect eigenvalues to other matrix invariants.
In applications, eigenvalues determine the stability of dynamical systems, the principal stresses in mechanics, the vibrational modes of structures, and the convergence rate of iterative algorithms. In a discrete dynamical system $$x_{n+1} = Ax_n$$, the system is stable if all eigenvalues satisfy $$|\lambda| < 1$$, marginally stable if $$|\lambda| = 1$$, and unstable if $$|\lambda| > 1$$. In differential equations $$\dot{x} = Ax$$, stability requires all eigenvalues to have negative real parts.
This calculator computes both eigenvalues along with the trace, determinant, and discriminant. It classifies the eigenvalues as distinct real, repeated real, or complex conjugate, providing complete information about the spectral properties of the 2×2 matrix. For complex eigenvalues, the real and imaginary parts are displayed separately.
The eigenvalues of $$A = \begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix}$$ satisfy the characteristic equation:
$$\det(A - \lambda I) = 0$$
$$\begin{vmatrix} a_{11} - \lambda & a_{12} \\ a_{21} & a_{22} - \lambda \end{vmatrix} = 0$$
$$(a_{11} - \lambda)(a_{22} - \lambda) - a_{12}a_{21} = 0$$
$$\lambda^2 - (a_{11} + a_{22})\lambda + (a_{11}a_{22} - a_{12}a_{21}) = 0$$
$$\lambda^2 - \text{tr}(A)\lambda + \det(A) = 0$$
By the quadratic formula:
$$\lambda_{1,2} = \frac{\text{tr}(A) \pm \sqrt{\text{tr}(A)^2 - 4\det(A)}}{2}$$
The eigenvalue type reveals the nature of the linear transformation. Two distinct real eigenvalues: the matrix stretches along two independent eigenvector directions by different factors. Repeated eigenvalue: the matrix is either a scalar multiple of identity (complete eigenspace) or a shear-like transformation (deficient eigenspace). Complex eigenvalues $$\alpha \pm \beta i$$: the transformation involves rotation by angle $$\theta = \arctan(\beta/\alpha)$$ combined with scaling by $$r = \sqrt{\alpha^2 + \beta^2}$$. The sign of eigenvalues indicates growth (positive) or decay (negative) in dynamical systems.
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Trace = 4+3 = 7, det = 12−2 = 10. Characteristic equation: λ² − 7λ + 10 = 0, so (λ−5)(λ−2) = 0. Eigenvalues: λ₁ = 5, λ₂ = 2. Check: 5+2 = 7 = trace, 5×2 = 10 = det ✓
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This is a 90° rotation matrix. Trace = 0, det = 1. Discriminant = 0 − 4 = −4 < 0. Eigenvalues: λ = ±i (purely imaginary). The rotation has no real eigenvectors — no direction is preserved.
An eigenvalue $$\lambda$$ is the scaling factor along its corresponding eigenvector direction. If $$\lambda = 2$$, vectors along that direction are stretched to twice their length. If $$\lambda = -1$$, vectors are reflected (reversed). If $$\lambda = 0$$, vectors in that direction are collapsed to the origin. Complex eigenvalues indicate rotation rather than pure scaling.
Eigenvalues are complex when the discriminant $$\Delta = \text{tr}^2 - 4\det < 0$$. This happens when the determinant is large relative to the trace squared, i.e., when $$\det > \text{tr}^2/4$$. Geometrically, complex eigenvalues indicate the transformation involves rotation. Rotation matrices always have complex eigenvalues (except for 0° and 180° rotations).
For any $$n \times n$$ matrix, the trace equals the sum of all eigenvalues and the determinant equals their product. For a 2×2 matrix: $$\lambda_1 + \lambda_2 = \text{tr}(A)$$ and $$\lambda_1 \cdot \lambda_2 = \det(A)$$. These are the coefficients of the characteristic polynomial and provide a quick way to verify eigenvalue calculations.
The characteristic polynomial is $$p(\lambda) = \det(A - \lambda I)$$. For a 2×2 matrix, it is a degree-2 polynomial: $$\lambda^2 - \text{tr}(A)\lambda + \det(A)$$. Its roots are the eigenvalues. The Cayley-Hamilton theorem states that every matrix satisfies its own characteristic equation: $$A^2 - \text{tr}(A) \cdot A + \det(A) \cdot I = 0$$.
A repeated (double) eigenvalue $$\lambda$$ occurs when the discriminant is zero. The matrix may be diagonalizable (if it equals $$\lambda I$$, the eigenspace is 2-dimensional) or defective (if it is not a scalar matrix, the eigenspace is only 1-dimensional). Defective matrices require Jordan normal form rather than standard diagonalization.
For discrete systems $$x_{n+1} = Ax_n$$: stable if $$|\lambda| < 1$$ for all eigenvalues. For continuous systems $$\dot{x} = Ax$$: stable if $$\text{Re}(\lambda) < 0$$ for all eigenvalues. The eigenvalue with largest magnitude (spectral radius) or largest real part dominates the long-term behavior. This makes eigenvalue analysis the fundamental tool for stability theory.
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