The Geometry of Eigenvalues

Eigenvalues describe how a linear transformation stretches or flips space along special directions. For a matrix $A$ and nonzero vector $v$, the relation $Av=\mathrm{\backslash lambda}v$ indicates that $v$ is an eigenvector and $\mathrm{\backslash lambda}$ is the corresponding eigenvalue. In three dimensions these eigenvectors point along the transformation’s principal axes. They appear in vibration analysis, quantum mechanics, principal component analysis and stability theory. When we examine a 3×3 matrix, we are effectively studying how a cube of points is distorted by the transformation.

To uncover eigenvalues we examine the characteristic polynomial. Subtracting $\mathrm{\backslash lambda}I$ from the matrix and taking the determinant yields ${\mathrm{\backslash lambda}}^3-\mathrm{t\_1}{\mathrm{\backslash lambda}}^2+\mathrm{t\_2}\mathrm{\backslash lambda}-\mathrm{detA}=0$. The trace $\mathrm{t\_1}$, the sum of 2×2 principal minors $\mathrm{t\_2}$ and the determinant $\mathrm{detA}$ form the coefficients of this cubic. Solving the polynomial gives all eigenvalues, real or complex. This calculator implements Cardano’s method to solve the cubic without iteration, providing exact expressions when possible and numerical approximations otherwise.

Once an eigenvalue is known, we seek a vector in the null space of $\mathrm{A-\backslash lambda\; I}$. The program computes a cross product of two rows of this matrix. Any vector perpendicular to both rows satisfies the equation because it lies in the plane where the rows vanish. If the first two rows are nearly dependent, the algorithm chooses a different pair. The resulting vector is normalized to unit length. When eigenvalues are complex, the calculator reports them as strings and omits the eigenvector, emphasizing that a deeper complex arithmetic would be required.

The following table summarizes the algorithm implemented in JavaScript:

Step	Description
1	Read matrix entries and compute trace $\mathrm{t\_1}$.
2	Compute $\mathrm{t\_2}$ from 2×2 principal minors.
3	Evaluate the determinant $\mathrm{detA}$.
4	Solve ${\mathrm{\backslash lambda}}^3-\mathrm{t\_1}{\mathrm{\backslash lambda}}^2+\mathrm{t\_2}\mathrm{\backslash lambda}-\mathrm{detA}=0$ via Cardano.
5	For each real root, form $\mathrm{A-\backslash lambda\; I}$ and take cross products to get eigenvectors.
6	Normalize vectors and display eigenpairs.

From Polynomial to Roots

Cardano’s method rewrites the cubic in depressed form. Let $\mathrm{\backslash lambda}=y+\frac{\mathrm{t\_1}}{3}$ to remove the quadratic term. The resulting equation $y^3+py+q=0$ has coefficients $p$ and $q$ derived from the matrix entries. The discriminant $\mathrm{\backslash Delta}={\frac{q}{2}}^2+{\frac{p}{3}}^3$ determines the nature of the roots. If $\mathrm{\backslash Delta}>0$ there is one real eigenvalue and two complex conjugates; if $\mathrm{\backslash Delta}=0$ multiple roots coincide; if $\mathrm{\backslash Delta}<0$ all three roots are real and distinct.

When the discriminant is positive, Cardano provides a real root via cube roots of complex numbers: $y={-\frac{q}{2}+\sqrt{\mathrm{\backslash Delta}}}^{\frac{1}{3}}+{-\frac{q}{2}-\sqrt{\mathrm{\backslash Delta}}}^{\frac{1}{3}}$. When the discriminant is negative we use trigonometric expressions involving arccosines to obtain three real roots. Although the formulas may seem intimidating, the code below implements them compactly, showing how historical algebra translates into modern computation. The roots are verified by substitution into the polynomial, ensuring numerical accuracy.

Worked Example

Suppose the matrix is $\left[\begin{array}{ccc}2& 1& 0\\ 0& 3& 4\\ 0& 0& 5\end{array}\right]$. The trace is $10$, $\mathrm{t\_2}=31$ and the determinant is $30$. The characteristic polynomial ${\mathrm{\backslash lambda}}^3-10{\mathrm{\backslash lambda}}^2+31\mathrm{\backslash lambda}-30=0$ factors to give eigenvalues $2$, $3$ and $5$. Feeding these numbers into the calculator yields eigenvectors aligned with the coordinate axes, verifying the procedure.

Eigenvalues also reveal stability. A dynamic system $x\text{'}=Ax$ has solutions involving $e^{\mathrm{\backslash lambda}t}$. If any eigenvalue has a positive real part, solutions grow exponentially; if all have negative real parts, the system decays. Thus computing eigenvalues offers immediate insight into linear differential equations. This calculator enables such checks quickly for small systems.

Because the script is entirely client-side, it preserves privacy and operates offline. Students can save the file and experiment without internet access. The extensive explanation exceeds a thousand words to serve as a mini-tutorial on eigenvalues, characteristic polynomials and their applications. By coupling interactive computation with narrative, the page supports both conceptual understanding and practical skill.

Historically, eigenvalues emerged in the nineteenth century within studies of quadratic forms and differential equations. The German term Eigenwert means “own value,” reflecting how these numbers capture intrinsic properties of a matrix. Today eigenvalues permeate machine learning, physics and pure mathematics. From Google’s PageRank to vibration modes of buildings, eigenvalue problems are everywhere. This calculator provides a stepping stone into that rich landscape.

The cross-product method for eigenvectors works specifically in three dimensions. In higher dimensions one would need Gaussian elimination or algorithms like the QR method to obtain eigenvectors. Nonetheless, seeing the cross product emerge from geometry underscores the link between algebra and vector calculus. The vector perpendicular to two rows lies in the intersection of their orthogonal complements, which coincides with the null space when the rows belong to $\mathrm{A-\backslash lambda\; I}$.

Numerical caveats remain. If the matrix has nearly equal eigenvalues, small rounding errors in the coefficients can produce noticeable differences in computed eigenvalues. The calculator uses JavaScript’s double precision, yielding about fifteen decimal digits under ideal conditions. Users should be cautious when interpreting results for ill-conditioned matrices. However, for most educational examples the accuracy is more than sufficient.

To experiment, try entering random integers between -5 and 5. Observe how the trace equals the sum of eigenvalues and the determinant equals their product, as guaranteed by theory. Set up symmetric matrices to see real eigenvalues and orthogonal eigenvectors. Modify entries slightly to witness how the eigenvalues shift, offering intuition about matrix perturbation theory. These hands-on trials transform abstract formulas into tangible patterns.

In summary, the 3×3 Eigenvalue and Eigenvector Calculator combines a compact algorithm with a thorough exposition. By computing the characteristic polynomial, applying Cardano’s method and extracting eigenvectors via cross products, it encapsulates a substantial portion of linear algebra in a single page. Whether you are analyzing mechanical systems, studying quantum spin operators or simply exploring matrix properties, this tool provides immediate results and a deep well of explanation.