# Eigenvalue and Eigenvector Calculator > Compute eigenvalues, eigenvectors, and characteristic polynomial for square matrices **Category:** Math **Keywords:** eigenvalue, eigenvector, matrix, linear algebra, characteristic polynomial, determinant, diagonalization **URL:** https://complete.tools/eigenvalue-eigenvector-calculator ## How eigenvalues work An eigenvalue of a square matrix A is a scalar λ such that there exists a non-zero vector v satisfying: ``` A · v = λ · v ``` In plain terms: when you multiply the matrix by the eigenvector v, the result is simply the vector scaled by λ. The eigenvector does not change direction — it only gets stretched or flipped. This property is central to dozens of applications: - **Principal Component Analysis (PCA)**: eigenvalues reveal how much variance each component captures - **Vibration analysis**: eigenvalues represent natural frequencies of mechanical systems - **Quantum mechanics**: observable measurements correspond to eigenvalues of operators - **Google PageRank**: the ranking vector is an eigenvector of the web link matrix - **Image compression**: eigenvectors of covariance matrices form optimal bases for compression ## The characteristic polynomial To find eigenvalues, we solve the characteristic equation: ``` det(A - λI) = 0 ``` where I is the identity matrix. This determinant expands into a polynomial in λ called the characteristic polynomial. Its degree equals the matrix size — degree 2 for a 2x2 matrix, degree 3 for a 3x3 matrix. **For a 2x2 matrix [[a, b], [c, d]]:** ``` λ² - (a + d)λ + (ad - bc) = 0 ``` The roots of this quadratic are the two eigenvalues. **For a 3x3 matrix:** ``` -λ³ + tr(A)λ² - (sum of 2x2 minors)λ + det(A) = 0 ``` The three roots of this cubic are the eigenvalues. If the discriminant of the characteristic polynomial is negative, eigenvalues are complex numbers of the form a + bi. For real matrices, complex eigenvalues always appear in conjugate pairs. ## Finding eigenvectors Once an eigenvalue λ is known, the corresponding eigenvector v satisfies: ``` (A - λI) · v = 0 ``` This is a homogeneous linear system. The eigenvector spans the null space of (A - λI). We find it using row reduction (Gaussian elimination) on the augmented matrix. For each eigenvalue there is at least one eigenvector. When the same eigenvalue appears multiple times (a repeated eigenvalue), there may be more than one independent eigenvector — the eigenspace dimension equals the algebraic multiplicity in the best case. This tool normalizes all eigenvectors to unit length, so the magnitude is 1. The direction is what matters for most applications. ## How to use 1. Choose matrix size: click "2 × 2" or "3 × 3" in the Matrix Size toggle. 2. Enter each matrix element in the grid. Tab between cells. 3. Click "Compute Eigenvalues" to calculate. 4. Review the characteristic polynomial, trace, and determinant. 5. Scroll to the Eigenvalues and Eigenvectors section to see each λ paired with its eigenvector. 6. Click "Reset to Identity" to start fresh with the identity matrix. ## Interpreting the results **Real eigenvalues** appear as plain numbers (e.g., 3.0, -1.5). These represent pure scaling along the eigenvector direction. **Complex eigenvalues** appear in the form a + bi (e.g., 2 + 3i). They represent rotation combined with scaling, and always come in conjugate pairs for real matrices. A complex eigenvalue a + bi corresponds to a spiral behavior — the matrix rotates vectors while scaling them. **The trace** equals the sum of all eigenvalues. **The determinant** equals the product of all eigenvalues. These are quick sanity checks — if your trace and determinant match what you computed by hand, the eigenvalues are correct. **Degenerate matrices** (those with a zero determinant) have at least one eigenvalue of zero. If the matrix is close to degenerate numerically, small floating-point errors may appear — the tool rounds values smaller than 1e-10 to zero. ## FAQs **Q:** What are eigenvalues? **A:** Eigenvalues are special scalars associated with a square matrix. When a matrix multiplies its eigenvector, the result is simply that vector scaled by the eigenvalue. They reveal the matrix's intrinsic stretching or compression behavior along each eigenvector direction. **Q:** What are eigenvectors? **A:** Eigenvectors are non-zero vectors that only get scaled (not rotated) when a matrix is applied to them. Each eigenvector points in a direction that the matrix preserves, making them natural coordinate axes for understanding the transformation. **Q:** Why might eigenvalues be complex? **A:** A real matrix can still have complex eigenvalues. This happens when the characteristic polynomial has a negative discriminant. Geometrically, complex eigenvalues correspond to rotational behavior — the matrix spirals vectors rather than purely stretching them. **Q:** What does it mean when two eigenvalues are the same? **A:** Repeated (degenerate) eigenvalues mean the characteristic polynomial has a repeated root. The matrix may or may not be diagonalizable depending on how many independent eigenvectors exist for that eigenvalue. The identity matrix, for example, has a repeated eigenvalue of 1 but is still diagonalizable. **Q:** How accurate are the results? **A:** The calculator uses IEEE 754 double-precision floating point (about 15-16 significant digits). Values smaller than 1e-10 in magnitude are rounded to zero to suppress floating-point noise. For matrices with entries in the range of typical textbook problems, results are accurate to at least 8 significant figures. **Q:** Can this handle matrices larger than 3x3? **A:** Currently the tool supports 2x2 and 3x3 matrices. Larger matrices require iterative numerical methods (like the QR algorithm) rather than closed-form formulas, which goes beyond this tool's scope. **Q:** What is the characteristic polynomial used for? **A:** The characteristic polynomial encodes all eigenvalues as its roots. By Cayley-Hamilton theorem, every square matrix satisfies its own characteristic polynomial — substituting the matrix itself for λ gives the zero matrix. This is useful in control theory and signal processing. --- *Generated from [complete.tools/eigenvalue-eigenvector-calculator](https://complete.tools/eigenvalue-eigenvector-calculator)*