Enter a 3x3 Hermitian Matrix
Only the upper triangle is required. Lower entries are built automatically as complex conjugates.
Analyze Hermitian matrices with intuitive complex input fields. See ordered eigenvalues, invariants, and chart insights. Save tables, summaries, plots, and exports for reports quickly.
Only the upper triangle is required. Lower entries are built automatically as complex conjugates.
For a 3x3 Hermitian matrix A, the calculator forms the characteristic equation:
λ³ − (tr A)λ² + c₂λ − det(A) = 0
With upper-triangle entries a12, a13, and a23, the second coefficient is:
c₂ = a11a22 + a11a33 + a22a33 − |a12|² − |a13|² − |a23|²
The determinant is:
det(A) = a11a22a33 + 2Re(a12·a23·conj(a13)) − a11|a23|² − a22|a13|² − a33|a12|²
Because Hermitian matrices have real eigenvalues, the cubic is solved in real form and returned in ascending order.
| Example | a11 | a22 | a33 | a12 | a13 | a23 | Eigenvalues |
|---|---|---|---|---|---|---|---|
| Sample 3x3 Hermitian matrix | 4 | 5 | 6 | 1 + 2i | 0 - 1i | 2 + 1i | 1.1769, 5.8926, 7.9305 |
A Hermitian matrix equals its own conjugate transpose. Diagonal entries are real, and each lower-triangle entry is the complex conjugate of the matching upper-triangle entry.
Hermitian matrices have a special symmetry in complex inner-product spaces. That structure guarantees every eigenvalue is real, which makes them important in physics, optimization, and numerical analysis.
The lower triangle is determined automatically by Hermitian symmetry. Entering only the upper-triangle complex values avoids mismatched pairs and keeps the matrix valid by construction.
The trace is the sum of diagonal entries. For any square matrix, it also equals the sum of the eigenvalues, so it is a quick correctness check.
The spectral radius is the largest absolute eigenvalue. It helps judge growth, stability, and dominant modes in many matrix-based systems.
If every eigenvalue is positive, the matrix is positive definite. If the smallest eigenvalue is zero or positive, it is positive semidefinite.
The spread is the largest eigenvalue minus the smallest eigenvalue. It shows how wide the spectrum is and how separated the extreme modes are.
This page is designed for 3x3 Hermitian matrices. Larger matrices usually need numerical methods such as QR, divide-and-conquer, or iterative eigensolvers.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.