Formula used
For matrix A, the eigenvalues are the roots of the characteristic equation.
For a 3x3 matrix, the expanded monic form is:
Here, tr(A) is the diagonal sum. S is the sum of the three principal 2x2 minors. det(A) is the determinant.
How to use this calculator
- Enter all nine values of your 3x3 matrix.
- Choose the decimal precision for output.
- Press the calculate button.
- Read eigenvalues, trace, determinant, and polynomial details.
- Use the CSV or PDF button to save your result.
Example data table
| Matrix | Expected eigenvalues | Use case | Action |
|---|---|---|---|
| [[2,0,0],[0,3,0],[0,0,5]] | 2, 3, 5 | Diagonal checking | |
| [[4,1,1],[1,4,1],[1,1,4]] | 6, 3, 3 | Symmetric matrix | |
| [[0,-1,0],[1,0,0],[0,0,2]] | 2, i, -i | Complex pair test |
Understanding 3x3 Eigenvalues
Eigenvalues describe how a square matrix stretches space. A 3x3 matrix can scale three independent directions. Each value belongs to a hidden direction called an eigenvector. When a matrix acts on that direction, the direction stays the same. Only the length and possible orientation change.
Why These Values Matter
Eigenvalues appear in linear algebra, physics, engineering, graphics, finance, and data science. They help measure stability in systems. They reveal vibration frequencies in structures. They support principal component analysis. They also help solve differential equation systems. For a real 3x3 matrix, the answers may be all real. They may also include one real value and a complex conjugate pair. This calculator reports both cases.
How The Calculator Works
The tool first reads the nine matrix entries. It then forms the characteristic polynomial. This polynomial comes from det(λI − A). Its coefficients are built from trace, principal minors, and determinant. After that, a cubic solver finds the three roots. These roots are the eigenvalues. The page also shows the trace check. The sum of eigenvalues should match the matrix trace. The product should match the determinant, after normal rounding limits.
Using The Results
Start by checking the characteristic equation. It gives a compact summary of the matrix. Next, review each eigenvalue. Real parts show growth, shrinkage, or balance in many models. Imaginary parts show rotation or oscillation. If complex parts are near zero, they may be rounding noise. Use the precision box to choose clearer output.
Export And Review
The CSV button saves a spreadsheet friendly file. The PDF button creates a printable report. These exports are useful for homework, audit notes, and technical records. The example table gives ready matrices for testing. Try a diagonal matrix first. Its eigenvalues are the diagonal entries. Then test mixed matrices to see how off diagonal terms change the roots.
Accuracy Notes
Good interpretation depends on context. Positive real values can indicate expansion. Negative values can indicate reversal or decay. Repeated values may mean repeated directions, or they may hide fewer independent eigenvectors. Always compare output with the original problem. Small input changes can move roots quickly when matrices are nearly singular.
FAQs
What is an eigenvalue?
An eigenvalue is a number that shows how a matrix scales a special direction. That direction is called an eigenvector. The direction stays aligned, while its size changes.
Can a 3x3 matrix have complex eigenvalues?
Yes. A real 3x3 matrix can have one real eigenvalue and two complex conjugate eigenvalues. Complex parts often indicate rotation or oscillation in the modeled system.
Why are there three eigenvalues?
The characteristic polynomial of a 3x3 matrix has degree three. Counting repeated and complex roots, a cubic equation has three roots.
What does the trace check mean?
The sum of all eigenvalues equals the trace of the matrix. Small differences can appear because decimal calculations are rounded.
What does the determinant check mean?
The product of all eigenvalues equals the determinant. This is useful for checking whether the result is reasonable and consistent.
Can I enter decimal values?
Yes. You can enter integers, decimals, and negative values. The calculator validates every entry before it calculates the characteristic equation.
What is spectral radius?
Spectral radius is the largest absolute eigenvalue. It helps describe growth, convergence, stability, and long term behavior in many matrix models.
Why use CSV and PDF exports?
CSV is useful for spreadsheets and later analysis. PDF is useful for reports, assignments, records, and sharing formatted results.