The characteristic polynomial packages a square matrix’s eigenvalue equation into one determinant expression.
Use it to derive or verify eigenvalues algebraically.
The relationship
The defining relationship
pA(λ)=det(λI−A)
See the structure
What the calculation is doing
v→Av = λvsame direction, scaled length
Worked interpretation
Read the result in context
pA(λ)=det(λI−A)
Interpret with care
Important boundary
Sign conventions vary in textbooks; this page states its convention explicitly.
Use the Matrix Calculator to experiment with small exact matrix operations before applying a more specialized route.
Extended domain
Complex-ready characteristic polynomial
The polynomial coefficients can be complex even though its definition remains det(λI−A). Its two roots are the eigenvalues.
Try an example
EIGENVALUESr1 = 1ir2 = -1i
p(λ) = λ² − (0)λ + (1)
Quick guide
How to use this calculator
Set the matrix dimensions required by the task.
Fill every visible entry and any operation-specific control.
Read the result type, factors, classification, and disclosed limitations.
Calculation method
Apply the stated linear-algebra contract
p_A(λ) = det(λI − A)
Undefined, singular, dimensionally incompatible, dependent, or unsupported inputs receive an explicit message instead of a fabricated numeric result.
Worked example
Example interpretation
For [[1,2],[3,4]], p(λ)=λ²−5λ−2.
p_A(λ) = det(λI − A)
Supported inputs
Precision and limits
Matrix size
Input matrices are limited to 1–6 rows and columns; operation-specific square, rank, or compatibility rules still apply.
Entry precision
Enter signed finite decimals with at most 30 digits and 15 decimal places.
Exactness
Finite decimals become reduced rational numbers and intermediate structural calculations are not rounded.
Complex 2 × 2 extension
The established panel retains its exact or larger real-matrix workflow. The extended panel accepts complex entries for a bounded 2 × 2 analysis, reports both eigenvalues, and states branch or diagonalizability limitations.
