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THE FEUTEX TOOLKIT

Symmetric 2×2 Matrix Spectrum

Calculate the two real eigenvalues and principal-axis angle of a symmetric two-by-two matrix.

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01 Your input
02 Your result
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FormulaEigenvalue High = (a+d+hypot(a-d,2*b))/2; Eigenvalue Low = (a+d-hypot(a-d,2*b))/2; Rotation Angle = atan2(2*b,a-d)/2
Sources and model references

Worked example

Inputs

A
3 number
B
1 number
D
3 number

Expected results

Eigenvalue High
4 number
Eigenvalue Low
2 number
Rotation Angle
0.7853981634 rad

Load the example in the workspace to reproduce these results. Display rounding does not change the underlying calculation.

How to use this tool

  1. Enter the labeled inputs using consistent units.
  2. Check the stated assumptions and calculate.
  3. Review each output and compare another input scenario.

Try this example

{
  "a": 3,
  "b": 1,
  "d": 3
}

Accuracy and scope

Matrix rows are [a,b] and [b,d]. Repeated eigenvalues have nonunique directions. Finite floating-point arithmetic; results can lose precision near singular or extreme inputs.

Is my data uploaded?

This tool runs locally in your browser. Input and output are not sent to a processing server. Copy or download your result before leaving the page.