Eigenvalue and Eigenvector Calculator (2x2)
MathCalculate the eigenvalues and eigenvectors of a 2x2 matrix.
Eigenvalue 1
The larger eigenvalue, found from the characteristic equation lambda^2 - (a+d)*lambda + (ad-bc) = 0.
Eigenvalue 2
2.0000
The smaller eigenvalue.
Eigenvector 1 Direction (y/x)
1.0000
The y/x ratio of the eigenvector for eigenvalue 1 (assuming the off-diagonal entry b is nonzero), e.g. a ratio of 1 means the eigenvector points in the direction (1, 1).
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Odeh Ahwal0people find this calculator helpful
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Frequently Asked Questions
What are eigenvalues and eigenvectors?
An eigenvector of a matrix is a special direction that the matrix only stretches or shrinks (never rotates), and the eigenvalue is the factor by which it gets stretched or shrunk along that direction.
How are the eigenvalues of a 2x2 matrix found?
They are the two solutions to the characteristic equation lambda^2 - (trace)*lambda + (determinant) = 0, where trace = a+d and determinant = ad-bc, solved with the quadratic formula.
Why is only one eigenvector ratio shown?
This calculator is scoped to the direction ratio (y/x) of the eigenvector for the first eigenvalue, assuming the off-diagonal entry b is nonzero, rather than a fully general eigenvector solver covering every edge case (repeated eigenvalues, b=0, complex eigenvalues).
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