Matematică

Eigenvalue Calculator

🔢Eigenvalue Calculator (2×2 Matrix)

a (top-left)

b (top-right)

c (bottom-left)

d (bottom-right)

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We're working on a comprehensive educational guide for the Eigenvalue Calculator in your language. The content below is shown in English.

Ce este Eigenvalue Calculator?

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Eigenvalues and eigenvectors are fundamental concepts in linear algebra. For a square matrix A, an eigenvector v is a non-zero vector that only scales (not rotates) when A is applied: Av = λv. The scalar λ is the corresponding eigenvalue. Eigenvalues reveal the fundamental behaviour of linear transformations and appear throughout physics, engineering, statistics (PCA), and machine learning.

Calkulon makes complex calculations simple — built for students and everyday problem-solvers.

Cum să Eigenvalue Calculator

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1For 2×2 matrix [[a,b],[c,d]], eigenvalues satisfy the characteristic equation: det(A − λI) = 0
2det(A − λI) = λ² − (a+d)λ + (ad−bc) = 0
3Trace = a+d, Determinant = ad−bc
4λ = [Trace ± √(Trace² − 4·Det)] / 2
5Discriminant > 0: two real eigenvalues; < 0: complex conjugate pair

Exemple rezolvate

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Exemplu 1

Date:Matrix [[4,1],[2,3]]

Rezultat:λ₁=5, λ₂=2

Trace=7, Det=10, Disc=9

Exemplu 2

Date:Matrix [[0,−1],[1,0]] (rotation 90°)

Rezultat:λ = ±i (complex)

Pure rotation has no real eigenvalues

Aplicații practice

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Academic homework and exam preparation

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Engineering and architecture calculations

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Everyday measurement tasks around the home

Eigenvalue Interpretation

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Eigenvalue Type	Geometrical Meaning
λ > 1	Stretches in eigenvector direction
0 < λ < 1	Compresses in eigenvector direction
λ = 1	No change along eigenvector
λ = 0	Matrix is singular (det = 0); projects onto a subspace
λ < 0	Reflects and possibly scales
Complex λ	Rotation component; no real invariant direction

Întrebări frecvente

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What is Eigenvalue Calc?

Eigenvalues and eigenvectors are fundamental concepts in linear algebra. For a square matrix A, an eigenvector v is a non-zero vector that only scales (not rotates) when A is applied: Av = λv

How accurate is the Eigenvalue Calc calculator?

The calculator uses the standard published formula for eigenvalue calc. Results are accurate to the precision of the inputs you provide. For financial, medical, or legal decisions, always verify with a qualified professional.

What units does the Eigenvalue Calc calculator use?

This calculator works with inches. You can enter values in the units shown — the calculator handles all conversions internally.

What formula does the Eigenvalue Calc calculator use?

The calculator applies the standard formula for this type of calculation. See the 'How It Works' steps above for the detailed formula breakdown.

Referințe

Formula-verified for precision

Reviewed October 2026

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Eigenvalue Calculator