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Eigenvalue Calculator

🔢আইগেনমান ক্যালকুলেটর (২×২ ম্যাট্রিক্স)

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কী 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.

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কীভাবে Eigenvalue Calculator

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

সমাধান করা উদাহরণ

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উদাহরণ 1
প্রদত্ত:Matrix [[4,1],[2,3]]
ফলাফল:λ₁=5, λ₂=2

Trace=7, Det=10, Disc=9

উদাহরণ 2
প্রদত্ত:Matrix [[0,−1],[1,0]] (rotation 90°)
ফলাফল:λ = ±i (complex)

Pure rotation has no real eigenvalues

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Eigenvalue Interpretation

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

সচরাচর জিজ্ঞাসা

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Q

What is Eigenvalue Calc?

A

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

Q

How accurate is the Eigenvalue Calc calculator?

A

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.

Q

What units does the Eigenvalue Calc calculator use?

A

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

Q

What formula does the Eigenvalue Calc calculator use?

A

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

Mathematically verified
Reviewed October 2026
Our methodology

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