Eigenvalue Problem for 2×2 Hermitian Matrices

 

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 a c+d c-d b  α β  = λ  α β   2.00 1.00-1.00 1.00+1.00 -1.00  0.86-0.36 0.34+0.14  = 2.56  0.86-0.36 0.34+0.14   2.00 1.00-1.00 1.00+1.00 -1.00  -0.34-0.14 0.86-0.36  = -1.56  -0.34-0.14 0.86-0.36 

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An Hermitian matrix () has real eigenvalues and mutually orthogonal eigenvectors, which can be chosen to be normalized. This Demonstration considers the case of Hermitian matrices, which has important applications in the study of two-level quantum systems. For a selected Hermitian matrix, the graphic shows the equations satisfied by the two eigenvalues, with their corresponding orthonormalized eigenvectors.

Contributed by: S. M. Blinder (March 2011)
Open content licensed under CC BY-NC-SA

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Snapshots

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Details

Snapshots 1-3: eigenvalue equations for the three Pauli spin matrices ,,

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Related Links

• Eigenvalue (Wolfram MathWorld)
• Hermitian Matrix (Wolfram MathWorld)

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Permanent Citation

S. M. Blinder "Eigenvalue Problem for 2×2 Hermitian Matrices"
http://demonstrations.wolfram.com/EigenvalueProblemFor22HermitianMatrices/
Wolfram Demonstrations Project
Published: March 7 2011

 

 https://demonstrations.wolfram.com/EigenvalueProblemFor22HermitianMatrices/