Math Problem Statement
Solution
Let's solve both parts systematically:
Part 1: Definition of Hermitian Matrix and Proof of Real Eigenvalues
Definition of Hermitian Matrix:
A Hermitian matrix in satisfies: where denotes the conjugate transpose of . This means: for all .
Proof: Eigenvalues of Hermitian Matrices are Real
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Let be a Hermitian matrix, and let be an eigenvalue of with eigenvector . Thus:
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Taking the inner product of with , we get: Since is a scalar:
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The Hermitian property implies:
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Substituting , this becomes:
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Since (the norm of is nonzero), divide through by :
Hence, is real.
Part 2: Proof of Nilpotency of
Given:
- satisfies for all .
Proof:
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Consider the Jordan canonical form of . Since is a linear transformation over a field of characteristic zero, it is similar to a block diagonal matrix consisting of Jordan blocks.
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The trace of is the sum of over all eigenvalues of . Since for all , this implies:
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This system implies that all eigenvalues of are zero. To see this, note that if any eigenvalue , then .
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Since all eigenvalues are zero, is nilpotent. Specifically, all Jordan blocks of correspond to the eigenvalue , meaning for some .
Do you have further questions or need detailed clarification on any part?
Related Questions:
- Can you explain why Hermitian matrices appear in quantum mechanics?
- How does the characteristic polynomial confirm the nilpotency of ?
- Can a Hermitian matrix have complex eigenvalues in any case?
- What are examples of fields with ?
- How is the trace of a matrix invariant under similarity transformations?
Tip:
In linear algebra, analyzing eigenvalues often simplifies the problem significantly—always check their properties first!
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Matrix Theory
Hermitian Matrices
Nilpotent Matrices
Formulas
Hermitian condition: A = A* (A* is the conjugate transpose)
Trace property: tr(T^i) = sum of eigenvalues raised to the power i
Theorems
Eigenvalues of Hermitian matrices are real
Nilpotent matrices have eigenvalue 0 and satisfy T^k = 0 for some k
Suitable Grade Level
Undergraduate Mathematics