Proof of Positive Definite Matrices: Symmetric & 2x2 w/Tr & Det

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The discussion focuses on proving that symmetric matrices are positive definite under specific conditions. For an nxn symmetric matrix A with distinct eigenvalues all greater than zero, it is established that A is positive definite, demonstrated by the inequality z'Az > 0 for any non-zero vector z. Additionally, for a 2x2 symmetric matrix A with a positive trace and positive determinant, it is also proven that A is positive definite. The spectral decomposition of A is a key concept in these proofs.

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  • Understanding of symmetric matrices
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  • Familiarity with the concepts of trace and determinant
  • Comprehension of spectral decomposition
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  • Explore the relationship between eigenvalues and matrix definiteness
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(i) Let A=A' be an nxn symmetric matrix with distinct eigenvalues la1, la2, ..., lan. Suppose that all eigenvalues lai > 0. Prove that A is positive definite: That is, prove that z'Az > 0 whenever z ne 0. (Hint: Consider the spectral decomposition of A.)

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(ii) Let A=A' be a 2x2 symmetric matrix with tr(A)>0 and det(A)>0. Prove that A is positive definite. (Hint: Consider the spectral decomposition of A.)

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i looked at this problem forever, nothing doing for me :cry: :confused:
 
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anyone? please help!
 
What is the definition of spectral decomposition (and no, I'm not asking out of ignorance)?
 

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