Theorem 6.4 If the matrix A=(aij)n×n is symmetric,then:

• All the n eigenvalues λ1, . . . , λn are real.
  
• Eigenvectors that corresponds to different     eigenvalues are orthogonal.
  
• There exists an orthogonal matrix P (i.e., PT = P−1)     such that
  

The columns v1, v2,…, vn of the matrix P=[v1v2⋯vn] are eigenvectors of unit length correspondingto the eigenvalues λ1,λ2,…,λn.

Proof

• Exercise: show this for the 2×2 matrix case.
  
• Omitted.
  
• Omitted.
  

Exercise Let a 2×2 symmetricmatrix A be given below.

Compute the matrix P describedin the above Theorem.

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