Worksheet: Orthonormal Basis

In this worksheet, we will practice finding the eigenvalues and the orthonormal basis of eigenvectors for a matrix.

Q1:

Find an orthonormal basis of eigenvectors for the matrix

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Q2:

Find an orthonormal basis of eigenvectors for the matrix

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Q3:

Find an orthonormal basis of eigenvectors for the matrix

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Q4:

Find an orthonormal basis of eigenvectors for the matrix

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Q5:

Given that find the eigenvalues and an orthonormal basis of eigenvectors for .

  • Aeigenvalues: 2, 6, and 12, eigenvectors:
  • Beigenvalues: 6, 12, and 18, eigenvectors:
  • Ceigenvalues: 6, 12, and 18, eigenvectors:
  • Deigenvalues: 6, 12, and 18, eigenvectors:
  • Eeigenvalues: 2, 6, and 12, eigenvectors:

Q6:

Find an orthonormal basis of eigenvectors for the matrix

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Q7:

The two level surfaces and intersect in a subspace of . Find a basis for this subspace, and then find an orthonormal basis for this subspace.

  • Abasis: , , orthonormal basis: ,
  • Bbasis: , , orthonormal basis: ,
  • Cbasis: , , orthonormal basis: ,
  • Dbasis: , , orthonormal basis: ,
  • Ebasis: , , orthonormal basis: ,

Q8:

Apply the Gram–Schmidt process to the vectors , , and to find an orthonormal basis for their span.

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