Symmetric Matrices and the Spectral Theorem · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.
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Symmetric matrices and their perpendicular axes

Mathematics · Linear Algebra · ages 20-21
Name ______________________   Date ____________

  1. Let H = [[1, 1], [1, -1]]/sqrt(2) and R = [[1, 1], [1, -1]]. For A = [[2, 1], [1, 2]], which pair satisfies Q^T Q = I and A = Q D Q^T?

    • Q = H, D = diag(1, 3)
    • Q = R, D = diag(3, 1)
    • Q = H, D = diag(3, 1)

  2. Every eigenvalue of a real symmetric matrix is real.

    Circle one:   True   False

  3. What is the largest eigenvalue of the symmetric matrix [[5, 2], [2, 5]]?

    Answer: ______________

  4. A is real symmetric and 2-by-2. Q = [u1 u2] has orthonormal columns forming a basis of R^2, with Au1 = -2u1 and Au2 = 8u2. Which D makes A = Q D Q^T?

    • diag(8, -2)
    • diag(2, 8)
    • diag(-2, 8)
    • diag(-2, -8)

  5. How is f(x, y) = 3x^2 classified?

    • Positive semidefinite
    • Negative semidefinite
    • Strictly positive definite
    • Indefinite with both signs

  6. Q has perpendicular eigenvector columns of lengths 2 and 1, in that order. Which statement is false?

    • Each column still satisfies its eigenvector equation
    • Q^T Q = I, so Q^T is Q's inverse
    • Rescaling either column preserves its eigenvalue

  7. For A = 5I, the independent eigenvectors u = (1, 0) and v = (2, 1) must be orthogonal because A is symmetric.

    Circle one:   True   False

  8. For A = [[7, 2], [2, 7]], v = (1, -1) is an eigenvector. What is its eigenvalue?

    Answer: ______________

  9. Which matrix refutes the claim that every real square matrix has an orthonormal eigenbasis?

    • [[2, 1], [0, 2]]
    • [[2, 0], [0, 5]]
    • [[1, 0], [0, 1]]

  10. An eigenvector is v = (0, 5, 12). What positive number must you divide v by to make it a unit column of Q?

    Answer: ______________

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Answer key

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Symmetric matrices and their perpendicular axes W1-mt_sdQv4m7Nbk-s1

  1. Q = H, D = diag(3, 1) · H has perpendicular unit columns. A scales its columns by 3 and 1, respectively, so D = diag(3, 1) gives the complete orthogonal diagonalisation.
  2. True · The spectral theorem guarantees that every eigenvalue of a real symmetric matrix is real.
  3. 7 · The trace is 10 and determinant is 21. The characteristic polynomial factors with roots 7 and 3.
  4. diag(-2, 8) · Q is a complete square orthogonal eigenvector matrix. Matching its column eigenvalues gives D = diag(-2, 8), so AQ = QD and A = Q D Q^T.
  5. Positive semidefinite · The eigenvalues are 3 and 0. The form is nonnegative but vanishes on the nonzero y-axis.
  6. Q^T Q = I, so Q^T is Q's inverse · Column dot products give Q^T Q = diag(4, 1). Since this is not I, Q is not orthogonal.
  7. False · Both vectors have eigenvalue 5, but u dot v = 2. Symmetry forces orthogonality for distinct eigenvalues, not for every pair sharing an eigenvalue.
  8. 5 · Av = (5, -5) = 5v, so this direction has eigenvalue 5.
  9. [[2, 1], [0, 2]] · [[2, 1], [0, 2]] has only eigenvalue 2 and only one independent eigenvector direction.
  10. 13 · The length is sqrt(0^2 + 5^2 + 12^2) = sqrt(169) = 13.
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