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

Symmetric matrices and their right angle axes

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

  1. The spectral theorem says every real symmetric matrix can be diagonalised by what?

    • Any invertible matrix
    • A diagonal matrix only
    • An orthogonal matrix

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

    Circle one:   True   False

  3. The symmetric matrix with rows (5, 2) and (2, 8) has eigenvalues 9 and 4. Type the largest one.

    Answer: ______________

  4. The spectral theorem says every real symmetric matrix can be diagonalised by...

    • An orthogonal matrix
    • Any invertible matrix
    • A rotation matrix only
    • A diagonal matrix only

  5. The quadratic form Q(x, y) = 3x^2 + 2y^2 is...

    • Positive definite
    • Negative definite
    • Indefinite
    • Negative semidefinite

  6. You stacked unit eigenvectors as Q but skipped normalising one of them. What breaks?

    • Nothing, Q stays orthogonal
    • Q loses orthogonality, so transpose is no longer the inverse
    • The eigenvalues change value

  7. Eigenvectors for distinct eigenvalues of a symmetric matrix are always orthogonal. True or false?

    Circle one:   True   False

  8. What is the largest eigenvalue of the symmetric matrix [[5, 2], [2, 8]]?

    Answer: ______________

  9. A classmate claims any square matrix has an orthonormal eigenbasis. Which example refutes them?

    • A non symmetric matrix with too few eigenvectors
    • A symmetric matrix with distinct eigenvalues
    • The identity matrix

  10. For a symmetric 3 by 3, one eigenvector is (2, 1, 0) and another is (-1, 2, 0). What is their dot product?

    Answer: ______________

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Answer key

For grown-ups. Fold this page away before handing over the rest.

Symmetric matrices and their right angle axes W1-mt_sdQv4m7Nbk-s1

  1. An orthogonal matrix · The orthonormal eigenbasis stacks into an orthogonal Q with A = Q D Q transpose.
  2. True · Reality of the spectrum is the first half of the spectral promise.
  3. 9 · Trace 13 and determinant 36 give the pair 9 and 4.
  4. An orthogonal matrix · Symmetric matrices admit A = Q D Q^T with Q orthogonal: the spectral decomposition.
  5. Positive definite · Both eigenvalues 3 and 2 are positive, so Q > 0 away from zero.
  6. Q loses orthogonality, so transpose is no longer the inverse · Orthogonal Q needs unit columns; without them the transpose trick fails.
  7. True · True: symmetry forces it, and the short proof uses x^T A y two ways.
  8. 9 · Characteristic: λ^2 - 13λ + 36 = 0, roots 9 and 4.
  9. A non symmetric matrix with too few eigenvectors · The guarantee belongs to symmetric matrices alone; others may fall short.
  10. 0 · -2 + 2 + 0 = 0, as the spectral theorem predicts.
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