The Singular Value Decomposition · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

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

  1. What is the largest singular value of the diagonal matrix [[3, 0], [0, 4]]?

    Answer: ______________

  2. The diagonal matrix with rows (3, 0) and (0, 4) has singular values 3 and 4. Type the largest one.

    Answer: ______________

  3. What does the largest singular value tell you about the map?

    • Its biggest stretch factor over all directions
    • Its determinant exactly
    • The number of its rows

  4. Rows (3, 4) and (0, 0) give A transpose times A with eigenvalues 25 and 0. Type the largest singular value.

    Answer: ______________

  5. A matrix has singular values 6, 2, 0. What is its rank?

    • 1
    • 2
    • 3
    • 0

  6. Why must the eigenvalues of A transpose times A be nonnegative?

    • Every matrix has only positive eigenvalues
    • The symmetric product form forces real nonnegative eigenvalues
    • Eigenvalues are always square roots

  7. A matrix has singular values 5, 2, and 0. What is its rank?

    • 3
    • 0
    • 2

  8. A = [[3, 4], [0, 0]]. What is its largest singular value?

    Answer: ______________

  9. You compress a matrix by keeping only its top singular value and direction. What did you build?

    • The best rank one approximation of the matrix
    • An exact copy with fewer entries
    • A matrix with all new eigenvalues

  10. A matrix has singular values 3 and 4. What is the sum of squares of all its entries?

    Answer: ______________

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Answer key

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

Stretch directions, ranked by size W1-mt_UsUZbM9VRK-s1

  1. 4 · A^T A = diag(9, 16); square roots give singular values 3 and 4.
  2. 4 · A transpose times A is diagonal with 9 and 16, whose roots are 3 and 4.
  3. Its biggest stretch factor over all directions · Singular values rank stretch sizes, and the top one is the maximum.
  4. 5 · Square roots of 25 and 0 are 5 and 0, so the largest is 5.
  5. 2 · Rank counts nonzero singular values: 6 and 2 qualify, so rank 2.
  6. The symmetric product form forces real nonnegative eigenvalues · The product is symmetric with a sum-of-squares structure behind it.
  7. 2 · Rank counts nonzero singular values, and here two are nonzero.
  8. 5 · A^T A = [[9, 12], [12, 16]] has eigenvalues 25 and 0; sqrt(25) = 5.
  9. The best rank one approximation of the matrix · Truncation keeps the dominant stretch and drops what adds little.
  10. 25 · Sum of squared entries equals the sum of squared singular values: 9 + 16 = 25.
Worksheet · LightMySky