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Mathematics · Linear Algebra · ages 20-21

Name ______________________ Date ____________

A = [[0, 0, 7]] is a 1-by-3 matrix. How many zero eigenvalues does A^T A have?
Answer: ______________
What is the largest singular value of A = [[-3, 0], [0, 4]]?
Answer: ______________
What does the largest singular value tell you about a real matrix map?
- The greatest output length for a unit input
- The signed determinant of any rectangular matrix
- The number of independent rows without further calculation
A has rows (3, 4) and (0, 0). A transpose times A has eigenvalues 25 and 0. What is the largest singular value?
Answer: ______________
Let A be real, with A^T A v_i = sigma_i^2 v_i and orthonormal v_i. For positive sigma_i, define u_i = Av_i/sigma_i. Which statement verifies orthonormality and handles zero singular values?
- Each u_i has squared length sigma_i^2, and distinct u_i have dot product 0; the division formula also works for zero values.
- The u_i have unit length and pairwise dot product 0; the formula cannot be used when sigma_i = 0.
- Every pair of constructed u_i has dot product 1, including distinct vectors; zero singular values must be removed from A.
- Distinct u_i are perpendicular only when their positive singular values differ; a zero value gives u_i = 0 as a unit column.
For real A, why are all eigenvalues of A^T A nonnegative?
- Every real matrix has only nonnegative eigenvalues
- Its quadratic form is the squared length ||Ax||^2
- Symmetry alone makes every eigenvalue strictly positive
A matrix has singular values 5, 2, and 0. What is its rank?
- 3
- 0
- 2
A^T A = 9I. Apply Gram-Schmidt to the eigenvectors (0, 1), then (2, 3), to form V. What is the first coordinate of the second unit vector?
Answer: ______________
Keep the largest positive singular value and its matching left and right singular vectors. What does this give in Frobenius norm?
- A closest matrix of rank at most one
- An exact copy of every nonzero matrix
- An inverse for every rectangular matrix
A matrix has singular values 6 and 8. What is the sum of the squares of all its entries?
Answer: ______________

Answer key

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2 · A^T A = diag(0, 0, 49), so its three eigenvalues include two zeros.
4 · A^T A = diag(9, 16). The nonnegative square roots are 3 and 4.
The greatest output length for a unit input · It is max ||Ax|| over all inputs with ||x|| = 1.
5 · Square roots of 25 and 0 are 5 and 0, so the largest is 5.
The u_i have unit length and pairwise dot product 0; the formula cannot be used when sigma_i = 0. · The inner-product formula gives 0 for distinct indices and 1 for equal indices, including repeated positive singular values. Division by a zero singular value is undefined; complete U with orthonormal vectors instead.
Its quadratic form is the squared length ||Ax||^2 · For an eigenvector v, lambda ||v||^2 = ||Av||^2. Dividing by the positive ||v||^2 gives lambda >= 0.
2 · Rank counts nonzero singular values, and here two are nonzero.
1 · The first vector is already unit length. Subtracting its projection from (2, 3) leaves (2, 0). Dividing by length 2 gives (1, 0), whose first coordinate is 1.
A closest matrix of rank at most one · The retained layer sigma1 u1 v1^T has rank one and minimises Frobenius error among matrices of rank at most one.
100 · The squared-entry sum equals the squared singular-value sum: 6^2 + 8^2 = 100.