Operators and matrices - basic matrix operations, determinant, spectrum of an operator, spectral theorem in finite-dimensional Hilbert spaces - Question Bank

1. What is the result of multiplying a matrix A = [[1, 2], [3, 4]] by its transpose A^T?
A) [[7, 10], [10, 16]]
B) [[1, 4], [2, 5], [3, 6]]
C) [[1, 2, 3], [4, 5, 6]]
D) [[1, 4], [3, 10]]
2. For a finite-dimensional complex vector space, a linear operator T is diagonalizable if and only if:
A) The geometric multiplicity of each eigenvalue equals its algebraic multiplicity.
B) The matrix is symmetric.
C) The matrix is invertible.
D) The trace is non-zero.
3. What is the definition of the spectrum of an operator T on a Hilbert space?
A) The set of complex numbers λ for which (T - λI) is not invertible.
B) The set of complex numbers λ for which (T - λI) is invertible.
C) The set of complex numbers λ for which T - λI = 0.
D) The set of complex numbers λ for which T = λI.
4. If A is a Hermitian matrix, what can be said about its eigenvalues?
A) They are always real.
B) They are always purely imaginary.
C) They are always complex.
D) They are always zero.
5. What is the relationship between the determinant and eigenvalues of an n x n matrix A?
A) det(A) = product of eigenvalues (counting multiplicity).
B) det(A) = sum of eigenvalues (counting multiplicity).
C) det(A) = average of eigenvalues (counting multiplicity).
D) det(A) = sum of absolute values of eigenvalues (counting multiplicity).
6. What is the relationship between the trace and eigenvalues of an n x n matrix A?
A) tr(A) = sum of eigenvalues (counting multiplicity).
B) tr(A) = product of eigenvalues (counting multiplicity).
C) tr(A) = average of eigenvalues (counting multiplicity).
D) tr(A) = sum of absolute values of eigenvalues (counting multiplicity).
7. For a finite-dimensional Hilbert space, the spectral theorem guarantees that any normal operator is unitarily equivalent to:
A) A diagonal operator.
B) A triangular operator.
C) A scalar operator.
D) An identity operator.
8. What is the spectrum of a scalar operator T(x) = cx on a Hilbert space?
A) {c}
B) {0}
C) {1}
D) The set of all complex numbers.
9. If A is a matrix, det(cA) for an n x n matrix and scalar c is equal to:
A) c^n det(A)
B) c det(A)
C) det(A)
D) n c det(A)
10. What is the definition of a matrix A being singular?
A) det(A) = 0
B) det(A) ≠ 0
C) A is the zero matrix.
D) A is the identity matrix.
11. Consider a projection operator P. What are its possible eigenvalues?
A) 0 and 1
B) 0 and -1
C) 1 and -1
D) Only 0
12. If A and B are n x n matrices, then det(AB) is equal to:
A) det(A)det(B)
B) det(A) + det(B)
C) det(A) - det(B)
D) det(A) / det(B)
13. What is the spectrum of a nilpotent operator T (i.e., T^k = 0 for some k)?
A) {0}
B) {1}
C) {0, 1}
D) The set of all complex numbers.
14. What property must a matrix satisfy to be diagonalizable by a unitary matrix?
A) It must be a normal matrix.
B) It must be a symmetric matrix.
C) It must be an orthogonal matrix.
D) It must be a skew-symmetric matrix.
15. If A is a matrix, det(A^T) is equal to:
A) det(A)
B) -det(A)
C) 1/det(A)
D) det(A)^2
16. What is the spectral decomposition of a self-adjoint operator T in a finite-dimensional Hilbert space?
A) T = Σ λ_i P_i, where P_i are orthogonal projection operators.
B) T = Σ λ_i U_i, where U_i are unitary operators.
C) T = Σ λ_i A_i, where A_i are arbitrary operators.
D) T = Σ λ_i v_i v_i^T, where v_i are arbitrary vectors.
17. The set of eigenvalues of a real symmetric matrix is:
A) Always a subset of the real numbers.
B) Always a subset of the imaginary numbers.
C) Always a subset of the complex numbers.
D) Always empty.
18. If T is a linear operator on a finite-dimensional vector space, what is an eigenvector?
A) A non-zero vector v such that Tv = λv for some scalar λ.
B) A vector v such that Tv = 0.
C) A vector v such that Tv = v.
D) Any vector in the vector space.
19. What is the spectrum of the identity operator I on a finite-dimensional Hilbert space?
A) {1}
B) {0}
C) {1, 0}
D) The set of all complex numbers.
20. If A is an invertible matrix, then det(A^-1) is equal to:
A) 1 / det(A)
B) det(A)
C) -det(A)
D) 1 / det(A^T)
21. What is the identity matrix I_n?
A) A square matrix with 1s on the main diagonal and 0s elsewhere.
B) A square matrix with all entries equal to 1.
C) A square matrix with all entries equal to 0.
D) A rectangular matrix with 1s on the main diagonal.
22. If A is an m x n matrix and B is an n x p matrix, what are the dimensions of the product AB?
A) m x p
B) n x n
C) m x n
D) p x m
23. What is the condition for two matrices A and B to be conformable for multiplication AB?
A) Number of columns in A equals the number of rows in B.
B) Number of rows in A equals the number of columns in B.
C) A and B have the same dimensions.
D) A and B are square matrices.
24. If A is an n x n matrix, how many eigenvalues does it have (counting multiplicities)?
A) Exactly n.
B) At most n.
C) At least n.
D) Infinitely many.
25. What is the geometric interpretation of a matrix determinant?
A) The scaling factor of the volume/area under the linear transformation.
B) The sum of the lengths of the vectors.
C) The angle of rotation.
D) The distance from the origin.
26. If A is a real symmetric matrix, what are its eigenvalues?
A) Always real.
B) Always complex.
C) Always positive.
D) Always negative.
27. The eigenvalues of a matrix are the roots of its:
A) Characteristic polynomial.
B) Minimal polynomial.
C) Identity polynomial.
D) Zero polynomial.
28. What is the characteristic polynomial of a 2x2 matrix A = [[a, b], [c, d]]?
A) λ^2 - tr(A)λ + det(A)
B) λ^2 + tr(A)λ + det(A)
C) λ^2 - det(A)λ + tr(A)
D) λ^2 + det(A)λ + tr(A)
29. Consider the matrix A = [[0, 1], [0, 0]]. What is its spectrum?
A) {0}
B) {1}
C) {0, 1}
D) {}
30. What does the spectral theorem imply for a compact normal operator on a Hilbert space?
A) It can be represented by a diagonal matrix with eigenvalues on the diagonal.
B) It can be represented by a matrix with real eigenvalues.
C) It can be represented by a matrix with positive eigenvalues.
D) It can be represented by a matrix with zero eigenvalues.
31. If A is a unitary operator on a finite-dimensional Hilbert space, what is true about its eigenvalues?
A) Their absolute value is 1.
B) They are all real.
C) They are all purely imaginary.
D) They are all zero.
32. For a finite-dimensional normal operator T, the spectrum is equal to:
A) The set of eigenvalues.
B) The set of singular values.
C) The set of characteristic values.
D) The set of eigenvectors.
33. What is the spectral radius of an operator T?
A) The maximum absolute value of the eigenvalues.
B) The minimum absolute value of the eigenvalues.
C) The sum of the absolute values of the eigenvalues.
D) The product of the absolute values of the eigenvalues.
34. Which property holds for the trace of matrices A and B (where multiplication is defined): tr(AB) = tr(BA)?
A) True
B) False
C) Only if A and B are symmetric
D) Only if A and B are square
35. Find the trace of the matrix B = [[5, 2, 8], [1, 9, 3], [4, 6, 7]].
A) 21
B) 30
C) 10
D) 15
36. What is the trace of a square matrix A?
A) The sum of the diagonal elements.
B) The product of the diagonal elements.
C) The determinant of the matrix.
D) The sum of all elements.
37. A matrix A is called Hermitian if:
A) A = A^T
B) A = A*
C) A = -A^T
D) A = -A*
38. What is the adjoint of a matrix A = [[1+i, 2], [3, 4-i]]?
A) [[1-i, 3], [2, 4+i]]
B) [[1+i, 3], [2, 4-i]]
C) [[1-i, 2], [3, 4+i]]
D) [[2, 1+i], [4-i, 3]]
39. If A is a square matrix, when is A an orthogonal matrix?
A) When A^T = A^-1
B) When A^T = A
C) When det(A) = 1
D) When A = I
40. What is the transpose of a matrix A = [[1, 2, 3], [4, 5, 6]]?
A) [[1, 4], [2, 5], [3, 6]]
B) [[6, 5, 4], [3, 2, 1]]
C) [[1, 2, 3], [4, 5, 6]]
D) [[1, 3, 2], [4, 6, 5]]
41. If A is a normal operator on a finite-dimensional complex Hilbert space, what can be said about its eigenvectors?
A) There exists an orthonormal basis of eigenvectors.
B) Eigenvectors corresponding to distinct eigenvalues are orthogonal.
C) All eigenvectors are linearly independent.
D) Eigenvectors may not exist for all eigenvalues.
42. What is the spectral theorem for a self-adjoint operator T on a finite-dimensional Hilbert space?
A) T can be diagonalized by a unitary matrix, and its eigenvalues are real.
B) T can be diagonalized by an orthogonal matrix, and its eigenvalues are complex.
C) T can be triangularized by a unitary matrix, and its eigenvalues are real.
D) T can be diagonalized by any invertible matrix, and its eigenvalues can be any complex number.
43. In a finite-dimensional Hilbert space, what is the relationship between the spectrum of an operator and its eigenvalues?
A) The spectrum is always a subset of the eigenvalues.
B) The spectrum is always equal to the set of eigenvalues.
C) The spectrum is always a superset of the eigenvalues.
D) There is no direct relationship.
44. What is the spectrum of a linear operator T on a finite-dimensional vector space?
A) The set of all eigenvalues of T.
B) The set of all singular values of T.
C) The set of all possible outputs of T.
D) The set of all vectors T can map to.
45. For a square matrix A, what does det(A) = 0 imply?
A) The matrix is invertible.
B) The matrix is singular.
C) The matrix is the identity matrix.
D) The matrix has all zero entries.
46. Find the determinant of the matrix M = [[2, 3], [4, 5]].
A) 22
B) -2
C) 10
D) 14
47. What is the determinant of a 2x2 matrix A = [[a, b], [c, d]]?
A) ad + bc
B) ac + bd
C) ad - bc
D) ab - cd
48. Calculate the product of matrices P = [[1, 0], [0, 1]] and Q = [[a, b], [c, d]].
A) [[a, c], [b, d]]
B) [[a, b], [c, d]]
C) [[1, 0], [0, 1]]
D) [[ad, bc]]
49. If matrix C = [[2, 1], [1, 3]] and scalar k = 3, what is kC?
A) [[6, 3], [3, 9]]
B) [[5, 4], [4, 6]]
C) [[2, 3], [3, 3]]
D) [[6, 1]]
50. What is the result of adding two matrices A and B, where A = [[1, 2], [3, 4]] and B = [[5, 6], [7, 8]]?
A) [[6, 8], [10, 12]]
B) [[5, 12], [21, 32]]
C) [[4, 4], [4, 4]]
D) [[7, 9], [11, 13]]
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