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

1. What is the adjoint of a matrix A = [[1+i, 2], [3, 4-i]]? — [[1-i, 2], [3, 4+i]]
2. What is the transpose of a matrix A = [[1, 2, 3], [4, 5, 6]]? — [[1, 4], [2, 5], [3, 6]]
3. If matrix C = [[2, 1], [1, 3]] and scalar k = 3, what is kC? — [[6, 3], [3, 9]]
4. What is the result of adding two matrices A and B, where A = [[1, 2], [3, 4]] and B = [[5, 6], [7, 8]]? — [[6, 8], [10, 12]]
5. What is the result of multiplying a matrix A = [[1, 2], [3, 4]] by its transpose A^T? — [[7, 10], [10, 16]]
6. Calculate the product of matrices P = [[1, 0], [0, 1]] and Q = [[a, b], [c, d]]. — [[a, b], [c, d]]
7. Consider the matrix A = [[0, 1], [0, 0]]. What is its spectrum? — {0}
8. What is the spectrum of a nilpotent operator T (i.e., T^k = 0 for some k)? — {0}
9. What is the spectrum of the identity operator I on a finite-dimensional Hilbert space? — {1}
10. What is the spectrum of a scalar operator T(x) = cx on a Hilbert space? — {c}
11. Consider a projection operator P. What are its possible eigenvalues? — 0 and 1
12. If A is an invertible matrix, then det(A^-1) is equal to: — 1 / det(A)
13. Find the trace of the matrix B = [[5, 2, 8], [1, 9, 3], [4, 6, 7]]. — 21
14. Find the determinant of the matrix M = [[2, 3], [4, 5]]. — -2
15. A matrix A is called Hermitian if: — A = A*
16. For a finite-dimensional Hilbert space, the spectral theorem guarantees that any normal operator is unitarily equivalent to: — A diagonal operator.
17. If T is a linear operator on a finite-dimensional vector space, what is an eigenvector? — A non-zero vector v such that Tv = λv for some scalar λ.
18. What is the identity matrix I_n? — A square matrix with 1s on the main diagonal and 0s elsewhere.
19. What is the determinant of a 2x2 matrix A = [[a, b], [c, d]]? — ad - bc
20. The set of eigenvalues of a real symmetric matrix is: — Always a subset of the real numbers.
21. If A is a real symmetric matrix, what are its eigenvalues? — Always real.
22. If A is a matrix, det(cA) for an n x n matrix and scalar c is equal to: — c^n det(A)
23. The eigenvalues of a matrix are the roots of its: — Characteristic polynomial.
24. If A is a matrix, det(A^T) is equal to: — det(A)
25. What is the definition of a matrix A being singular? — det(A) = 0
26. What is the relationship between the determinant and eigenvalues of an n x n matrix A? — det(A) = product of eigenvalues (counting multiplicity).
27. If A and B are n x n matrices, then det(AB) is equal to: — det(A)det(B)
28. If A is an n x n matrix, how many eigenvalues does it have (counting multiplicities)? — Exactly n.
29. What does the spectral theorem imply for a compact normal operator on a Hilbert space? — It can be represented by a diagonal matrix with eigenvalues on the diagonal.
30. What property must a matrix satisfy to be diagonalizable by a unitary matrix? — It must be a normal matrix.
31. If A is an m x n matrix and B is an n x p matrix, what are the dimensions of the product AB? — m x p
32. What is the condition for two matrices A and B to be conformable for multiplication AB? — Number of columns in A equals the number of rows in B.
33. What is the spectral decomposition of a self-adjoint operator T in a finite-dimensional Hilbert space? — T = Σ λ_i P_i, where P_i are orthogonal projection operators.
34. What is the spectral theorem for a self-adjoint operator T on a finite-dimensional Hilbert space? — T can be diagonalized by a unitary matrix, and its eigenvalues are real.
35. For a finite-dimensional complex vector space, a linear operator T is diagonalizable if and only if: — The geometric multiplicity of each eigenvalue equals its algebraic multiplicity.
36. For a square matrix A, what does det(A) = 0 imply? — The matrix is singular.
37. What is the spectral radius of an operator T? — The maximum absolute value of the eigenvalues.
38. What is the geometric interpretation of a matrix determinant? — The scaling factor of the volume/area under the linear transformation.
39. What is the spectrum of a linear operator T on a finite-dimensional vector space? — The set of all eigenvalues of T.
40. What is the definition of the spectrum of an operator T on a Hilbert space? — The set of complex numbers λ for which (T - λI) is not invertible.
41. For a finite-dimensional normal operator T, the spectrum is equal to: — The set of eigenvalues.
42. In a finite-dimensional Hilbert space, what is the relationship between the spectrum of an operator and its eigenvalues? — The spectrum is always equal to the set of eigenvalues.
43. What is the trace of a square matrix A? — The sum of the diagonal elements.
44. If A is a unitary operator on a finite-dimensional Hilbert space, what is true about its eigenvalues? — Their absolute value is 1.
45. If A is a normal operator on a finite-dimensional complex Hilbert space, what can be said about its eigenvectors? — There exists an orthonormal basis of eigenvectors.
46. If A is a Hermitian matrix, what can be said about its eigenvalues? — They are always real.
47. What is the relationship between the trace and eigenvalues of an n x n matrix A? — tr(A) = sum of eigenvalues (counting multiplicity).
48. Which property holds for the trace of matrices A and B (where multiplication is defined): tr(AB) = tr(BA)? — True
49. If A is a square matrix, when is A an orthogonal matrix? — When A^T = A^-1
50. What is the characteristic polynomial of a 2x2 matrix A = [[a, b], [c, d]]? — λ^2 - tr(A)λ + det(A)
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