Let \(x_{1}, x_{2}, x_{3}, x_{4}, x_{5}\) be a system of orthonormal vectors in \(\mathbb{R}^{10}\). Consider the matrix \(A = x_{1}x_{1}^{\top} + \ldots + x_{5}x_{5}^{\top}\). Which of the following statements is/are correct?
Answer
(A), (B)
Explanation
Because the \(x_{i}\) are orthonormal, \(Ax_{j} = \sum_{i} x_{i}(x_{i}^{\top}x_{j}) = x_{j}\) for \(j = 1, \ldots, 5\), while \(Av = 0\) for every \(v\) orthogonal to all five. So \(A\) is the orthogonal projection onto their 5-dimensional span: it is symmetric, \(A^{2} = A\), and its eigenvalues are 1 (five times) and 0 (five times).
- A holds: the singular values of a symmetric matrix are the absolute values of its eigenvalues, and here the eigenvalues 0 and 1 are already non-negative.
- B holds: so the singular values are 0 or 1.
- C and D fail: 0 is an eigenvalue, so \(\det A = 0\) and \(A\) is not invertible.
Answer: A, B