91影视

To show that if B is a symmetric nonsingular matrix, then B虏 is positive definite, we prove that B虏 has the following properties: 1. Symmetric: Since B is symmetric (B = B岬�), we find that \((B^2)^T = (B \times B)^T = B^T \times B^T = B \times B = B^2\), so B虏 is also symmetric. 2. Nonsingular: Since B is nonsingular, its determinant is nonzero: \(\det(B) \neq 0\). Using the property \(\det(B^2) = \det(B \times B) = \det(B)\det(B)\), we conclude that \(\det(B^2) \neq 0\) and B虏 is nonsingular. 3. Positive eigenvalues: B is symmetric, so it has real eigenvalues and an orthogonal matrix P such that B = PDP岬�, where D is a diagonal matrix of its eigenvalues. Then, B虏 = PD虏P岬�. As B is nonsingular, all eigenvalues are either positive or negative. When squared, they become positive, making all eigenvalues of B虏 positive. Thus, B虏 is positive definite, as it is symmetric, nonsingular, and has positive eigenvalues.