Since I can't comment, I wish to add this: the Cholesky decomposition (or its variant, LDL T, L a unit lower triangular matrix and D a diagonal matrix) can be used to verify if a symmetric matrix is positive/negative definite: if it is positive definite, the elements of D are all positive, and the Cholesky decomposition will finish successfully without taking the square root of a negative number. As a hint, I will take the determinant of another 3 by 3 matrix. But it's the exact same process for the 3 by 3 matrix that you're trying to find the determinant of. So here is matrix A. Here, it's these digits. This is a 3 by 3 matrix. And now let's evaluate its determinant.

May 27, 2010 · Let "A" be an orthogonal matrix. Then by definition of an orthogonal matrix, the transpose of A is equal to the inverse of A: A^T = A^(-1) Then remember what the definition of an inverse matrix is: A*A^(-1) = I "I" will be the identity matrix. Now you can substitute in A^(T) for A^(-1) since A^(T) = A^(-1): A*A^(T) = I. Take the determinant of ... Keywords: symplectic matrix, determinant, transvection, isometry 2010 MSC: 15B57, 65F40, 11C20, 51A50 1. Introduction Let K be a ﬁeld and n 2N := f1;2;:::g. A matrix S 2K2n 2n is called J-symplectic if ST JS = J (1) for regular and skew-symmetric J 2K2n 2n, i.e., JT = J. If the characteristic

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