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Basic Computations
Published in Jhareswar Maiti, Multivariate Statistical Modeling in Engineering and Management, 2023
If the determinant of a matrix A is zero, A is said to be singular. Johnson and Wichern (2002) mentioned singularity as follows: for a non-zero vector x, a matrix A is said to be singular if the relationship Ax=0 holds.
Determinants
Published in Jeff Suzuki, Linear Algebra, 2021
So what about larger matrices? Note that we proved Proposition 15 without using any preselected formula for the determinant. Thus, regardless of what formula we use to compute a determinant, we can always find the determinant of a matrix by choosing a row or column, then finding the sum of the product of the row or column entries with the corresponding cofactors.
Determinants
Published in C.W. Evans, Engineering Mathematics, 2019
A determinant is a number which is calculated from the elements in a square matrix. If we have a square matrix, we may represent its determinant using the row and column notation of matrices. To distinguish the two concepts we enclose the elements of a determinant between vertical parallel lines. We can write the determinant of the square matrix A as either |A| or det A.
New gradient methods with adaptive stepsizes by approximate models
Published in Optimization, 2023
Zexian Liu, Hongwei Liu, Ting Wang
According to (1), we know , which implies that there exist at least mutually orthogonal vectors such that As a result, which implies that at least eigenvalues of all equal to . Now we calculate the other two eigenvalues and . According to the fact that the trace of a matrix equals to the sum of all eigenvalues of the matrix, by some simple algebraic operations we have Using the fact that the determinant of a matrix equals the product of all eigenvalues of the matrix, we can easily obtain Combining (13) and (14), after some simple algebraic operations we obtain that which completes the proof.