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Introduction to Linear Algebra
Published in Timothy Bower, ®, 2023
Another interesting result is that the trace of S=ATA, denoted as Tr(S), is the sum of the squares of all the elements of A. The trace of a matrix is the sum of its diagonal elements.
Linear Algebra Problems
Published in Dingyü Xue, YangQuan Chen, Scientific Computing with MATLAB®, 2018
i.e., the trace of a matrix is defined as the sum of diagonal elements. From linear algebra theory, the trace of a matrix equals the sum of the eigenvalues. The trace of matrix A can be obtained using the MATLAB function trace(), such that t = (A). The trace of the matrix in Example 4.10 can be obtained directly from trace(A) = 34.
Eigenvalues and Eigenvectors
Published in Sohail A. Dianat, Eli S. Saber, ®, 2017
Sohail A. Dianat, Eli S. Saber
The product and sum of the eigenvalues of any matrix are equal to the determinant and trace of that matrix, respectively. We first show that the product of eigenvalues is equal to the determinant of the matrix. Let A be an n × n matrix with characteristic polynomial P(λ). Then:
Application of Tikhonov regularization to reconstruct past climate record from borehole temperature
Published in Inverse Problems in Science and Engineering, 2021
Jia Liu, Tingjun Zhang, Gary D. Clow, Elchin Jafarov
The Generalized Cross-Validation (GCV) method employs the GCV function to find a regularization parameter α. The optimal regularization parameter α is the one that minimizes the GCV function: where is a matrix that generates the regularized solution when multiplied by d, i.e.. Here, In is the identity matrix of dimension n×n. trace(·) is a mathematical operator on a square matrix, defined to be the sum of the elements on the matrix diagonal.
An optimal data assimilation method and its application to the numerical simulation of the ocean dynamics
Published in Mathematical and Computer Modelling of Dynamical Systems, 2018
Konstantin Belyaev, Andrey Kuleshov, Natalia Tuchkova, Clemente A.S. Tanajura
where are two given vectors. The matrix KQK’ is symmetric and positive definite. Its trace is equalled to the sum of its diagonal elements as well as to the sum of its eigenvalues. Function J(K) is the convex, quadratic and smooth function of variable K and its derivative is easily calculated. Hence, the problems (6) and (7) may be solved with the help of classical Lagrangian Multipliers Method [20,21].
Generalized formulation of extended cross-section adjustment method based on minimum variance unbiased linear estimation
Published in Journal of Nuclear Science and Technology, 2019
Kenji Yokoyama, Takanori Kitada
where denotes the matrix trace, which is the sum of the diagonal elements. To derive Equation (50), we have used the following equations for arbitrary matrices and :