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A matrix polynomial identity is a matrix polynomial equation which holds for all matrices A in a specified matrix ring M n (R). Matrix polynomials are often demonstrated in undergraduate linear algebra classes due to their relevance in showcasing properties of linear transformations represented as matrices, most notably the Cayley–Hamilton ...
In mathematics, matrix calculus is a specialized notation for doing multivariable calculus, especially over spaces of matrices.It collects the various partial derivatives of a single function with respect to many variables, and/or of a multivariate function with respect to a single variable, into vectors and matrices that can be treated as single entities.
To solve a matrix ODE according to the three steps detailed above, using simple matrices in the process, let us find, say, a function x and a function y both in terms of the single independent variable t, in the following homogeneous linear differential equation of the first order,
The conjugate gradient method with a trivial modification is extendable to solving, given complex-valued matrix A and vector b, the system of linear equations = for the complex-valued vector x, where A is Hermitian (i.e., A' = A) and positive-definite matrix, and the symbol ' denotes the conjugate transpose.
By the Rouché–Capelli theorem, the system of equations is inconsistent, meaning it has no solutions, if the rank of the augmented matrix (the coefficient matrix augmented with an additional column consisting of the vector b) is greater than the rank of the coefficient matrix. If, on the other hand, the ranks of these two matrices are equal ...
A matrix difference equation is a difference equation in which the value of a vector (or sometimes, a matrix) of variables at one point in time is related to its own value at one or more previous points in time, using matrices. [1] [2] The order of the equation is the maximum time gap between any two indicated values of the variable vector. For ...
The Cholesky decomposition is commonly used in the Monte Carlo method for simulating systems with multiple correlated variables. The covariance matrix is decomposed to give the lower-triangular L. Applying this to a vector of uncorrelated observations in a sample u produces a sample vector Lu with the covariance properties of the system being ...
where B and C are n × n matrices. If, for an arbitrary n × n matrix M, M has nonnegative entries, we write M ≥ 0. If M has only positive entries, we write M > 0. Similarly, if the matrix M 1 − M 2 has nonnegative entries, we write M 1 ≥ M 2. Definition: A = B − C is a regular splitting of A if B −1 ≥ 0 and C ≥ 0. We assume that ...