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5 days ago · Every matrix can be considered as an array or vectors whose entries are algebraic entries. A matrix is the next generalization of a vector. In this section, you will learn how to define matrices with Mathematica as well as some other manipulation tools.
1 day ago · In mathematics, a matrix ( pl.: matrices) is a rectangular array or table of numbers, symbols, or expressions, arranged in rows and columns, which is used to represent a mathematical object or a property of such an object. For example, is a matrix with two rows and three columns. This is often referred to as a "two by three matrix", a " matrix ...
2 days ago · Vector Fields in ℝ2. A vector field in ℝ2 can be represented in either of two equivalent ways. The first way is to use a vector with components that are two-variable functions: ⇀ F(x, y) = P(x, y), Q(x, y) . The second way is to use the standard unit vectors: ⇀ F(x, y) = P(x, y)ˆi + Q(x, y)ˆj.
4 days ago · The bivector magnitude (denoted by (a, b) ∧ (c, d)) is the signed area, which is also the determinant ad − bc. [2] If an n × n real matrix A is written in terms of its column vectors , then. This means that maps the unit n -cube to the n -dimensional parallelotope defined by the vectors the region.
5 days ago · A lower index only, Ai, means the i ‘th column vector; an upper index only, Aj, means the j ‘th row vector. Therefore: A=(A1 ⋯ Ai ⋯ A∣U ∣) = ⎝⎛ A1 ⋮ Aj ⋮ A∣V ∣⎠⎞. We can think of a particular column of a matrix like A1 as its action on a particular basis vector from U: A1 = A(u1)= A1jvj.
3 days ago · A classical context of the Floquet theory was discovered in 1883 by a French mathematician Gaston Floquet (1847--1920). It was intended to describe the behavior of a set of linear differential equations with a time-periodic coefficients, which was in turn originating from the problem of the stability of periodic orbits in classical mechanics.
3 days ago · Subtract the empirical mean vector u from each row of the data matrix X. Store mean-subtracted data in the \( n\times p \) matrix B. \[ \mathbf{B} = \mathbf{X} - \mathbf{h}\mathbf{u^{T}} \] where h is an \( n\times 1 \) column vector of all 1s: \[ h[i] = 1, i = 1, ..., n \] Find the covariance matrix
