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The Gram matrix of any orthonormal basis is the identity matrix. Equivalently, the Gram matrix of the rows or the columns of a real rotation matrix is the identity matrix. Likewise, the Gram matrix of the rows or columns of a unitary matrix is the identity matrix. The rank of the Gram matrix of vectors in or equals the dimension of the space ...
with N B and N G being the molarities 1-column matrices of the batch and glass components respectively, and B being the batching matrix. [ 1 ] [ 2 ] [ 3 ] The symbol " T " stands for the matrix transpose operation, " −1 " indicates matrix inversion , and the sign "·" means the scalar product .
In general, the mass matrix M depends on the state q, and therefore varies with time. Lagrangian mechanics yields an ordinary differential equation (actually, a system of coupled differential equations) that describes the evolution of a system in terms of an arbitrary vector of generalized coordinates that completely defines the position of ...
In mathematics, a Euclidean distance matrix is an n×n matrix representing the spacing of a set of n points in Euclidean space. For points x 1 , x 2 , … , x n {\displaystyle x_{1},x_{2},\ldots ,x_{n}} in k -dimensional space ℝ k , the elements of their Euclidean distance matrix A are given by squares of distances between them.
It follows from application of the Eckart conditions that the matrix G −1 gives the kinetic energy in terms of arbitrary linear internal coordinates, while F represents the (harmonic) potential energy in terms of these coordinates. The GF method gives the linear transformation from general internal coordinates to the special set of normal ...
Mass fraction can also be expressed, with a denominator of 100, as percentage by mass (in commercial contexts often called percentage by weight, abbreviated wt.% or % w/w; see mass versus weight). It is one way of expressing the composition of a mixture in a dimensionless size ; mole fraction (percentage by moles , mol%) and volume fraction ...
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For example, 50 g of zinc will react with oxygen to produce 62.24 g of zinc oxide, implying that the zinc has reacted with 12.24 g of oxygen (from the Law of conservation of mass): the equivalent weight of zinc is the mass which will react with eight grams of oxygen, hence 50 g × 8 g/12.24 g = 32.7 g.