Vector matrix multiplier

Damman grating to replicate the source inputs. The use of the CGH to replicate the inputs comes from the CGH property that the spots in the replay field are the Fourier transform of input illumination. Since only one channel is likely to be required at each output, those not required can be blocked using liquid crystal shutters. Such switches are based on the Stanford vector matrix multiplier (SVMM) [50] related switching devices [51]. When implemented using a CGH to fan out, and with a 2D array of inputs (rather than the 1D arrays of the SVMM) to simplify the free space optics, these are called matrix - matrix switches [52]. This kind of structure is found in a range of optical processing architectures (see Sec. 2.4). For a symmetrical switch with n inputs and n outputs, an array of nxn shutters is required. 

The most significant development, which led to the use of liquid crystal devices in neural networks, was the Stanford vector matrix multiplier (SVMM) by Goodman in 1978 [50, 63]. The basic structure of the system is shown in Fig. 56. 

If ns absorbance measurements, y. ..yns, are taken for ns different mixtures of the nc components, then ns equations of the kind (3.8) can be written. Again, it is possible and more convenient to use the vector/matrix notation the vector y is the product of a matrix C that contains, as rows, the concentrations of the components in each solution, multiplied by the same column vector a with the molar absorptivities. 

These transformations can be thought of in terms of a matrix multiplying a vector... 

Positive definite means that the matrix when left- and right-multiplied by an arbitrary vector will yield a nonnegative scalar. If the matrix multiplied by a vector composed of forces is proportional to a flux, it implies that the flux always has a positive projection on the force vector. Technically, one should say that Lap is nonnegative definite but the meaning is clear. 

If a set of independent vectors is multiplied by an orthogonal matrix, the resulting set is still independent. Thus, the ranks of A and 2 are the same. Consequently, the rank of a matrix is the number of non-zero singular values. 

In calculating the product of a matrix (or vector) by a scalar quantity, each element of the matrix (or vector) is multiplied by the scalar. 

The purpose of this factorisation is to help understand what happens when a vector is multiplied by the same matrix repeatedly, as happens in subdivision. 

What happens when a general vector is multiplied by a matrix ... 

Eigenfactorisation of a matrix helps us to understand what happens when vectors are multiplied by the matrix repeatedly. 

To calculate the abundance of the population at the next time step (t + 1), the vector is multiplied against the matrix as follows ... 

All of the quantities in Eqs. (39)-(42) are evaluated using vector instructions and the matrix multiply is evduated using a fast matrix multiply routine. It should be noted that the coordinates specified by x (i) and x (i) all depend on R, R, ... 

We can extend the algebra of matrices to include products formed by multiplying two matrices together. The product of the matrices must give a third matrix, since we know, for example, that the combined 20- operation in the C2V point group is equivalent to the other vertical reflection CTv. The multiplication of the two matrices can be carried out by treating the columns of the second matrix as vectors and multiplying each one by the rows... 

The left-side coefficient matrix multiplying the unknown vector p is said to be banded because its elements fall within diagonal bands. The product shown equals the nonzero right side in Equation 20-17, which contains the delta-p pressure drop (P/ - P ) that drives the Darcy flow. This delta-p, applied across the entire core, mathematically manifests itself by controlling the top and bottom rows of the governing tridiagonal matrix equation. 

The outer product of two vectors is obtained by multiplying the first element of a column vector by a row vector, thus forming the first row of a matrix. Subsequently, the second element of the coluirm vector is multiplied by the row vector, and so forth. In the example of Eq. (20.7) this would lead to ... 

The i-th column of the matrix vw is the vector v multiplied by the i-th component of the vector w. Hence vw is a rank-one matrix. 

All of the iterative diagonalization methods involve broadly similar operations to those outlined above starting from an initial seed vector and proceeding via matrix multiply... 

There is a linear algebra equivalent of the differential eigenvalue equation. Instead of a differential operator acting on a function, a square matrix multiplies a column vector, and this is equal to a constant, the eigenvalue, times the vector ... 

In contrast to this, if x is the eigenvector of A, then the multiplication of the eigenvector x by matrix A yields the same vector x multiplied by a scalar k, that is, the same vector but of different length ... 

A square matrix has the eigenvalue A if there is a vector x fulfilling the equation Ax = Ax. The result of this equation is that indefinite numbers of vectors could be multiplied with any constants. Anyway, to calculate the eigenvalues and the eigenvectors of a matrix, the characteristic polynomial can be used. Therefore (A - AE)x = 0 characterizes the determinant (A - AE) with the identity matrix E (i.e., the X matrix). Solutions can be obtained when this determinant is set to zero. 

The symbol when interposed between two vectors means that a matrix is to be formed. Tlie ijth element of the matrix u v is obtained by multiplying u, by Vy. 

---