Hadamard product

For your information, relative gain array can be computed as the so-called Hadamard product, Ay = KjjKrH, which is the element-by-element product of the gain matrix K and the transpose of its inverse. You can confirm this by repeating the examples with MATLAB calculations. 

The Hadamard product of two matrices A and B of same dimension is defined as ... 

The -> adjacency matrix A of a molecular graph G is an example of binary sparse matrix, only the off-diagonal entries i-j, where v, and Vy are adjacent vertices, i.e. vertices connected by a bond, being equal to one. Using the adjacency matrix as multiplier in the Hadamard product it follows ... 

The Cluj matrices defined above, both symmetric and unsymmetrical, can be either path-Cluj matrices (U CJ and SC J ) when all the pairs of vertices of the graph are accounted for in the matrix calculation or edge-Cluj matrices (UCJ and SCJ ) if the only nonzero elements correspond to edges, that is, only pairs of adjacent vertices are accounted for. The edge-Cluj matrices can be obtained by the Hadamard product of the path-Cluj matrices and the adjacency matrix A SCJ, = SCJp A UCJ, = UCJp A... 

The most popular expanded matrices are —> expanded distance matrices, D M, derived as the Hadamard product between the —> distance matrix D and some different graph-theoretical matrix M, such as the —> Wiener matrix, —> Cluj matrices, Szeged matrix, and walk matrices. Moreover, —> expanded reciprocal distance matrices, D M, were defined by analogy with the expanded distance matrices by using the —> reciprocal distance matrix instead of the... 

The sparse Wiener matrix of mth order ""W is derived from the path-Wiener matrix setting to zero all entries except those corresponding to paths " py of length m. This matrix can be calculated by the Hadamard product of the Wiener matrix and the geodesic matrix B whose elements corresponding to paths of length m are equal to 1, or else zero ... 

In order to concisely describe multi-way models, the usual matrix product is not sufficient. Three other types of matrix products are introduced the Kronecker ( ), Hadamard ( ) and Khatri-Rao (O) product [McDonald 1980, Rao Mitra 1971, Schott 1997], The Kronecker product allows a very efficient way to write Tucker models (see Chapter 4). Likewise, the Khatri-Rao product provides means for an efficient way to write a PARAFAC model (see Chapter 4). The Hadamard product can, for instance, be used to formalize weighted regression (see Chapter 6). 

In the example used earlier for the Kronecker product, the Hadamard product of A and B becomes ... 

Some properties of the Hadamard product are [Magnus Neudecker 1988] ... 

The IMP is based on the structure of the Hadamard product, which is related to the result of multiplying two sums, retaining the diagonal crossterms only. In this way, the Hadamard (or inward) product of two sums... 

IMP and classic matrix products coincide within the diagonal matrix subspaces. From now on, the IMP and Hadamard products will be synonyms of an operation, which can be applied not only to matrix spaces but also to a wide variety of mathematical objects. Keep in mind that the IMP main characteristic is the result, defined as producing another mathematical object of the same kind as the objects involved in the operation. 

Pointwise multiplication of two vectors, also called the Schur or Hadamard product, and denoted in this work by Q (U + 2299), is defined as the multiplication of two vectors by taking each entry of the two vectors and multiplying them together, that is, Z). = Xkyk> where k are the index locations. 

Where the operator represents an element by element, or Hadamard, product. 

It is important to note that Equation 9.10 indicates an element by element multiplication of the corresponding elements of the two matrices G and R. This type of multiplication is called the Hadamard product of two matrices [2], and is not the normal matrix product. 

Now, the Hadamard product of the two matrices must be calculated where A. = gijtij. For example ... 

The Hadamard product of two matrices (A°B) produces another matrix where each element ij is the product of the elements ij of the original matrices. The condition is that A and B matrices must have the same dimensions. For instance, in the given examples of Figure 2 the Hadamard product could be done between A and B or even A and (where the superindex T denotes the transpose of the matrix), but not between A and C (Figure 3). The Kronecker product... 

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