Dual spaces

We can consider a dual space V with respect to V. For an arbitrary fixed element u G V the functional u —> u u) can be defined, where u G V. This functional is linear and continuous on V and therefore is an element of the space V. For every u gV the functional u G V can be pointed out such that... 

Here ( , ) means a duality pairing between and its dual space... 

Postulate B.—There exists a set of vectors indicated by the bra symbol < in one-to-one correspondence with the vectors of 3/f, forming a dual Hilbert space 3 . As a matter of notation we use the symbol , etc. This dual space must be such that a meaning can be given to the scalar product of any vector with the following properties ... 

J, where u and v are arbitrary complex numbers. The dual space is... 

In the illustration of Fig. 29.4 we regard the matrix X as either built up from n horizontal rows of dimension p, or as built up from p vertical columns x,.of dimension n. This exemplifies the duality of the interpretation of a matrix [9]. From a geometrical point of view, and according to the concept of duality, we can interpret a matrix with n rows and p columns either as a pattern of n points in a p-dimensional space, or as a pattern of p points in an n-dimensional space. The former defines a row-pattern P" in column-space 5, while the latter defines a column-pattern P in row-space S". The two patterns and spaces are called dual (or conjugate). The term dual space also possesses a specific meaning in another... 

After preprocessing of a raw data matrix, one proceeds to extract the structural features from the corresponding patterns of points in the two dual spaces as is explained in Chapters 31 and 32. These features are contained in the matrices of sums of squares and cross-products, or cross-product matrices for short, which result from multiplying a matrix X (or X ) with its transpose ... 

The matrix X defines a pattern P" of n points, e.g. x, in which are projected perpendicularly upon the axis v. The result, however, is a point s in the dual space S". This can be understood as follows. The matrix X is of dimension nxp and the vector V has dimensions p. The dimension of the product s is thus equal to n. This means that s can be represented as a point in S". The net result of the operation is that the axis v in 5 is imaged by the matrix X as a point s in the dual space 5". For every axis v in 5 we will obtain an image s formed by X in the dual space. In this context, we use the word image when we refer to an operation by which a point or axis is transported into another space. The word projection is reserved for operations which map points or axes in the same space [11]. The imaging of v in S into s in S" is represented geometrically in Fig. 29.9a. Note that the patterns of points P" and P are represented schematically by elliptic envelopes. 

Using the same argument as above, we can see that the product is of dimension p, since the matrix X has dimensions pxn and the vector u possesses dimension n. Hence, the vector can be imaged as a single point in the dual space S . We state that the vector u in S" is imaged by the matrix X into the point 1 in the dual space S . 

In a general way, we can state that the projection of a pattern of points on an axis produces a point which is imaged in the dual space. The matrix-to-vector product can thus be seen as a device for passing from one space to another. This property of swapping between spaces provides a geometrical interpretation of many procedures in data analysis such as multiple linear regression and principal components analysis, among many others [12] (see Chapters 10 and 17). 

The geometrical interpretation of MLR is given in Fig. 29.10. The n rows (objects) of X form a pattern P" of points (represented by x,) which is projected upon an (unknown) axis b. This causes the axis b in S to be imaged by X in the dual space S" at the point y. The vector of observed measurements y has dimension n and, hence, is also represented as a point in 5". Is it possible then to define an axis b in S " such that the predicted y coincides with the observed y Usually this will not be feasible. One may propose finding the best possible b such that y comes as close to y as possible. A criterion for closeness is to ask for the distance between y and y, which is equal to the normlly - yll, to be as small as possible. 

From the above we conclude that the product of a matrix with a vector can be interpreted geometrically as an operation by which a pattern of points is projected upon an axis. This projection produces an image of the axis at a point in the dual space. The concept can be extended to the product of a matrix with another matrix. In this case we can conceive of the latter as a set of axes, each of which produces image points in the dual space. In the special case when this matrix has only two columns, the product can be regarded as a projection of a pattern of points upon the plane formed by the two axes. As a result one obtains two image points (one for each axis that defines the plane of projection) in the dual space. 

In the general case we use the symbols U and V to represent projection matrices in 5" and S , each containing r projection vectors, and the symbols S and L to represent their images in the dual space ... 

We have seen above that the r columns of U represent r orthonormal vectors in row-space 5". Hence, the r columns of U can be regarded as a basis of an r-dimensional subspace 5 of 5". Similarly, the r columns of V can be regarded as a basis of an r-dimensional subspace S of column-space 5. We will refer to S as the factor space which is embedded in the dual spaces S" and SP. Note that r

factor-spaces will be more fully developed in the next section. 

In Section 29.3 it has been shown that a matrix generates two dual spaces a row-space S" in which the p columns of the matrix are represented as a pattern P , and a column-space S in which the n rows are represented as a pattern P". Separate weighted metrics for row-space and column-space can be defined by the corresponding metric matrices and W. This results into the complementary weighted spaces and S, each of which can be represented by stretched coordinate axes using the stretching factors in -J v and, where the vectors w and Wp contain the main diagonal elements of W and W. ... 

In the case of an nxp matrix X we can define two sets of weighted measures of location and spread, one for each of the two dual spaces and in terms of weighted means and weighted variances ... 

The pattern of points produced by Z is centred in both dual spaces and S, since the weighted row- and column-means m and are zero ... 

Correspondence factor analysis can be described in three steps. First, one applies a transformation to the data which involves one of the three types of closure that have been described in the previous section. This step also defines two vectors of weight coefficients, one for each of the two dual spaces. The second step comprises a generalization of the usual singular value decomposition (SVD) or eigenvalue decomposition (EVD) to the case of weighted metrics. In the third and last step, one constructs a biplot for the geometrical representation of the rows and columns in a low-dimensional space of latent vectors. 

J. Liu, P. Cheung, L. Guibas, and F. Zhao, A dual-space approach to tracking and sensor management in wireless sensor networks , in The First A CM International Workshop on Wireless Sensor Networks and Applications, Atlanta, USA, September 2002. 

Exercise 2.14 (Used in Section 5.5) Let V denote a complex vector space. Let y denote the set of complex linear transformations from V to C. Show that y is a complex vector space. Show that ifV is finite dimensional then dim y = dim V. The vector space y is called the dual vector space or, more simply, the dual space. 

Another way to make the dual space (C ) concrete is to use the complex scalar product and think of elements of the dual space as column vectors. Recall the notation for the conjugate transpose of a vector. In this interpretation... 

With this complex scalar product on the dual space V in hand, we can make the relationship between the dual and the adjoint clear. The definition... 

A method has been developed to identify the nodes which will not be immediately approached by the event and can be turned off to save energy [Liu 02], The method is based on the dual space transformation [O R 98], Figure 2 shows the dual space. Points from the primal space are transformed into lines in the dual space. Lines from the primal space are transformed into points in the dual space. As a result, the dual space is partitioned into cells. The e point, the shadow edge, is contained in the shaded cell. Since the e point can not intersect the n2 line, before it crosses one of the cell boundaries, the N2 node can stay turned off as long as none of Nl, N3 and N4 senses a transition. This method may provide a substantial power reduction for a large sensor field. However, if nodes that line the perimeter around the event misbehave and declare a transition, it will force several other nodes to wake up and waste energy. 

Figure 2. Dual Space Indicates the Sequence of Transitions...

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