Weighing matrix
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In mathematics, a weighing matrix of order and weight is a matrix with entries from the set such that:
Where is the transpose of and is the identity matrix of order . The weight is also called the degree of the matrix. For convenience, a weighing matrix of order and weight is often denoted by .[3]
Weighing matrices are so called because of their use in optimally measuring the individual weights of multiple objects. When the weighing device is a
Properties
Some properties are immediate from the definition. If is a , then:
- The rows of are pairwise orthogonal. Similarly, the columns are pairwise orthogonal.
- Each row and each column of has exactly non-zero elements.
- , since the definition means that , where is the inverseof .
- where is the determinant of .
A weighing matrix is a generalization of Hadamard matrix, which does not allow zero entries.[3] As two special cases, a is a Hadamard matrix[3] and a is equivalent to a conference matrix.
Applications
Experiment design
Weighing matrices take their name from the problem of measuring the weight of multiple objects. If a measuring device has a statistical variance of , then measuring the weights of objects and subtracting the (equally imprecise) tare weight will result in a final measurement with a variance of .
An order matrix can be used to represent the placement of objects—including the tare weight—in trials. Suppose the left pan of the balance scale adds to the measurement and the right pan subtracts from the measurement. Each element of this matrix will have:
Let be a column vector of the measurements of each of the trials, let be the errors to these measurements each
Assuming that is
The variance of the estimated vector cannot be lower than , and will be minimum if and only if is a weighing matrix.[4][5]
Optical measurement
Weighing matrices appear in the engineering of spectrometers, image scanners,[6] and optical multiplexing systems.[5] The design of these instruments involve an optical mask and two detectors that measure the intensity of light. The mask can either transmit light to the first detector, absorb it, or reflect it toward the second detector. The measurement of the second detector is subtracted from the first, and so these three cases correspond to weighing matrix elements of 1, 0, and −1 respectively. As this is essentially the same measurement problem as in the previous section, the usefulness of weighing matrices also applies.[6]
Examples
Note that when weighing matrices are displayed, the symbol is used to represent −1. Here are some examples:
This is a :
This is a :
This is a :
Another :
Which is circulant, i.e. each row is a cyclic shift of the previous row. Such a matrix is called a and is determined by its first row. Circulant weighing matrices are of special interest since their algebraic structure makes them easier for classification. Indeed, we know that a circulant weighing matrix of order and weight must be of square weight. So, weights are permissible and weights have been completely classified.[7] Two special (and actually, extreme) cases of circulant weighing matrices are (A) circulant Hadamard matrices which are conjectured not to exist unless their order is less than 5. This conjecture, the circulant Hadamard conjecture first raised by Ryser, is known to be true for many orders but is still open. (B) of weight and minimal order exist if is a
Equivalence
Two weighing matrices are considered to be equivalent if one can be obtained from the other by a series of permutations and negations of the rows and columns of the matrix. The classification of weighing matrices is complete for cases where as well as all cases where are also completed.[8] However, very little has been done beyond this with exception to classifying circulant weighing matrices.[9][10]
Open questions
This section possibly contains synthesis of material which does not verifiably mention or relate to the main topic. (November 2019) |
There are many open questions about weighing matrices. The main question about weighing matrices is their existence: for which values of and does there exist a ? A great deal about this is unknown. An equally important but often overlooked question about weighing matrices is their enumeration: for a given and , how many 's are there?
This question has two different meanings. Enumerating up to equivalence and enumerating different matrices with same n,k parameters. Some papers were published on the first question but none were published on the second important question.
References
- ^ ISSN 0003-4851.
- ^ ISBN 978-3-319-59031-8.
- ^ S2CID 122560830.
- ^ ISBN 978-0471704850.
- ^ S2CID 122205953.
- ^ PMID 20155192.
- )
- S2CID 1004492.
- .
- .