![BalanceB](https://upload.wikimedia.org/wikipedia/commons/7/75/BalanceB.jpg)
In
mathematics, a weighing matrix of order
and weight
is a matrix
with entries from the set
such that:
:
Where
is the
transpose
In linear algebra, the transpose of a matrix is an operator which flips a matrix over its diagonal;
that is, it switches the row and column indices of the matrix by producing another matrix, often denoted by (among other notations).
The tr ...
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
.
Weighing matrices are so called because of their use in optimally measuring the individual weights of multiple objects. When the weighing device is a
balance scale
A scale or balance is a device used to measure weight or mass. These are also known as mass scales, weight scales, mass balances, and weight balances.
The traditional scale consists of two plates or bowls suspended at equal distances from a ...
, the
statistical variance of the measurement can be minimized by weighing multiple objects at once, including some objects in the opposite pan of the scale where they subtract from the measurement.
Properties
Some properties are immediate from the definition. If
is a
, then:
* The rows of
are pairwise
orthogonal (that is, every pair of rows you pick from
will be 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
inverse of
*
where
is the
determinant
In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if a ...
of
A weighing matrix is a generalization of
Hadamard matrix
In mathematics, a Hadamard matrix, named after the French mathematician Jacques Hadamard, is a square matrix whose entries are either +1 or −1 and whose rows are mutually orthogonal. In geometric terms, this means that each pair of rows ...
, which does not allow zero entries.
As two special cases, a
is a
Hadamard matrix
In mathematics, a Hadamard matrix, named after the French mathematician Jacques Hadamard, is a square matrix whose entries are either +1 or −1 and whose rows are mutually orthogonal. In geometric terms, this means that each pair of rows ...
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
Tare weight , sometimes called unladen weight, is the weight of an empty vehicle or container.
By subtracting tare weight from gross weight (laden weight), one can determine the weight of the goods carried or contained (the net weight).
Etymolo ...
will result in a final measurement with a variance of
.
It is possible to increase the accuracy of the estimated weights by measuring different subsets of the objects, especially when using a
balance scale
A scale or balance is a device used to measure weight or mass. These are also known as mass scales, weight scales, mass balances, and weight balances.
The traditional scale consists of two plates or bowls suspended at equal distances from a ...
where objects can be put on the opposite measuring pan where they subtract their weight from the measurement.
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
independent and identically distributed
In probability theory and statistics, a collection of random variables is independent and identically distributed if each random variable has the same probability distribution as the others and all are mutually independent. This property is usual ...
with variance
, and let
be a column vector of the true weights of each of the
objects. Then we have:
:
Assuming that
is
non-singular
In the mathematical field of algebraic geometry, a singular point of an algebraic variety is a point that is 'special' (so, singular), in the geometric sense that at this point the tangent space at the variety may not be regularly defined. In ca ...
, we can use the
method of least-squares to calculate an estimate of the true weights:
:
The variance of the estimated
vector cannot be lower than
, and will be minimum if and only if
is a weighing matrix.
Optical measurement
![Measuring Spectrum of Light Using a Weighing Matrix Optical Mask](https://upload.wikimedia.org/wikipedia/commons/e/e9/Measuring_Spectrum_of_Light_Using_a_Weighing_Matrix_Optical_Mask.svg)
Weighing matrices appear in the engineering of spectrometers, image scanners,
and optical multiplexing systems.
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.
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 cyclic, namely, 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.
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, 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 power of a prime and such a circulant weighing matrix can be obtained by signing the complement of a finite projective plane.
Since all
for
have been classified, the first open case is
.
The first open case for a general weighing matrix (certainly not a circulant) is
.
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
≤ 5 as well as all cases where
≤ 15 are also completed.
However, very little has been done beyond this with exception to classifying circulant weighing matrices.
Open Questions
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
{{Matrix classes
Matrix theory
Combinatorics
Design of experiments
Combinatorial design