In mathematics, a weighing matrix is a matrix such that

is the transpose is the identity matrix is also called the degree.

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, 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

, then:

• are pairwise orthogonal. Similarly, the columns are pairwise orthogonal.

• non-zero elements.

• , since the definition means that is the inverse of

• is the determinant of

A weighing matrix is a generalization of a Hadamard matrix, which does not allow zero entries. is a Hadamard matrix is equivalent to a conference matrix.

Applications

Experimental design

objects and subtracting the (equally imprecise) tare weight. It is possible to increase the accuracy of the estimated weights by measuring different subsets of the objects, especially when using a balance scale where objects can be put on the opposite measuring pan where they subtract their weight from the measurement.

will have:

be the errors to these measurements each independent and identically distributed objects. Then we have:

has full rank, we can use the method of least squares to calculate an estimate of the true weights:

, and will be minimum if and only if is a weighing matrix.

Optical measurement

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.

Orthogonal designs

An orthogonal design are commuting variables. Additionally, an orthogonal design must satisfy:

.. An orthogonal design of one variable is a weighing matrix, and so the two fields of study are connected. Because of this connection, new orthogonal designs can be discovered by way of weighing matrices.

Examples

is used to represent −1. Here are some examples:

:

:

:

:

which is circulant, i.e. each row is a cyclic shift must be of square 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, the circulant Hadamard conjecture first raised by Ryser, is known to be true for many orders but is still open is a prime power and such a circulant weighing matrix can be obtained by signing the complement of a finite projective plane.

Equivalence

Two weighing matrices are considered to be equivalent. However, very little has been done beyond this with exception to classifying circulant weighing matrices.

Existence

:

1. is the sum of two integer squares.

2. .

3. is the sum of three integer squares.

4. .

5. an integer.

.

's are there?

References