Orthogonal Matrix


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orthogonal matrix

[ȯr′thäg·ən·əl ′mā·triks]
(mathematics)
A matrix whose inverse and transpose are identical.

Orthogonal Matrix

 

An orthogonal matrix of order n is a matrix

whose product with the transpose A′ gives the identity matrix, that is, AA′ = E and AA = E. The elements of an orthogonal matrix satisfy the relations

or the equivalent relations

The determinant ǀAǀ of an orthogonal matrix is equal to +1 or – 1. The product of two orthogonal matrices is an orthogonal matrix. All orthogonal matrices of order n form, with respect to the operation of multiplication, a group called the orthogonal group. In conversion from one rectangular coordinate system to another, the coefficients aij in the coordinate transformation equations

form an orthogonal matrix.

References in periodicals archive ?
Fortunately, LR is capable of transforming an ill-conditioned matrix closer to an orthogonal matrix by identifying a set of basis that is closer to orthogonality than the columns of a matrix that span the same lattice.
27] orthogonal matrix is formed as described in the Table 4.
END FOR Compute the orthogonal matrix Q in terms of W by Gram-Schmidt orthogonalization [31].
The purpose of this study was to optimize the extraction conditions to simultaneously obtain the pectic polysaccharide from oriental tobacco leaves using orthogonal matrix method.
14) is not guaranteed to be orthogonal, we perform a QR factorization of P to obtain an orthogonal matrix O.
Where, Q(n) and R(n) denote (M-1)*(M-1) orthogonal matrix and (M-1)*(M-1) upper triangular matrix respectively.
Envisaged as the president's dream city, 'Djibloho' uses an orthogonal matrix that takes advantage of the existing road network to connect nearby villages.
In order to verify the precision of designing the experiments based on the L8 orthogonal matrix, a full factorial simulation was also performed
T] = F with PZP = Z constraint, where M = (A, C)U, N = (B, D)U, and P related to U is a given symmetric and orthogonal matrix, then Problem I and Problem II can be transformed equivalently into Problems A and B (which will be stated rearwards), respectively.
Then, there exist an (k +1) x (k +1) orthogonal matrix Q(n) such that
This codebook construction method maintains one full set of original orthogonal matrix in one bin.
For example: Lecturing on "orthogonal matrices" which were introduced by the French mathematician Charles Hermite in 1854 we went further back to 1770 when Euler for the first time considered a system of linear equations in which an orthogonal matrix was used implicitly without knowing anything about matrices in general or orthogonal matrices in particular.

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