Orthogonal Matrix

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orthogonal matrix

[ȯr′thäg·ən·əl ′mā·triks]
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 ?
Then, the optimal source training sequences are obtained from the first [N.sub.S] rows of the circular orthogonal matrix while the optimal relay ones are obtained from the next [N.sub.R] rows, i.e.,
In addition, given an arbitrary orthogonal matrix M [member of] [R.sup.rxr], it holds J(VM) = J(V), so J(V) is a cost function defined on Gr(r, n), and J(V) = J(V) = J [omicron] [pi](V).
Similarly, when wGK[L.sub.l] starts with [x.sub.1] and wGK[L.sub.u] starts with [y.sub.1]= [y.sub.1] = M[x.sub.1]/[[PARALLEL]M[x.sub.1][PARALLEL].sub.K], we have Y = Y and X = XQ with the orthogonal matrix Q satisfying B = QB.
The next step of the algorithm was to actually make use of the orthogonal matrixes created above and match the input PDF with the target PDF.
Taguchi [L.sub.27] orthogonal matrix is formed as described in the Table 4.
Then we always have C = UH where H is positive semidefinite and positive definite if C has full rank, and U is [m.sub.1] x [m.sub.2] column orthogonal matrix. In that last case, U is unique; see Higham [5].
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.
It is shown how to build the orthogonal matrix operator and how to use it in a process of curve reconstruction.
(1) Unit Orthogonal Matrix is a unitized square matrix, the transpose of which equals to its inverse, which can be expressed as Q = [Q.sup.-1], QQ = QQ = E, where E is the identity matrix.
Let H be an M x N real matrix; then there exist an M x M orthogonal matrix U, an N x N orthogonal matrix V, and an M x N diagonal matrix S with diagonal entries [s.sub.1] [greater than or equal to] [s.sub.2] [greater than or equal to] ...
Theorem 2 (equivalence of an orthogonal matrix to a rotation matrix [31]).
The first one is to find the best orthogonal matrix H to remove the couplings among features and build new features.

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