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Linear Systems
Published in Jeffery J. Leader, Numerical Analysis and Scientific Computation, 2022
and since R is a square upper triangular matrix this can be solved by back substitution, provided that no diagonal entry of R is zero. Since an orthogonal matrix has determinant ±1, we have det(A)=det(QR)=det(Q)det(R)=±det(R).
Geometry of Rotation
Published in Kenichi Kanatani, 3D Rotations, 2020
The basis {e1, e2, e3} is orthonormal, i.e., consisting of mutually orthogonal unit vectors. Since norms and angles are preserved by a rotation, {r1, r2, r3} is also an orthonormal system. A matrix with orthonormal columns is called an orthogonal matrix. Since {r1, r2, r3} has the same sense as {e1, e2, e3}, the matrix R has determinant 1. An orthogonal matrix of determinant 1 is called a rotation matrix. Thus, a rotation is represented by a rotation matrix. If {r1, r2, r3} of Eq. (2.3) are a left-handed system, R has determinant −1. As pointed out earlier, a linear mapping specified by an orthogonal matrix of determinant −1 is a composition of a rotation and a reflection, i.e., an improper rotation. Thus, an orthogonal matrix represents either a proper rotation or an improper rotation.
The Integer Discrete Cosine Transform
Published in Humberto Ochoa-Domínguez, K. R. Rao, Discrete Cosine Transform, 2019
Humberto Ochoa-Domínguez, K. R. Rao
Transformation is one popular technique for data compression. The first step is to decorrelate the pixels by applying a transform. The most significant transform coefficients are retained, with the rest set to zero, and inverse transformation is applied to recover the original image. Therefore, it is important that the transform process be simple and fast. The family of orthogonal transforms is well suited for this application because the inverse of an orthogonal matrix is its transpose. The DCTDCT is widely accepted as having a high efficiency [387]. Before H.264/AVCH.264/AVC, the transformation in video coding standards was specified in floating point operations. Nevertheless, errors were introduce in the recovered image because of the different floating point implementations of encoders and decoders. In H.264/AVCH.264/AVC [204] and HEVC [209]HEVC video coding standards, the inverse transforms are specified in integer arithmetic to induce almost the same reconstructed image at the decoder side. The normative is specified in [199]. In 2006 the moving picture experts group (MPEG)Moving picture experts group approved the replacement specification ISO/IEC 23002-1 [207].
Fractional-order two-input two-output process identification based on Haar operational matrix
Published in International Journal of Systems Science, 2021
Haar wavelet operational matrix of integration can be obtained using the generalised operational matrix of integration . Let the fractional integration of the is expressed by where square matrix denotes Haar wavelet operational matrix of FOI. Haar wavelet functions can be expanded into an M-terms block pulse functions, as they are piecewise constant, by Therefore, using (7) and (10), one can obtain We know, an orthogonal matrix is always invertible and its inverse is same as its transpose. As the M-square Haar matrix is an orthogonal matrix so its inverse always exists.
Multiple-image encryption scheme via compressive sensing and orthogonal encoding based on double random phase encoding
Published in Journal of Modern Optics, 2018
Dongming Huo, Xin Zhou, Luozhi Zhang, Yuanyuan Zhou, Huaidong Li, Shaoliang Yi
Orthogonal encoding involves orthogonal matrix, such as the identity matrix, the Hadamard matrix, the wavelet transform matrix, the Fourier transform matrix, etc. The order of the identity matrix can be any positive integer, however, the identity matrix as a key is so simple that it can be easily breached. Other types of orthogonal matrices are more complex than the identity matrix, which can be used as the key, but in general, the order of them is (p is a positive integer).
Recursive orthogonal least squares based adaptive control of a polymerisation reactor
Published in Indian Chemical Engineer, 2019
Sarthak Tiwari, Purushottam Sawant, Imran Rahman
Applying the gram-Schmidt method to orthogonally decompose as:where is an orthogonal matrix and is an upper triangular matrix.