In a previous paper we investigated Givens transformations applied to quaternion valued matrices. Since arithmetic operations with quaternions are very costly it is desirable to reduce the number of arithmetic operations with quaternions. We show that the Fast Givens transformation, known for the real case, can also be defined for quaternion valued matrices, and we apply this technique to the r...

In a previous paper we investigated Givens transformations applied to quaternion valued matrices. Since arithmetic operations with quaternions are very costly it is desirable to reduce the number of arithmetic operations with quaternions. We show that the Fast Givens transformation, known for the real case, can also be defined for quaternion valued matrices, and we apply this technique to the r...

A general representation of the quaternion gradients presented in the literature is proposed, and an universal update equation for QLMS-like algorithms is obtained. The general update law is used to study the convergence of widely linear (WL) algorithms. It is proved that techniques obtained with a gradient similar to the i-gradient are the fastest-converging in two situations: 1) When the corr...

Two-layered neural networks are well known as autoencoders (AEs) in order to reduce the dimensionality of data. AEs are successfully employed as pre-trained layers of neural networks for classification tasks. Most of the existing studies conceived real-valued AEs in real-valued neural networks. This study investigated complexand quaternion-valued AEs for complexand quaternion-valued neural netw...

Think of a quaternion Q as a vector augmented by a real number to make a four element entity. It has a real part Qcre and a vector part Qcve: If Qcre is zero, Q represents an ordinary vector; if Qcve is zero, it represents an ordinary real number. In any case, the ratio between the real part and the magnitude of the vector part jQcvej plays an important role in rotations, and is conveniently re...

We describe a fast algorithm for the 2-D left-side QDFT which is based on the concept of the tensor representation when the color or four-componnrnt quaternion image is described by a set of 1-D quaternion signals and the 1-D left-side QDFTs over these signals determine values of the 2-D left-side QDFT at corresponding subset of frequency-points. The efficiency of the tensor algorithm for calcu...

We introduce a total order and an absolute value function for dual numbers. The of numbers takes number values, has properties similar to those the real define magnitude quaternion, as number. Based upon these, we extend 1-norm, $$\infty$$ -norm, 2-norm quaternion vectors.

The first mathematician who studied quaternion extensions (H8-extensions for short) was Dedekind [6]; he gave Q( √ (2 + √ 2)(3 + √ 6) ) as an example. The question whether given quadratic or biquadratic number fields can be embedded in a quaternion extension was extensively studied by Rosenblüth [32], Reichardt [31], Witt [36], and Damey and Martinet [5]; see Ledet [19] and the surveys [15] and...