Building an Orthonormal Basis from a Unit Vector
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Abstract We show how to easily create a right-handed orthonormal basis, given a unit vector, in 2−, 3−, and 4-space.Keywords:
Basis (linear algebra)
Orthonormality
Unit vector
Subspace tracking is important for many communications and signal processing tasks. Many of the simplest subspace tracking methods, however, only approximately maintain the orthonormality of the subspace matrix estimate. In this paper, we describe a generalized procedure for designing principal subspace tracking algorithms that maintains the orthonormality of the subspace matrix estimate in a numerically-robust fashion. Our generalized algorithm families include two orthonormal update principal subspace tracking algorithms as special cases, and all but one of the new algorithms are computationally-simpler than these existing approaches. Moreover, we show how to modify these algorithms to perform minor subspace tracking in a numerically-stable fashion. Simulations verify the numerically-robust performances of the algorithms in principal and minor subspace tracking tasks, respectively.
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Tracking (education)
Signal subspace
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Matrix (chemical analysis)
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Reinforcement Learning with Orthonormal Basis Adaptation Based on Activity-Oriented Index Allocation
An orthonormal basis adaptation method for function approximation was developed and applied to reinforcement learning with multi-dimensional continuous state space. First, a basis used for linear function approximation of a control function is set to an orthonormal basis. Next, basis elements with small activities are replaced with other candidate elements as learning progresses. As this replacement is repeated, the number of basis elements with large activities increases. Example chaos control problems for multiple logistic maps were solved, demonstrating that the method for adapting an orthonormal basis can modify a basis while holding the orthonormality in accordance with changes in the environment to improve the performance of reinforcement learning and to eliminate the adverse effects of redundant noisy states.
Orthonormality
Basis (linear algebra)
Basis function
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Orthonormality
Basis (linear algebra)
Truncation (statistics)
Basis function
Truncation error
Constant (computer programming)
Orthogonal functions
Function Approximation
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Abstract We show how to easily create a right-handed orthonormal basis, given a unit vector, in 2−, 3−, and 4-space.
Basis (linear algebra)
Orthonormality
Unit vector
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Orthonormality
Unit vector
Cross product
Direction vector
Null vector
Inner product space
Basis (linear algebra)
Scalar multiplication
Vector-valued function
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Orthonormal basis (plural orthonormal bases): a set B of vectors in Euclidean or Hilbert space such that every vector can be written as a (finite or infinite) linear combination of vectors from B, while all vectors from B have length 1 and any two of them are orthogonal. The number of vectors in B then equals the dimension of the space, which can be finite or infinite.
Basis (linear algebra)
Orthonormality
Unit vector
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Abstract The method of orthonormality‐constrained variation is extended using a dual‐basis set instead of a single orthonormal basis. The complete and the partial variation methods are discussed and applied to electronic systems. It is found that the present formulation leads to the most general equation in the coupling operator method.
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Basis (linear algebra)
Variation (astronomy)
Operator (biology)
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Given an orthonormal basis $ {\mathcal V}= \{v_j\} _{j\in N}$ in a separable Hilbert space $H$ and a set of unit vectors $ {\mathcal B}=\{w_j\}_{j\in N}$, we consider the sets $ {\mathcal B}_N$ obtained by replacing the vectors $v_1, ...,\, v_N$ with vectors $w_1,\, ...,\, w_N$. We show necessary and sufficient conditions that ensure that the sets $ {\mathcal B}_N$ are Riesz bases of $H$ and we estimate the frame constants of the $ {\mathcal B}_N$. Then, we prove conditions that ensure that $ {\mathcal B}$ is a Riesz basis. Applications to the construction of exponential bases on domains of $ R^d$ are also presented.
Basis (linear algebra)
Riesz representation theorem
Unit vector
Riesz potential
Mutually unbiased bases
Orthonormality
Riesz transform
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Abstract The space‐vector theory, which allows to describe three linearly dependent quantities (e. g. three voltages or three currents) using only two linearly independent quantities represented in an orthonormal coordinate system, is widely used in the fields of AC machine theory, power definitions and active filtering. It can be used very effectively in case of circuits with only three terminals. Problems occur in case of four terminals, when four linearly dependent quantities (e. g. four currents, the sum of which is always zero) exist. In this case a zero‐sequence quantity can be additionally introduced, which has to be treated separately with special equations. Vector operations like cross product or dot product, which are very useful to calculate e. g. power quantities, can no longer be used in the general case. This paper presents a method transforming any system represented by four linearly dependent quantities or by three linearly independent quantities into a three‐dimensional orthonormal coordinate system. All four members of a set of linearly dependent quantities are treated equally. We suggest to call this representation Hyper Space Vector (HSV). A lot of vector operations can be applied without problems to HSV, making calculations much easier and more graphic.
Orthonormality
Sequence (biology)
Unit vector
Representation
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Orthonormality
Inner product space
Unit vector
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Vector-valued function
Basis (linear algebra)
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