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From the definition, it is clear that a displacement vector is a polar vector. The velocity vector is a displacement vector (a polar vector) divided by time (a scalar), so is also a polar vector. Likewise, the momentum vector is the velocity vector (a polar vector) times mass (a scalar), so is a polar vector.
Since any signal vector that resides in the signal subspace must be orthogonal to the noise subspace, , it must be that for all the eigenvectors {} = + that spans the noise subspace. In order to measure the degree of orthogonality of e {\displaystyle \mathbf {e} } with respect to all the v i ∈ U N {\displaystyle \mathbf {v} _{i}\in {\mathcal ...
The cross product with respect to a right-handed coordinate system. In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here ), and is denoted by the symbol .
Here the location parameter is a n-dimensional complex vector; the covariance matrix is Hermitian and non-negative definite; and, the relation matrix or pseudo-covariance matrix is symmetric. The complex normal random vector Z {\displaystyle \mathbf {Z} } can now be denoted as Z ∼ C N ( μ , Γ , C ) . {\displaystyle \mathbf {Z} \ \sim ...
The vector can be characterized as a right-singular vector corresponding to a singular value of that is zero. This observation means that if A {\displaystyle \mathbf {A} } is a square matrix and has no vanishing singular value, the equation has no non-zero x {\displaystyle \mathbf {x} } as a solution.
Using Zorn's lemma and the Gram–Schmidt process (or more simply well-ordering and transfinite recursion), one can show that every Hilbert space admits an orthonormal basis; [7] furthermore, any two orthonormal bases of the same space have the same cardinality (this can be proven in a manner akin to that of the proof of the usual dimension ...
where is the four-dimensional totally antisymmetric Levi-Civita symbol;; is the relativistic angular momentum tensor operator ();; is the four-momentum operator.; In the language of exterior algebra, it can be written as the Hodge dual of a trivector, [7]
Since the dual of the pseudoscalar is the product of two "pseudo-quantities", the resulting tensor is a true tensor, and does not change sign upon an inversion of axes. The situation is similar to the situation for pseudovectors and anti-symmetric tensors of order 2. The dual of a pseudovector is an anti-symmetric tensor of order 2 (and vice ...