realignment using the AIR program35 and inspected for motion-related or technically induced artifacts; multivariate linear fitting of the 64 elements of the diffusion ten- sor at each voxel; tensor diagonalizat
where for the third equality we have used the fact that the Pauli matrices have zero trace, and for the last one that they are mutually orthogonal. It is possible to write the above expressions in a more compact form by introducing the polar angle\(\theta \)between\(\mathbf{n}\)and\(...
In addition to lexico-semantic tasks, the involvement of motor and somatosensory cortices in tasks of motor imagery- mentally simulated movements that do not lead to overt motor production25- has also been explored. The mental rotation of body parts, for instance, has been used to investigate wh...
realignment using the AIR program35 and inspected for motion-related or technically induced artifacts; multivariate linear fitting of the 64 elements of the diffusion ten- sor at each voxel; tensor diagonalization by matrix rotation to obtain the three eigenvalues and eigenvectors that were used to ...
Do 90^o rotation matrices have eigenvectors? Do non square matrices have eigenvalues? For what value of x will the following matrices be inverses of one another: [9 16] and [-7 16] 4 7 x -9 Determine if matrices A and B are inverses of one another. A=\begin{bmatrix} 1 &0 &3...
(b)ForsomerotationmatricesR,thefunctionRPYRINV(R)isnumericallyill-defined.Whatis wrong? (c)That[abc]=RPYRINV(RPYR([abc]))forsomea,b,c—e.g.try[4π00]. (d)R=RPYR(RPYRINV(R))foralla,b,cwhereRPYRINV()iswell-defined.
the eigenvectors are not necessarily orthogonal. Furthermore, one typically normalizes the eigenvec- tors prior to setting up a rotation matrix. Com- puting the normalization factor is somewhat messy but, as seen below, not necessary. We find the fol- lowing matrices will diagonalize A: S ...
The most general form of the rotation R p is R p = a p n p × a p n p , (9) where |a p | = 1, a p ⊥ n p and we treat vectors in (9) as column matrices. Vectors |p, λ are eigenvectors of the four-momentum operators ˆ P µ |p, λ = p µ |p, λ ,...
Spin Dynamics: Basics of Nuclear Magnetic Resonance, Second Edition: - Follows a multi-disciplinary approach, concentrating on basic principles and concepts rather than the more practical aspects - Assumes only a basic knowledge of complex numbers and matrices - Takes a strong pedagogical approach, ...
), the 90-degree rotation matrix acting on R 2 . It is not diagonalizable since there are no eigenvectors: a rotation in R 2 sends no nonzero vector to a scalar multiple of itself. This geometric reason is complemented by an algebraic reason: the characteristic polynomial T 2 +1 of R ...