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Linear Algebra for Quantum Mechanics
Published in Caio Lima Firme, Quantum Mechanics, 2022
If a pair of vectors is orthonormal, they are orthogonal and they are unit vectors. Let us suppose a pair of 2-dimensional orthonormal vectors U(u1, u2) and V(v1,v2). Then, we have: u1v1+u2v2=0u12+u22=1v12+v22=1
Techniques
Published in Aditi Majumder, M. Gopi, Introduction to Visual Computing, 2018
If the rank of A is r < n, then only the first r of the diagonal entries of D are non‐zero as shown in Figure 2.7. Also, U and V are orthonormal matrices, i.e. they represent unit vectors that are orthogonal to each other. One property of orthonormal matrices that is important here is that their inverse is their transpose. Therefore, U-1 = UT and V-1 = VT. Now, let us consider a matrix D⋆ $ D^{{ \star }} $ given by inverting the r × r submatrix of D while the rest of the entries remain zero as in D. Note that D⋆ $ D^{{ \star }} $ is again a diagonal matrix whose top left r × r submatrix has diagonal elements that are reciprocal of those of D. It can be shown that AVD⋆UTb=b $ AVD^{{ \star }} U^{T} b = b $ . Therefore, x=VD⋆UTb $ x = VD^{{ \star }} U^{T} b $ is the solution of Ax = b.
Overview of Digital Communications
Published in M.P. Kennedy, R. Rovatti, G. Setti, Chaotic Electronics in Telecommunications, 2018
Géza Kolumbán, Michael Peter Kennedy
High bandwidth efficiency requires a large signal set. The main advantage of using orthonormal basis functions is that a huge signal set can be generated from a small number of basis functions. Typically, a pair of quadrature sinusoidal signals (a cosine and a sine) is used as the set of basis functions. Since quadrature sinusoidal signals can be generated using a simple phase shifter, it is sufficient to know (or recover) only one sinusoidal signal at the receiver.
A fully-actuated quadcopter representation using quaternions
Published in International Journal of Control, 2022
J. Cariño, H. Abaunza, P. Castillo
A rotation operation can be seen as a rigid transformation on a real three-dimensional space with the following properties: The position of the origin does not change.It is linear.It can be represented by an orthonormal rotation matrix.It does not affect the magnitude.
Detailed wheel/rail geometry processing using the planar contact approach
Published in Vehicle System Dynamics, 2022
Rotation matrices are orthonormal, that is, each column has unit norm and is orthogonal to the other columns, such that . The inverse of a rotation is thus described by the transpose matrix .