Direction cosine matrix

The transpose of the direction-cosine matrix is also its inverse, Nr = N 1. Therefore... 

The relatively simple inversion of the direction-cosine matrix can be seen from trigonometric identities among direction cosines, stated as... 

Number of molecules with energy n and at least m quanta in the critical vibrational mode Extensive property of a system Direction cosine matrix... 

From the close relationship between the direction cosine matrix elements diagonal in J, matrix elements of the components of the angular momentum, J,K,M Jv J,K My, we have ... 

This justifies the general statement of Eq. (2.3.9) that, for all operators A obeying anomalous molecule-fixed commutation rules, Eq. (2.3.6) (any operator that includes R), the matrix elements of A are real and positive but that A+ and A- act exactly as lowering and raising operators, respectively. This reversal of the lowering/raising roles of A 1 may be viewed as arising from the direction cosine matrix elements [Eq. (2.3.17)] rather than any unusual property of the operator A. 

For a linear molecule, the position of the symmetry axis (the molecule-fixed. z-axis) in space is specified by only two Euler angles, / and 7, which are respectively identical to the spherical polar coordinates 6 and (see Fig. 2.4). The third Euler angle, a, which specifies the orientation of the molecule-fixed x- and y-axes, is unaffected by molecular rotation but appears explicitly as an O- dependent phase factor in the rotational basis functions [Eq. (2.3.41)]. Cartesian coordinates in space- and molecule-fixed systems are related by the geometrical transformation represented by the 3x3 direction cosine matrix (Wilson et al., 1980, p. 286). The direction cosine matrix a given by Hougen (1970, p. 18) is obtained by setting a = 7t/2 (notation of Wilson et al, 1980 6 fi,4)=, x = oi 7t/2). The direction cosine matrix is expressed in terms of sines and cosines of 9 and 4>. Matrix elements (J M O la JMQ), evaluated in the JMQ) basis, of the direction cosines, are expressed in terms of the J, M, and quantum numbers. The direction cosine matrix elements of Hougen (1970, p. 31), Townes and Schawlow (1955, p. 96), and Table 2.1 assume the basis set definition derived from Eq. (2.3.40) and the phase choice a = 7t/2 ... 

Equations (6.3.22a - 6.3.22b) can be simplified, for the case of isotropically oriented molecules, unpolarized radiation, and zero external magnetic or electric fields, by summing over M (see Hougen, 1970, p. 39).f The resultant M-independent (f2 J a f]J) direction cosine matrix elements are listed in Table 6.1. Note that the a Ail = Tl matrix elements have opposite signs for P versus R transitions, whereas the az Afl = 0 matrix elements have the same signs for P and R transitions. 

Table 6.1 M-Independent (fl J) Direction Cosine Matrix Elements...

This expression separates the vibrational dependence, incorporated in p from the rotational dependence in the direction cosine matrix elements /a. Our ability to apply the transformation of Eq. (34) lies in our ability to rewrite both the vibrational contributions ptt and the rotational contributions /a to the right side of Eq. (36) in terms of raising and lowering operators (50). 

Let the bond make the polar and azimuthal angles 6 and 0 with respect to the axes OX1X2X2 of the unit and choose the Oxi axis so that it lies in the 0X1X2 plane. This can always be done because of the cylindrical symmetry of the tensor. The direction-cosine matrix linking the sets of axes OV1V2X3 and OX 1X2X2 is then... 

If now the rotational quantization were neglected, rnne-, ivhich assumes both positive and negative values, might be omitted from (12) and (15) and the direction cosine matrix elements replaced b their classical average over all directions. 

Here b means the IMU body frame e denotes the ECEF frame i indicates the inertial frame Cl is the Direction Cosine Matrix (DCM) from body frame to ECEF frame, ft is the skew-symmetric matrix for angular rate measurements is the vector of acceleration measurements from the accelerometers. F is the system matrix applied in the ECEF frame is the distance from the earth geometric center to the earth surface g is the local gravity is the position of the IMU in ECEF. The noise vector w contains, in the indicated order, gyroscope bias, acceleration bias, acceleration noise, angular rate noise, receiver clock error and receiver clock rate noise. These noise terms are described by the error covariance matrix Q in the Kalman filter routine ... 

The perturbation connects adjacent J states. Evaluation of the direction cosine matrix elements in the basis J, M) and the energy level differences gives... 

The matrix elements < n I p I n" ) are primarily associated with vibrational transitions. Since, however, the rotational quantum numbers R and R" also change, in a consistent treatment the transition probabilities of all rotational components of a vibrational absorption band must be evaluated. Usually, an approximate approach is adopted [4] and the direction cosine matrix elements replaced by classical averages over the cosines. These are equal to [3-5]. 

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