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When a matrix has the same number of rows and columns (ie, n = m), the matrix is said to be square The elements ai,i for are referred to as the diagonal elements of A (ie, diag(A) = [a11 , , an,n ] A square matrix A is called diagonal when aij = 0 for i = j The notation A = diag([a1 , , an ]) will be used to state that A is a diagonal n n matrix with main diagonal elements speci ed by the vector [a1 , , an ] The identity matrix is an example of a diagonal matrix A square matrix A is symmetric if and only if it is square and aij = aji for all i, j [1, n] A square matrix A is skew-symmetric if and only if aij = aji for all i, j [1, n] This implies that a skew-symmetric matrix has diagonal elements all equal to zero Skew symmetric matrices are useful for representing the cross product as a matrix operation, see eqn (B14) There are various special classes of matrices One type of matrix is a permutation matrix A matrix P is a permutation matrix when each row and column of P contains exactly one 1 and all other elements of P are 0 The term vector can have both physical and algebraic meanings For example, the line segment from point P1 to point P2 can be represented as the vector d = P2 P1 Note that this vector is well-de ned without any discussion of reference frames Also, the velocity vector v of a point has well-de ned physical interpretation The velocity vector points in the direction of travel and has magnitude equal to the speed of the point If a frame-of-reference is speci ed, then either d or v can be represented algebraically as a special matrix that only has one column When represented with respect to a speci c reference frame, the i-th element of a vector v is the projection of v onto the i-th coordinate axis of the reference frame Alternatively, for convenience, data or variables are sometimes organized into a column matrix (or vector) without the vector having a clear physical interpretation with respect to any single frame of reference An example of this is the vector of Euler angles and the vector of Euler angle derivatives discussed in Section 2721 When we want to refer to the component in the i-th row of the vector v, we use the notation vi instead of vi,1 Throughout the majority of this text all vectors will be represented as column vectors The only time that.

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Eff = RR/(RR + RL )

following are formal de nitions for scalars and vectors [90]:

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Scalars are quantities that are invariant under coordinate transformation (eg, mass); Vectors are sets of quantities which if organized as v = [v1 , v2 , v3 ] transform between coordinate frames according to vb = Rb va where Rb is a rotation matrix and a a the superscript on v indicates the frame in which it is represented This strict de nition of vectors is required to distinguish which items arranged as [a, b, c] can be transformed between coordinate systems via vector transformation operations (ie, rotations) Some quantities although convenient to organize in a vector-like notation, are not vectors in this strict sense In this strict sense, Euler angle 3-tuple ( , , ) and its derivative d , d , d are not vectors Note for example that each element of dt dt dt the Euler angle derivative 3-tuple is de ned in a di erent reference frame

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this rule will be violated is when, in the middle of text, considerable space would be wasted by writing out the column vector

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which is the ratio of the radiation resistance to the total antenna system resistance As a percentage, Eff% = 100 RR/(RR + RL) High efficiency in a transmitting antenna is obtained when the radiation resistance is much larger than the loss resistance Then most of the power goes into useful EM radiation, and relatively little is dissipated as heat in the earth and in objects surrounding the antenna When the opposite situation exists radiation resistance comparable to, or smaller than, the loss resistance, a transmitting antenna is inefficient This is often the case for extremely short radiators, because they always have low radiation resistance To some extent this can be overcome by careful antenna design and location to minimize loss resistance, but even the most concerted efforts rarely reduce the loss resistance to less than a few ohms If an antenna system has a high loss resistance, it can nevertheless work efficiently if its radiation resistance is high enough When an antenna radiator is just the right length at a given frequency, and if it is constructed of wire or thin tubing, its radiation resistance can be in excess of 1000 This makes it easy to construct an efficient antenna even when the loss resistance is fairly high

The transpose operation, which is indicated by a superscript , simply exchanges the rows and columns of a matrix Therefore, for B = [bi,j ] Rn m , B = [bj,i ] Rm n Two examples may clarify this operation First, 020 171 000 A = 000 314 200 000 is a matrix in R3 2 Second, for v = 200 , 714 v = 000 200 714

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