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is a matrix in R1 3 which is often referred to as a row vector When a matrix is symmetric, it is straightforward to show that A = A When a matrix is skew-symmetric, A = A The addition of matrices is straightforward and is de ned componentwise When we say that C = A + B for A, B, C Rn m we mean that ci,j = ai,j + bi,j for i [1, n] and j [1, m] In order to add two matrices, their dimensions must be identical Therefore, with the matrices given previously A and v cannot be added If we de ne 100 171 B = 003 200 014 214 then A and B cannot be added or subtracted, but 120 000 A + B = 003 200 300 414 Subtraction of matrices is de ned similarly The symmetric part of a square matrix A is de ned by As = 1 A+A 2 1 A A 2 (B1)

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A half wavelength in free space is given by the equation: Lft = 492/fo where Lft is the linear distance in feet, and fo is the fundamental frequency, in megahertz, at which the antenna exhibits resonance A half wavelength in meters, Lm, is given by: Lm = 150/fo For ordinary wire, the results as obtained above should be multiplied by a velocity factor, v, of 095 (95 percent) For tubing or large-diameter wire, v can range down to about 090 (90 percent)

The anti-symmetric part of A is de ned by Aa = (B2)

Open Dipole An open dipole or doublet is a half-wavelength ( /2) radiator fed at the center (Fig 27-2A) Each leg of the antenna is a quarter wavelength ( /4) long For a straight wire radiator, the length Lft, in feet, at a design frequency fo, in megahertz, for a center-fed, /2 dipole is approximately:

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With these de nitions, any square matrix can be represented as the sum of its symmetric and anti-symmetric parts A = As + Aa (B3)

Matrix multiplication is particularly important When A Rn m and B Rm p , then their product C = AB Rn p is de ned as ci,j = m k=1 ai,k bk,j Note that the number of columns in A and the number of rows in B, referred to as the inner dimension of the product, must be identical; otherwise the two matrices cannot be multiplied Using the vectors and matrices de ned above, we have the following: AB A B = = 02396 63776 14300 13559 not a valid operation

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These values assume v = 095 In free space, the impedance at the feed point is a pure resistance of approximately 73 This represents the radiation resistance alone, in the absence of reactance at the resonant frequency

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Matrix multiplication does not commute: AB = BA Matrix division is unde ned Matrix multiplication does have the following useful formulas: (AB) C = (A + B) C = (ABC) = A (BC) AC + BC C B A Associative property Distributive property (B4) (B5) (B6)

In the certain derivations it will be convenient to insert appropriately dimensioned identity matrices The reader should be comfortable with formulas such as the following: ABC = AIm BIp C Ax = AIm x (B7) (B8)

The subscript denoting the dimension will typically be dropped to simplify the notation, as it is usually clear from the context A square matrix A is an orthogonal matrix if and only if AA = A A is a diagonal matrix A square matrix A is an orthonormal matrix if and only if AA = A A = I Often authors use the term orthogonal to mean orthonormal Let Rb represent the vector transformation from reference a frame a to reference frame b The inverse transformation from reference frame b to reference frame a can be shown to be Ra = (Rb ) 1 = (Rb ) a a b Then (Rb ) Rb = Rb (Rb ) = I This shows that Rb and Ra are orthonora a a a a b mal matrix The scalar product of two vectors u, v RN is de ned to be

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