The collected mathematical papers: Geometrical optics by Hamilton W.R.

By Hamilton W.R.

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Extra info for The collected mathematical papers: Geometrical optics

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89) for −π/2 < θ < π/2. 5). This means that the jω-axis poles of September 9, 2015 32 8:50 Broadband Matching... - 9in x 6in b1862-ch01 BROADBAND MATCHING A(s) are simple, and that K denotes the matrix of the residues of the elements of A(s) evaluated at the pole jω0 . 6). 8). Thus, condition (iii) is satisfied. To show that condition (iv) cannot be violated, we observe that since A(s) is positive-real, X ∗Ah (s)X is nonnegative, and is a continuous function of σ and ω for all s in the open RHS.

Page 16 September 9, 2015 8:50 Broadband Matching... - 9in x 6in FOUNDATIONS OF NETWORK THEORY b1862-ch01 page 17 17 ˆ First, we show that each column sum of Y(s) is equal to zero. 38) to yield n+1 n+1 n+1 yˆ jiV i = i=1 j=1 I m = 0. 40) m=1 The last equation is obtained by applying Kirchhoff’s current law. Setting all the terminal voltages to zero except the kth one gives n+1 yˆ jk = 0. 41) j=1 Since V k = 0, it follows that the sum of the elements of each column ˆ equals zero. Thus, the indefinite-admittance matrix is singular.

The example illustrates a procedure for determining whether or not a given hermitian matrix is positive or nonnegative definite by converting a hermitian form into an expression involving squares of magnitudes. This process is long and tedious and really does not help very much. In the following, we present simple criteria for testing a given hermitian matrix for positive-definiteness and nonnegativedefiniteness in terms of the elements of the matrix. September 9, 2015 8:50 26 Broadband Matching...