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Algebraic Operations. Transpose of a Matrix
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We have found that, in general, matrix multiplication is not commutative; that is, in general, AB does not equal BA (eqs. (486) and (489) are exceptions to this general rule). Other than the restriction on multiplication, however, most of the other rules of ordinary algebra do also apply to matrix algebra, as follows. First, as pointed out in section 11.1, matrix addition is commutative; that is, A B B A. Also, of course, A B B A. Second, matrix multiplication is distributive with respect to addition; that is, A B C AB AC. Third, matrix multiplication is associative; that is, AB C A BC . Next, with regard to matrix equations, we may add or subtract the same matrix from both sides of such equations without upsetting the equality of the two sides; for example, if A B, then, A C B C. (This assumes, of course, that the matrices all have the same
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CHAPTER 11 Matrix Algebra. Networks
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number of m rows and the same number of n columns, as laid down in the requirement for matrix addition in section 11.1.) We may also multiply both sides of a matrix equation by the same matrix; thus, if B C, then AB AC, provided, of course, that A is conformable with B and C, and that the order of multiplication is the same on both sides of the equation. Let us next de ne that the TRANSPOSE of any given matrix A is another matrix At ,* in which the rows of A are written as the columns of At ; that is, the rst row of A is the rst column of At , the second row of A is the second column of At , and so on. For example, if 2 3 3 1 0 9 6 7 A 42 4 7 6 5 5 then 2 8 3 1 3 5 87 7 7 3 5 1
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3 2 6 1 4 6 At 6 4 0 7 9 6
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Various relationships exist between a matrix and its transpose. The following are the most important ones that you should be aware of. At t A A B t At Bt AB t Bt At det A det At ( reversal rule ) 490 491 492 493
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Note that A can only be a square matrix in eq. (493), because a determinant is de ned as a square array of elements. Problem 237 (a) If A (b) If 2 4 3 9 2 5 7 4 2 1 1 9 0
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! ; then At 3
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Problem 238 Verify that eq. (491) is true for the following two given matrices: ! ! 5 2 3 3 and B A 4 3 7 6
* It should be noted that the alternative notations AT ( A, superscript T ) and A0 ( A prime ) are sometimes used to denote the transpose of a matrix A.
Matrix Algebra. Networks
Problem 239 Verify that eq. (492) is true for the given two matrices: 3 2 2 4 3 0 7 6 A 4 0 6 5 and B 2 1 7 3
Matrix Equations for the Two-Port Network
Let us imagine that we have any kind of linear bilateral network, either active or passive (meaning that it may contain generators, as well as passive elements of R, L, and C), enclosed in a box with a pair of INPUT TERMINALS (1, 1) and a pair of OUTPUT TERMINALS (2, 2), brought out as illustrated in Fig. 277.
Fig. 277
In the above, each pair of terminals is called a port ; thus Fig. 277 represents the general form of a TWO-PORT network, with the input port on the left and the output port on the right. It s also correct to speak of Fig. 277 as a four-terminal network. In all of our work here with such gures, it will be understood that the inputs and outputs are steady-state sinusoidal waves of voltage and current. The letters V, I, and Z that will appear in the equations, will, as usual, denote the complex numbers representing these quantities.* In working with two-port block diagrams, such as Fig. 277, we make use of four measurable external quantities, these being the four quantities V1 , I1 , V2 , and I2 , as shown in the gure. You ll recall that, in network analysis, the rst step is to draw voltage and current arrows to indicate the chosen positive reference directions in the network. Once chosen, the directions of the arrows must not, of course, be changed during the investigation of the network. You ll also recall that, in writing network equations, generator voltages are put on one side of the equations with voltage drops on the other side, the signs of the quantities being determined by the sense in which the arrows are traversed (eqs. (122) and (123) in section 5.8 illustrate this detail). Let us now emphasize that, in the case of two-port block diagrams, it is standard practice to always draw the voltage and current arrows in the directions shown in Fig. 277. The practical application of the notation will be taken up later on, as we progress. Let us assume that we do not know what is actually inside the box in Fig. 277, except that it is linear and bilateral in nature. Our PROBLEM is to nd an EQUIVALENT
" " * In our work here we ll dispense with the usual overscore notation, and simply write V, I, and Z instead of V , I , " and Z, since it will be understood that these quantities represent complex numbers.
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