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B1 SIMULTANEOUS EQUATIONS AND THE CHARACTERISTIC MATRIX Many engineering systems are described by a set of linearly independent simultaneous equations of the form y1 a11 x1 a12 x2 a13 x3 a1n xn y2 a21 x1 a22 x2 a23 x3 a2n xn ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ym am1 x1 am2 x2 am3 x3 amn xn where the xj are the independent variables, the yi the dependent variables, and the aij are the coe cients of the independent variables. The aij may be constants or functions of some parameter. A more convenient form may be obtained for the above equations by expressing them in matrix form. 2 3 2 32 3 y1 x1 a11 a12 a13 . . . a1n 6 y2 7 6 a21 a22 a23 . . . a2n 76 x2 7 6 7 6 76 7 4 . . . 5 4 . . . . . . . . . . . . . . . 54 . . . 5 ym am1 am2 am3 . . . amn xn or Y AX, by a suitable de nition of the product AX (see Section B3). Matrix A aij is called the characteristic matrix of the system; its order or dimension is denoted as d A m n where m is the number of rows and n is the number of columns.

(a) Let the subscript 1 denote quantities at T1 258C, and 2 denote them at any other temperature T2 . Then, according to Section 5.2, ICBO2 2 T2 25 =10 ICBO1 VBEQ2 VBEQ1 VBEQ VBEQ1 0:002 T2 25

A number line is a line on which distances from a point are marked off in equal units, positively in one direction and negatively in the other. The origin is the zero point from which distances are measured. Figure 12-1 shows a horizontal number line.

Row matrix. A matrix which may contain any number of columns but only one row; d A 1 n. Also called a row vector. Column matrix. A matrix which may contain any number of rows but only one column; d A m 1. Also called a column vector. Diagonal matrix. A matrix whose nonzero elements are all on the principal diagonal. 455

CHAP. 5]

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Unit matrix. A diagonal matrix having every diagonal element unity. Null matrix. A matrix in which every element is zero. Square matrix. A matrix in which the number of rows is equal to the number of columns; d A n n. Symmetric matrix. Given 2 3 a11 a12 a13 . . . a1n 6a a22 a23 . . . a2n 7 7 d A m n A 6 21 4 ... ... ... ... ... 5 am1 am2 am3 . . . amn the transpose of A is a11 6 a12 6 AT 6 a13 6 4 ... a1n 2 a21 a22 a23 ... a2n a31 a32 a33 ... a3n ... ... ... ... ... 3 am1 am2 7 7 am3 7 7 ... 5 amn vice versa.

+5 II +4 +3 +2 +1 5 4 3 2 1 3 Q( 4, 3) 4 1 2 3 4 5

Thus, the rows of A are the columns of AT , and symmetric matrix must then be square. Hermitian matrix. Given 2 a11 a12 6 a21 a22 A 6 4 ... ... am1 am2 the conjugate of A is a 11 6 a A 6 21 4 ... a m1 2 a 12 a 22 ... a m2

Fig. 12-1

Hence, by (2) of Problem 5.15, ICQ2 1 VEE VBEQ1 0:002 T2 25 2 T2 25 =10 ICBO1 RE 1

... ... ... ...

Fig. 12-2

(b) At T2 1258C, (1) gives us ICQ2 51 5 0:7 0:002 125 25 2 125 25 =10 0:5 10 6 1:53 0:512 2:042 mA 50 3 103

Matrix A is hermitian if A A T ; that is, a hermitian matrix is a square matrix with real elements on the main diagonal and complex conjugate elements occupying positions that are mirror images in the main diagonal. Note that A T AT . Nonsingular matrix. An n n square matrix A is nonsingular (or invertible) if there exists an n n square matrix B such that AB BA I where I is the n n unit matrix. The matrix B is called the inverse of the nonsingular matrix A, and we write B A 1 . If A is nonsingular, the matrix equation Y AX of Section B1 has, for any Y, the unique solution X A 1 Y

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