By Salas, Saturnino L.; Etgen, Garret J.; Hille, Einar
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Additional resources for Calculus One and Several Variables 10th Ed Textbook + Student Solutions Manual
Example text
Using previous notation, the operator L is as follows 7,(0) M dt »? 10). 14) where A is a constant n x n-matrix, b and c are constant matrices of dimensions nxm and n x / respectively. The asterisk * denotes the transposition in the real case and Hermitian conjugation in the complex case. 3 8 Linear electric circuits. Transfer functions and frequency responses of linear blocks For the sequel we are needed in the following notation ' xi j] = xi, 77 = 1 2 , ... ,?? 13) may be written by ii — x2, &n — 1 — ^n?
However the last relation cannot be true since Xo is an arbitrary vector fromIR". 3). Now we show that condition 2) yields condition 1). Suppose, condition 1) is not true. Then condition 1') is not also true. n such that z*Akb = 0, Vfc = 0 , . . , n - 1. We also prove that z*Anb= 0. By the Cayley identity we have An + S^xA"-1 + .. 9) where Sj are coefficients of characteristic polynomial of the matrix A. 9) z*Anb = - 16) if matrices A and b depend on t. Further we consider the case of a difference kernel only 7(<> r ) =l{t-r). T h e following transformation t j l{t-r)i{T)dr o is often called a convolution operator. 16) is most general. 14) we obtain a(t) = c*eMx0, l{t, T) = c*eA^-Th. Now we discuss the answer to the question: In what sense is there understood a linearity of all the above-mentioned descriptions of block i ? 14) the block is linear since all the equations mentioned are linear. 16). D e f i n i t i o n 2 .