Signal Processing and Linear Systems-B.P.Lathi copy

Thus an ordered n tuple xl x 2 x n represents an n

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Unformatted text preview: a column ( column v ector): B.6 Vectors and Matrices 35 A m atrix w ith m rows a nd n columns is called a matrix of t he order (m, n ) or an ( m X n ) m atrix. For t he special case where m = n , t he matrix is called a s quare m atrix of order n . I t should be stressed a t t his point t hat a m atrix is n ot a number such as a determinant, b ut a n array of numbers arranged in a particular order. I t is convenient t o a bbreviate the representation of matrix A in Eq. (B. 50) w ith the form ( aij ) mxn, implying a m atrix of order m x n with a ij as its i jth element. In practice, when the order m x n is understood or need not be specified, the notation can be abbreviated t o ( aij). Note t hat t he first index i of a ij indicates the row a nd t he second index j indicates t he column of t he element a ij i n m atrix A . T he simultaneous equations (B.4S) may now be expressed in a symbolic form as y =Ax x= (B.51) or YI au a 12 a ln Xl Y2 a 21 a 22 a 2n X2 Xn S imultaneous linear equations can be viewed as the transformation of one vector into another. Consider, for example, the n simultaneous linear equations (B.52) Y rn (B.4S) Y rn = a mlXI + a rn2X 2 + ., . + a mnxn B.6-1 I f we define two column vectors x and y as a mI a mn am2 Xn E quation (B. 51) is t he symbolic representation of Eq. (B.4S). As yet, we have not defined the operation of t he multiplication of a m atrix by a vector. T he q uantity A x is n ot meaningful until we define such an operation. Some Definitions and Properties A s quare matrix whose elements are zero everywhere except on the main diagonal is a d iagonal m atrix. An example of a diagonal matrix is YI Y2 x= y= (B.49) Ym t hen Eqs. (B.48) may be viewed as the relationship or the function t hat transforms vector x into vector y. Such a t ransformation is called the l inear t ransformation of vectors. In o rder t o perform a linear transformation, we need to define the array of coefficients a ij a ppearing in Eqs. (B.4S). T his array is called a m atrix a nd is denoted by A for convenience: A diagonal matrix with unity for all its diagonal elements is called an i dentity m atrix or a u nit m atrix, d enoted by I . Note t hat this is a s quare matrix: 100 o (B.50) 1 0 0 00 1 o 000 1= A= 0 1 (B.53) T he order of the unit m atrix is sometimes indicated by a subscript. Thus, In represents the n x n u nit m atrix (or identity matrix). However, we shall omit the subscript. T he order of the u nit m atrix will be understood from t he context. 36 B ackground B.6 Vectors a nd M atrices A m atrix h aving all its elements zero is a z ero m atrix. A s quare m atrix A is a s ymmetric m atrix if a ij = a ji ( symmetry a bout t he m ain diagonal). Two matrices o f t he same order are said t o b e e qual if they are equal element by element. Thus, i f A +B a nd ~ ~::: ( ami t hen A = B only i f a ij = bij for all i a nd j . I f t he rows a nd columns of a n m x n m atrix A are interchanged so t hat t he elements in t he i th row now become t he elements of t h...
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This note was uploaded on 04/14/2013 for the course ENG 350 taught by Professor Bayliss during the Spring '13 term at Northwestern.

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