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Appendix A Matrices and Determinants A.I DEFINITIONS A rectangular array of ordered elements (numbers, functions or just symbols) is known as a matrix. The literal form of a matrix in general is written as A= We use boldface type to represent a matrix, and we enclose the array itself in square brackets. The horizontal lines are called rows and the vertical lines are called columns. Each element is associated with its location in the matrix. Thus the element a;j is defined as the element located in the ith row and the jth column. Using this notation, we may also use the notation [a;jJmxn to identify a matrix of order m x n, i.e. a matrix having m rows (the number of rows is given first) and n columns. Some frequently used matrices have special names. A matrix of one column but any number of rows is known as a column matrix or a column vector. Frequently, for such a matrix, only a single subscript is used for the elements of the array. Another type of matrix which is given a special name is one which contains only a single row. This is called a row matrix, or a row vector. A matrix which has the same nu.mber of rows and columns, i.e. m = n, is a square matrix of order (n x n) or just of order n. The main or prin­ ciple diagonal of a square matrix consists of the elements all. a22 • .•.• ann. A square matrix in which all elements except those of the principal diagonal 688 Appendix A are zero is known as a diagonal matrix. If, in addition, all elements of a diag­ onal matrix are unity, the matrix is known as a unit or identity matrix, U. If all elements of a matrix are zero, aij = 0, the matrix is called a zero matrix, O. A subclass of a square matrix which is frequently encountered in circuit analysis is a symmetric matrix. The elements of such a matrix satisfy the equality au = aji for all values of i and j, or in other words, this matrix is symmetrical about the main diagonal. For instance, the matrices A-_[ 2 -1 IJ , -13 04] , -1 0 3 o 2 are of orders (2 x 3) for A, (3 x 3) for Band (3 x 1) for C. Matrix B is a square and symmetric matrix, and matrix C is a column matrix or vector. The matrix is a unit matrix (also square, diagonal and symmetric). Finally, it should be noted that a matrix of the order (1 x 1), which consists of a single element, is simply refered to as a scalar. A.2 MATRIXEQUALITY Two matrixes are equal if and only if (1) they are of the same order, and (2) each element of one matrix is equal to its associated (placed in the row of the same number and the colunm of the same number) element in the other matrix. Thus, for two matrices, A and B, of the same order and with elements aij and bi}> respectively, if A = B, then all the elements have to be equal, i.e. aij = bij for all values of i andj. Thus, for example, for the matrices a12 b12 [al1 al3 J and [bl1 bl3 J a21 a22 a23 b21 b22 b23 to be equal, we must have al1 = bll, al2 = b12 , a13 = b13 a21 = b2!. a22 = b22 , a23 = b23 · Appendix A 689 On the other hand, and can never be equal no matter what the relationships between aij and bij are, as condition (1) above is not satisfied. A.3 MATRIX EQUATIONS The definition of equality of matrices allows us to express sets of equations in a compact form. Thus, for instance, ifF denotes a column matrix alli! + a12i2 + a\3i3] F = [ a21~1 + a22~2 + a23~3 a31 11 + a32 12 + a33 13 (note that the sum of three addends in each row is actually one matrix element) and V denotes a vector then the matrix equation all il + a12i2 + a\3i3 ] [VI] [ a21~1 + a22~2 + a23~3 = V2 (A.la) a31 11 + a3212 + a33 13 V3 may be written simply as F=V. Moreover, either form of this matrix equation is equivalent to the three equations allil + a22i2 + al3i3 = VI a21 il + a22i2 + a23 i3 = V2 (A.lb) a3li l + a32 i2 + a33 i3 = V3 690 Appendix A A.4 ADDITION AND SUBTRACTION OF MATRICES If two matrices A and B are of the same order, i.e. have the same number of rows and the same number of columns, we may determine their sum by adding the corresponding elements. Thus if the elements of A are aij and those ofB are bij, then the elements of the resulting matrix Care for all i andj , (A.2a) and (A.2b) Clearly A + B = B + A for matrices. Note that if two matrices differ in their number of rows or their number of columns, their sum is not defined. As an example of matrix addition, consider the following 1 2 5] [2 3 -1] [3 5 4] [3 4 6 + 7 -8 1 = 10 -4 7 . Subtraction is similarly defined, i.e. for all i andj, (A.3a) and C=A-B (A.3b) For example, A-B=[ 1 20]_[2 1]=[-1 1 -1]. -1 3 4 3 2 -1 -4 1 5 A.5 MULTIPLICATION BY A SCALAR A real or complex number is referred to as a scalar to distinguish it from a matrix. The multiplication of a matrix by a scalar means that every element of the matrix is multiplied by the scalar. Thus, if k is a scalar and A is a matrix with elements ai}, the elements of the matrix kA are kai}: kall ka12 kal n ka21 ka22 ka2n kA= (A.4) Appendix A 691 Consider the following example: +Abl1 o 0] [all al3 ] b22 0 = a21 a23 , o b33 a31 a33 +A.b33 which in short form is A + A.B. A.6 MULTIPLICATION OF MATRICES We first consider multiplication of a square matrix by a column matrix. It was shown (see Section A.3) that the sets of simultaneous equations Al b might be represented by the matrix equation A.la. We ca.n next represent the three currents in equation A.l as a column matrix Then the matrix equation Ala can be written, by analogue to the scalar equation ai = v, as AI=V, (A5) where A is a square matrix of the coefficients of the equation, aij' In order to use the above notation, the mUltiplication operation, indicated as Al, must be defined in order to produce the correct members for equations A.lb. Therefore the matrix multiplication in equation A6 must be interpreted as requiring that each element of the first row of the first matrix A must be mUltiplied by the corresponding element of the second, column matrix I, and the sum of these products should be taken as a first: element of the resulting, i.e. of the product, matrix. Similarly, the second element of the resulting matrix will be the sum of the products of each element of the second row of matrix A with the corresponding element of the matrix I, and so forth. For the case where A is an nth-order square matrix and Y and X are column matrices with n rows, the matrix equation Y=AX (A.6) is simply defined by the relation n Yi = Laikxk, i = 1,2, ... , n (A 7) k=l 692 Appendix A It is very important to emphasize that the multiplication of an (n x n)­ order square matrix by a column matrix may be performed if and only if the column matrix has n elements, or, which is the same, n rows. In other words the number of columns of the first matrix, A, has to be the same as the number of rows of the second matrix, X. Note, that the product matrix, Y, is also the column matrix of the same order, n. Note also, that if these two numbers (columns in the first matrix and rows in the second matrix) are different, the product matrix AX does not exist. We may now generalize the multiplication rule for any two matrices. Thus, the multiplication of two matrices A and B is defined only if the number of columns of A is equal to the number of rows ofB. If A is of order (m x n) and B is of order (n x p) (such a pair of matrices is said to be conform­ able for multiplication), then the product AB is a matrix C of order (m x p): (A. 8) The elements of C are found from the elements of A and B by multiplying the ith row elements of A and the correspondingjth column elements of B and summing these products to give cij: n cij = ailblj + ai2b2j + ... + ainbnj = Laikbkj, (A.9) k=l where i = 1,2, ... , m,j = 1,2, ... , p. By contrast with scalar multiplication, where ab = ba, matrix multiplica­ tion is not commutative in general, i.e. AB#BA usually, even when BA is defined, which means that their orders satisfy equation A.8. Thus the order of matrix multiplications is of fundamental importance. Matrix C in equation A.8 is the product of matrix A into matrix B, i.e. B is said to be pre-multiplied by A, or A is said to be post-multi­ plied byB.
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