= [a ij ] 2 3. Square matrix A square matrix is one that has equal number of rows and columns, that is n = m. Some examples of square matrices are


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1 This document deals with the fundamentals of matrix algebra and is adapted from B.C. Kuo, Linear Networks and Systems, McGraw Hill, It is presented here for educational purposes. 1 Introduction In the study of modern network and system analysis, long and complicated expressions are encountered. To save time and space, it is often desirable to use shorthandsymbols in order to simplify the representation of these mathematical expressions. The simplified notation often makes the complex equations much easier to handle and to analyze. Consider the following set of simultaneous linear algebraic equations: a 11 x 1 +a 12 +a 13 x 3 y 1 (1) a 21 x 1 +a 22 +a 23 x 3 y 2 (2) a 31 x 1 +a 32 +a 33 x 3 y 3 (3) A matrix equation AX Y can be defined to represent the above set of equations. The symbols A, X and Y represent matrices which are related to the coefficients and variables presented in the original set of equations as follows:, X x 1 x 3, Y We can easily see that A, X and Y are a bracketed array of coefficients and numbers. Our original set of simultaneous equations may be written as The left side of the equation is a product two arrays of numbers and this operation is not governed by the normal laws of algebra. Therefore, we must define a new type of algebra to determine the manipulation of matrices. x 1 x 3 2 Notation and Definitions of a Matrix Here we describe and define some of the terms related to matrix algebra. These are not meant to be mathematically exhaustive definitions they are meant to be brief and relevant to the study of electrical network analysis. Matrix A matrix is a collection of numbers or elements arranged in a rectangular of square array. The most common way to represent this array is to enclose the array with square brackets. For example, [ One should not confuse the matrix with the determinant. The major difference between the determinant and a matrix is that the determinant has a value whereas the matrix does not. y 1 y 2 y 3 y 1 y 2 y 3 Dr. Vahe Caliskan 1 of 5 Posted: August 22, 2011
2 Also, the determinant is always a square array it has the same number of rows and columns. On the other hand, a matrix does not to be a square array it can have any number of rows and columns Matrix elements If a matrix is written in the form [ a11 a A 12 a 13 the term a ij is called the element of the matrix in the ith row and jth column. (4) Order of a Matrix The numbers of rows and columns of a matrix indicate the order of the matrix. For example, the matrix in (4) has two rows and three columns and is said to have an order 2 3. In general, if a matrix has n rows and m columns, the order is n m. Using the concepts of matrix elements and order, the matrix in (4) can be expressed as A [a ij n m [a ij 2 3 (5) Square matrix A square matrix is one that has equal number of rows and columns, that is n m. Some examples of square matrices are [ [a ij 2 2 [b 3 4 ij Column matrix A column matrix is formed when the matrix order is n 1 with n 1. A column matrix has a single column but more than one row. Some examples of column matrices are [ 1 x1 4 7 Row matrix A row matrix is formed when the matrix order is 1 m with m 1. A row matrix has a single row but more than one column. Some examples of row matrices are [ α β [ Diagonal matrix A diagonal matrix is a square matrix with a ij 0 for all i j. In other words, a diagonal matrix is a square matrix with all of the offdiagonal elements equal to zero. Some examples of diagonal matrices are a a a 3 Unit (identity) matrix A unit (or identity) matrix is a diagonal matrix with a ij 1 for all i j. In other words, a unit matrix is a diagonal matrix with all of the diagonal elements equal to one. A unit (or identity) matrix is usually denoted as I or U in linear algebra literature; however, since in network theory we use I for current we will use U for the unit matrix in this course. A 3 3 unit matrix is given by U Dr. Vahe Caliskan 2 of 5 Posted: August 22, 2011
3 Null matrix A null matrix is one in which all of the elements are zero. That is, a ij 0 for all i and j. Examples of null matrices are [ [ Symmetric matrix A symmetric matrix is a square matrix with elements that satisfy a ij a ji for all i and j. In other words, a symmetric matrix has the property that, if the rows are interchanged with columns, the same matrix is obtained. Examples of symmetric matrices are [ Determinant of a matrix ForeachsquarematrixA, thereisacorrespondingdeterminant. The determinant of A is denoted as det(a) or A. The determinant has the same elements and order as the original matrix. For example, given the square matrix A the determinant of A is given by A det(a) where the enclosure by the vertical bars ( ) denotes the determinant. In introductory circuit analysis and linear algebra, the determinant is often evaluated using Cramer s rule. Evaluating the determinant gives det(a) 6. Singular matrix A square matrix is said to be singular if the value of its determinant is zero. In other words, A is a singular matrix if det(a) 0. A singular matrix implies that the system of simultaneous equations associated with it have certain properties. In particular, a singular matrix implies that the not all of the rows or columns are independent of each other. To illustrate this, consider the following set of equations: x x 3 0 2x 1 x 3 0 6x x 3 0 In matrix form, these equations can be written as AX 0 where x X x 3 0 The determinant of A is given by det(a) Dr. Vahe Caliskan 3 of 5 Posted: August 22, 2011
4 Since the determinant is zero, matrix A is singular. This implies that the the matrix has linearly dependent rows or columns. In this case, the third row is two times the sum of the first two rows; therefore, the expression in row three does not have any new information that is not already contained in the first two rows. Nonsingular matrix A square matrix whose determinant is not zero is said to be a nonsingular matrix. This implies that all of the rows or columns that represent the simultaneous set of equations are independent. Transpose of a matrix The transpose of a matrix A is defined as the matrix obtained by interchanging the corresponding rows and columns in A. If A [a ij n m, the transpose of A (denoted by A T ) is given by A T [a ji m n Note that A does not have to be square matrix to have a transpose. Furthermore, if the order of A is n m, then the order of A T is m n. For example, given 1 3 [ 2 4 A T Cofactors and cofactor matrix Cofactors of a square matrix A are related to its determinant det(a). The (i, j) cofactor of det(a) is the determinant obtained by omitting the ith row and jth column and multiplying it by ( 1) i+j. For example, given A and det(a) det(a) The (1, 1) cofactor of det(a), denoted by cof(a 11 ), is given by cof(a 11 ) ( 1) (1+1) a 22 a 23 a 32 a 33 a 22a 33 a 23 a 32 Similarly, the (2, 3) cofactor of det(a), denoted by cof(a 23 ), is given by cof(a 23 ) ( 1) (2+3) a 11 a 12 a 31 a 32 (a 11a 32 a 12 a 31 ) The matrix which contains all of the cofactors is called the cofactor matrix and is given by cof(a 11 ) cof(a 12 ) cof(a 13 ) cof(a) cof(a 21 ) cof(a 22 ) cof(a 23 ) cof(a 31 ) cof(a 32 ) cof(a 33 ) For our 3 3 matrix, the complete cofactor matrix is given by a 22 a 33 a 23 a 32 (a 21 a 33 a 23 a 31 ) a 21 a 32 a 22 a 31 cof(a) (a 12 a 33 a 13 a 32 ) a 11 a 33 a 13 a 31 (a 11 a 32 a 12 a 31 ) a 12 a 23 a 13 a 22 (a 11 a 23 a 13 a 21 ) a 11 a 22 a 12 a 21 Dr. Vahe Caliskan 4 of 5 Posted: August 22, 2011
5 Adjoint matrix The adjoint of a square matrix A, denoted by adj(a), is defined as the transpose of its corresponding cofactor matrix and is given by adj(a) (cof(a)) T For our 3 3 matrix, the adjoint matrix is given by a 22 a 33 a 23 a 32 (a 21 a 33 a 23 a 31 ) a 21 a 32 a 22 a 31 T adj(a) (a 12 a 33 a 13 a 32 ) a 11 a 33 a 13 a 31 (a 11 a 32 a 12 a 31 ) a 12 a 23 a 13 a 22 (a 11 a 23 a 13 a 21 ) a 11 a 22 a 12 a 21 a 22 a 33 a 23 a 32 (a 12 a 33 a 13 a 32 ) a 12 a 23 a 13 a 22 adj(a) (a 21 a 33 a 23 a 31 ) a 11 a 33 a 13 a 31 (a 11 a 23 a 13 a 21 ) a 21 a 32 a 22 a 31 (a 11 a 32 a 12 a 31 ) a 11 a 22 a 12 a 21 Consider the 2 2 matrix A The adjoint of A is given by 3 Matrix Algebra adj(a) A [ [ T [ In working with matrices and matrix equations, it is necessary to define such matrix operations as addition, subtraction, subtractions, division, etc. Some of the matrix operations will be very similar to those in the algebra for scalars; however, others may have entirely different meaning in the context of matrices. Equality of matrices two matrices A and B are said to be equal if and only if they satisfy the following two conditions: 1. They are of the same order. 2. a ij b ij for every i and j. For example, [ a11 a 12 a 21 a 22 [ implies that a 11 1, a 12 2, a 21 3 and a It would be meaningless to write the following matrix equation x x since the order of left side (3 1) does not equal the order of the right side (3 2). Matrix addition/subtraction two matrices A and B can be added or subtracted if they are of the same order. If A [a ij n m and B [b ij n m, then C A±B [c ij n m [a ij n m ±[b ij n m Dr. Vahe Caliskan 5 of 5 Posted: August 22, 2011
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