Examining Matrices Of Relation

Examining Matrices Of RelationHistory of hyaloplasm had to be going back to the ancient times, because it is not appevasivenessd until 1850. intercellular substance is the Latin word for womb, and is same in English. It stub excessively mean something is stageed or produced. hyaloplasm was introdeced by James Joseph Sylvester,who have brief vocation at the University of Virginia, which came to an abrupt end after an enraged Sylvester, hit a newspaper-reading student with a sword stick and fled the country, believing he had killed the studentAn grievous Chinese text from between 300 BC and AD 200, Nine Chapters of the Mathematical Art (Chiu Chang Suan Shu), gives the use in matrix method to solve co-occurrent equations. And this is origins of matrix.Too much and not enough, is the concept of a determinant initiatory appears in the treatises seventh chapter. These concepts is invented nearly cardinal millennia before Japanese mathematician Seki Kowa in 1683 or his German cont emporary Gottfried Leibnitz (who is besides credited with the invention of differential calculus, separately from but simultaneously with Isaac Newton) found it and use it widely.In chapter eight Methods of rectangular arrays, use a counting board that is mathematically identical to the modern matrix method of solution to solve the simultaneous equation is more widely use. This is also called Gaussian elimination outlined by Carl Friedrich Gauss (1777-1855). Matrices has its serious in ancient China and today it is not plainly solve simultaneous equation, but also for designing the ready reckoner games graphics, describing the quantum mechanics of atomic structure, analysing relationships, and even plotting complicated dance stepsBackground of MatricesMore and larger with amount of numerical data, measurements of one division or another gathered from their lab is confronting the scientists. However the mere collecting and recording data have been collected, data must analyze a nd interpreted. And here, matrix algebra is useful in both simplifying and promoting much development of many analysis methods but also in organizing computer techniques to execute those methods and present its results.DefinitionAn M x N matrix is a rectangular array of members having m rows and n columns. The number comprising the array atomic number 18 called element of the matrix. The numbers m and n are called dimensions of the matrix. The set of all m x n matrices is denoted by Rm x n.We shall ordinarily denote a matrix by an upper case Latin or Greek letter, whenever possible, an element of a matrix will be denoted by the corresponding lower case Greek letter with devil subscripts, the first specifying the row that contains the element and the second the column.( )( )Thus the 3 x 3 matrix has the formA3x3( )The matrix is read as A with r rows and c columns has edict r x c (read as r by c) or Ar x cAnd 4 x 3 matrix has the form( )In some applications, notably those involvin g partitioned matrices, considerable notational simplification can achieved by permitting matrices with one or both its dimensions zero. Such matrices will be give tongue to to be void.Row and column matrixThe n x 1 matrix A has the formSuch matrix is called a column transmitter which has a star column completely, which looks exactly like a member of Rn. We shall not distinguish between n x 1 matrices and n-vectors they will de denoted by upper or lower case Latin letters as convenience dictates.Example the 1 x n matrix R has the formR= (11, 12, , 1n).R= (5, 6, 7, ,n)Such a matrix will be called a row vector.A well-organized notation is that of denoting matrices by uppercase letters and their elements by the lowercase counterparts with appropriate subscripts. Vectors are denoted by lowercase letters, often from the end of the alphabet, using the prime superscript to distinguish a row vector from a column vector. Thus A is a column vector and R is a row vector, is use for scala r whereby scalar represent a single number such as 2,-4Equal matricesFor two matrices to be check, every single element in the first matrix must be rival to the corresponding element in the other matrix.So these two matrices are equal=But these two are notOf course this means that if two matrices are equal, then they must have the same numbers of rows and columns as each other. So a 33 matrix could never be equal to a 24 matrix, for instance.Also remember that each element must be equal to that element in the other matrix, so its no true if all the values are there but in different placesCombining the ideas of subtraction and equality leads to the definition of zero matrix algebra. For when A=B , then aij =bijAnd soA B = aij bij = 0 =0Which mean in matrix areSquare MatrixA square matrix is a matrix which has the same number of rows and columns. An m x n matrix A is said to be a square matrix if m = nExample number of rows = number of columns.*provided no ambiguityIn the seque l the dimensions and properties of a matrix will often be determined by context. As an example of this, the statement that A is of order n carries the implication that A is square.An n-by-n matrix is drive inn as a square matrix of order n. Any two square matrices of the same order can be added and multiplied. A square matrix A is called invertible or non-singular if there exists a matrix B such thatAB = IThis is equivalent to BA = I Moreover, if B exists, it is unique and is called the inverse matrix of A, denoted A1.The entries Ai,i form the important diagonal of a matrix. The trace, TR(A) of a square matrix A is the sum of its diagonal entries. While, as mentioned above, matrix multiplication is not commutative, the trace of the product of two matrices is independent of the order of the factorsTR (AB) = TR (BA).Also, the trace of a matrix is equal to that of its transpose, i.e. TR(A) = TR(AT).If all entries outside the main diagonal are zero, A is called a diagonal matrix. If o nly all entries above (below) the main diagonal are zero, A is called a lower angulate matrix (upper triangular matrix, respectively). For example, if n = 3, they look like(Diagonal), (lower) and (upper triangular matrix).Properties of Square Matrix Any two square matrices of the same order can be added. Any two square matrices of the same order can be multiplied. A square matrix A is called invertible or non-singular if there exists a matrix B such thatAB = In.Examples for Square Matrix For example A = is a square matrix of order 3 - 3.Relations of matricesIf R is a relation from X to Y and x1, . . . , xm is an ordering of the elements of X and y1, . . . , yn is an ordering of the elements of Y , the matrix A of R is obtained by defining Aij = 1 if xi R yj and 0 otherwise. Note that the matrix of R depends on the orderings of X and Y.Example The matrix of the relationR = (1, a), (3, c), (5, d), (1, b)From X = 1, 2, 3, 4, 5 to Y = a, b, c, d, e relative to the orderings 1, 2, 3, 4, 5 and a, b, c, d, e isExample We see from the matrix in the first example that the elements (1, a), (3, c), (5, d), (1, b) are in the relation because those entries in the matrix are 1. We also see that the domain is 1, 3, 5 because those rows contain at least one 1, and the range is a, b, c, d because those columns contain at least one.Symmetric and anti-symmetricLet R be a relation on a set X, let x1, . . . , xn be an ordering of X, and let A be the matrix of R where the ordering x1, . . . , xn is used for both the rows and columns. hence R is involuntary if and only if the main diagonal of A consists of all 1s (i.e., Aii = 1 for all i). R is symmetric if and only if A is symmetric (i.e., Aij = Aji for all i and j). R is anti-symmetric if and only if for all i = j, Aij and Aji are not both equal to 1. R is transitive verb if and only if whenever A2 ij is nonzero, Aij is also nonzero.ExampleThe matrix of the relation R = (1, 1), (1, 2), (1, 3), (2, 2), (2, 3), (3, 3), (4, 3) on 1, 2, 3, 4 relative to the ordering 1, 2, 3, 4 is A =We see that R is not reflexive because As main diagonal contains a 0. R is not symmetric because A is not symmetric for example, A12 = 1, butA21 = 0. R is anti-symmetric because for all i = j, Aij and Aji are not both equal to 1.Reflexive MatricesIn functional analysis, reflexive operator is an operator that has enough invariant subspaces to characterize it. The matrices that obey the reflexive rules also called ref matrices. A relation is reflexive if and only if it contains (x,x) for all x in the base set. Nest algebras are examples of reflexive matrices. In dimensions or spaces of matrices, finite dimensions are the matrices of a given size whose nonzero entries lie in an upper-triangular pattern.This 2 by 2 matrices is NOT a reflexive matricesThe matrix of the relation which is reflexive isR=(a, a),(b,b),(c,c),(d,d),(b,c),(c,b)on a,b,c,d, relative to the ordering a,b,c,d isOrIn generally reflexive matrices are in the case if a nd only if it contains (x,x) for all x in the base set.Transitive MatricesWhen we talk about transitive matrices, we have to compare the A(matrix) to the A2(matrix). Whenever the element in the A is nonzero then the element in theA2 have to be nonzero or vice versa to show that the matrices is transitive.For examples of transitive matricesThen the A2 isNow we can have a look where all the element aij in A and A2 is either both nonzero or both are zero. other exampleConclusionIn conclusion, the matrix we are discussed previous is useful and powerful in the mathematical analysis and collecting data. Besides the simultaneous equations, the quality of the matrices are useful in the programming where we putting in array that is a matrix also to store the data. Lastly, the matrices are playing very important role in the computer science and applied mathematics. So we can manage well of matrix, then we can play easy in computer science but the matrix is not easy to understand whereby thes e few pages of discussion and characteristic just a minor part of matrix. With this mini project, we know more about matrix and if we need to know all about how it uses in the computer science subject, I personally think that it will be difficult as it can be very complicated.