Storing a partitioned matrix implies storing a set of submatrices. The partitionings of rows and columns need not be the same if they are, as often happens, the diagonal blocks are square. Alternatively, partitioning can be considered simply as a data management tool, which helps to organize the transfer of information between main memory and auxiliary devices. The greater flexibility of the concept of partitioning then brings useful computational advantages. Partitioning plays an important role in sparse matrix technology because many algorithms designed primarily for matrices of numbers can be generalized to operate on matrices of matrices. The blocks can be treated as if they were the elements of the matrix and the partitioned matrix becomes a matrix of matrices. The idea of dividing a large matrix into submatrices or blocks arises naturally. Sergio Pissanetzky, in Sparse Matrix Technology, 1984 1.11 Storage of block-partitioned matrices In particular, if D 0 and D 1 are the values of the determinant if z = 0 and z = 1, respectively, then the value of the element z at which the determinant vanishes is z k 2, where k 1 and k 2 are constants.That is, if z is any element, then the value of the determinant is D = k 1 The value of a determinant is a linear function of any of its elements (provided, of course, that all the other elements remain unchanged). If all the elements below (or above) the main diagonal are zero, the determinant's value is the product of the elements of the main diagonal. If the multiple of a row is added to any other row, the determinant's value does not change. If a row of a determinant is a multiple of any other row, the determinant's value is zero. If two rows are interchanged, the sign of the determinant's value is changed. If all elements of a row are multiplied by a constant c, the determinant's value is multiplied by c. Note that the cofactor A ij already includes the appropriate sign! Example 1-1įrom Definition 1-1, the following important properties of a determinant can be derived: Then we assign the sign (−1) i j to it (the “chessboard” rule: white squares are positive, black squares are negative) and the result is the cofactor denoted by A ij. To any element a ij of a determinant of order n can be assigned a subdeterminant of order n − 1, by omitting the ith row and the jth column of the determinant. Thus, for the general case we proceed as follows:įirst, we define the concept of cofactors. This technique can be generalized for any determinant of arbitrary order n, and, in fact, it is more useful, understandable, and much more practical to use than Definition 1-1. Expression ( 1-2) is called the expansion of the determinant by its first row. Note that the elements of the first row of | A| are multiplied by the second-order determinants, which are obtained by omitting the first row and the corresponding column of | A|, and then affixing a negative sign to the second determinant. The Determinants of Matrix in Matrices is Represented ByĤ.(1-2) | A | = a 11 ⋅ | a 22 a 23 a 32 a 33 | − a 12 ⋅ | a 21 a 23 a 31 a 33 | a 13 ⋅ | a 21 a 22 a 31 a 32 | According to the Determinant Properties, the Value of Determinant Equals to Zero if Row isĢ. This is the beauty of Maths, it never gets old!ġ. Thus, it is a very old concept and continues to hold such a high level of importance to this date. He devised it as a way of determining solutions for the quadratic equations. The Determinants are calculated byĭet \ = R.H.Sįun Fact: You might find it very interesting to know that Determinants were introduced by the great Mathematician and physicist Gauss in his book Disquisitiones arithmeticae while talking about quadratic equations in 1801. The Determinant of a square Matrix is a value ascertained by the elements of a Matrix. If X’ is a Matrix made by interchanging the positions of two rows, then det (X’) = -det (x) There is a change of sign under row swap. Invariance under row operations if X’ is a Matrix formed by summing up the multiple of any row to another row, then det (X) = det (X’). Invariance under transpose det (X) = det (X t ). Multiplicativity det (XY) = det (X) det (Y) The Determinant is considered an important function as it satisfies some additional properties of Determinants that are derived from the following conditions. The Determinants and its properties are useful as they enable us to obtain the same outcomes with distinct and simpler configurations of elements. The Determinants of a Matrix say K is represented as det (K) or, |K| or det K. In Linear Algebra, a Determinant is a unique number that can be ascertained from a square Matrix.
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