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Matrices and determinats

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Matrices and determinats

1. 1. DANIEL FERNANDO RODRIGUEZ COD: 2073410 PETROLEUM ENGINEERING Bucaramanga, Julio 2010 METODOS NUMERICOS MATRICES AND DETERMINATS MATRICES AND DETERMINATS MATRICES AND DETERMINATS
2. 2. Definition <ul><li>A matrix is a rectangular arrangement of numbers. For example, </li></ul><ul><li>An alternative notation uses large parentheses instead of box brackets: </li></ul>
3. 3. <ul><li>The horizontal and vertical lines in a matrix are called rows and columns , respectively. The numbers in the matrix are called its entries or its elements . To specify a matrix's size, a matrix with m rows and n columns is called an m -by- n matrix or m  ×  n matrix, while m and n are called its dimensions . The above is a 4-by-3 matrix. </li></ul>
4. 4. TYPES OF MATRICES <ul><li>Upper triangular matrix If a square matrix in which all the elements that are below the main diagonal are zeros. the matrix must be square. </li></ul><ul><li>Lower triangular matrix If a matrix in which all the elements that are above the main diagonal are zeros. the matrix must be square. </li></ul>
5. 5. TYPES OF MATRICES <ul><li>Determinant of a matrix. The determinant of a matrix A (n, n) is a scalar or polynomial, which is to obtain all possible products of a matrix according to a set of constraints, being denoted as [A]. The numerical value is also known as the matrix module. </li></ul><ul><li>EXAMPLE: </li></ul>
6. 6. TYPES OF MATRICES <ul><li>Band matrix: </li></ul><ul><li>In mathematics, particularly in the theory of matrices, a matrix is banded sparse matrix whose nonzero elements are confined or limited to a diagonal band: understanding the main diagonal and zero or more diagonal sides. Formally, an n * n matrix A = a (i, j) is a banded matrix if all elements of the matrix are zero outside the diagonal band whose rank is determined by the constants K1 and K2: Ai, j = 0 if j <i - K1 j> i + K2, K1, K2 ≥ 0. </li></ul>
7. 7. TYPES OF MATRICES <ul><li>Transpose Matrix If we have a matrix (A) any order mxn, then its transpose is another array (A) of order nxm where they exchange the rows and columns of the matrix (A). The transpose of a matrix is denoted by the symbol &quot;T&quot; and is, therefore, that the transpose of the matrix A is represented by AT. Clearly, if A is an array of size mxn, At its transpose will nxm size as the number of columns becomes row and vice versa.If the matrix A is square, its transpose is the same size. </li></ul><ul><li>EXAMPLE: </li></ul>
8. 8. TYPES OF MATRICES <ul><li>Two matrices of order n are reversed if your product is the unit matrix of order n. A matrix has inverse is said to be invertible or scheduled, otherwise called singular. Properties (A ° B) -1 = B-1 to-1 (A-1) -1 = A (K • A) -1 = k-1 to-1 (A t) -1 = (A -1) t </li></ul><ul><li>Inverse matrix calculation by determining </li></ul><ul><li>=Matrix Inverse </li></ul><ul><li>= Determinant of the matrix = Matrix attached = Matrix transpose of the enclosed </li></ul>
9. 9. BASIC OPERATIONS <ul><li>SUM OR ADITION: </li></ul><ul><li>Given the matrices m-by-n, A and B, their sum A + B is the matrix m-by-n calculated by adding the corresponding elements (ie (A + B) [i, j] = A [i, j] + B [i, j]). That is, adding each of the homologous elements of the matrices to add. For example: </li></ul>
10. 10. BASIC OPERATIONS <ul><li>SCALAR MULTIPLICATION Given a matrix A and a scalar c, cA your product is calculated by multiplying the scalar by each element of In (ie (cA) [I j] = cA [R, j]). </li></ul><ul><li>  Example </li></ul><ul><li>Properties </li></ul><ul><li>Let A and B matrices and c and d scalars. </li></ul><ul><li>Closure: If A is matrix and c is scalar, then cA is matrix. </li></ul><ul><li>Associativity: (cd) A = c (dA) </li></ul><ul><li>Neutral element: 1 ° A = A </li></ul><ul><li>Distributivity: To scale: c (A + B) = cA + cB Matrix: (c + d) A = cA + dA </li></ul>
11. 11. BASIC OPERATIONS <ul><li>The product of two matrices can be defined only if the number of columns in the left matrix is the same as the number of rows in the matrix right. If A is an m × n matrix B is a matrix n × p, then their matrix product AB is m × p matrix (m rows, p columns) given by: </li></ul><ul><li>for each pair i and j. </li></ul><ul><li>For example: </li></ul>
12. 12. BIBLIOGRAPHY <ul><li>http://es.wikipedia.org/wiki/Matriz_(matem%C3%A1tica) </li></ul><ul><li>http://www.fagro.edu.uy/~biometria/Estadistica%202/MATRICES%201.pdf </li></ul><ul><li>http://descartes.cnice.mec.es/materiales_didacticos/matrices/matrices_operaciones_II.htm </li></ul><ul><li>http://docencia.udea.edu.co/GeometriaVectorial/uni2/seccion21.html </li></ul>