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Matrices

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Matrices

1. 1. MATRICES<br />María Isabel Cadena <br />Métodos Numéricos<br />
2. 2. TYPES OF MATRICES<br />UPPER TRIANGULAR MATRIX:<br />The matrix A = (aij) a square matrix of order n. We say that A is upper triangular if all elements of A situated below the main diagonal are zero, ieaij = 0 for all i> j, i, j = 1 ,...., nFor example the matrices<br />
3. 3. LOWER TRIANGULAR MATRIX:<br />The matrix A = (aij) a square matrix of order n. We say that A is lower triangular if all elements of A located above the main diagonal are zero, ieaij = 0 for all i <j, i, j = 1 ,...., nFor example, arrays<br />
4. 4. MATRIX TRANSPOSE:<br /> Given a matrix A, is called the matrix transpose of the matrix A is obtained by changing sort rows by the columns.<br /><ul><li>(At)t = A
5. 5. (A + B)t = At + Bt
6. 6. (α ·A)t = α· At
7. 7. (A ·  B)t = Bt · At</li></li></ul><li>Symmetric matrix<br /> A symmetric matrix is a square matrix that verifies:A = At.<br />MATRIX INVERSE<br />The product of a matrix by its inverse equals the identity matrix.A · A-1 = A-1 ° A = I<br />PROPERTIES<br />(A ° B) -1 = B-1 to-1(A-1) -1 = A(K • A) -1 = k-1 to-1(A t) -1 = (A -1) t<br />
8. 8. OPERATIONS WITH MATRICES<br />SUM OF MATRICES:<br />Given two matrices of the same size, A = (aij) and B = (bij) is defined as the matrix sum: <br /> A + B = (aij + bij).The matrix sum is obtained by adding the elements of the two arrays that occupy the same same position.<br />
9. 9. Properties of matrixaddition:<br /><ul><li>Internal:The sum of two matrices of order mxn matrix is another dimension mxn.
10. 10. Associations:A + (B + C) = (A + B) + C
11. 11. Neutral element:A + 0 = AWhere O is the zero matrix of the same dimension as matrix A.
12. 12. Opposite element:A + (-A) = OThe matrix is opposite that in which all elements are changed in sign.
13. 13. Commutative:A + B = B + A</li></li></ul><li>Product of a scalar by a matrix:<br />Given a matrix A = (aij) and a real number kR, defines the product of a real number by a matrix: the matrix of the same order as A, in which each element is multiplied by k.<br />kA=(k aij)<br />
14. 14. Product Matrix:<br />Two matrices A and B are multiplied if the number of columns of A matches the number of rows of B.Mm Mn x x n x m x p = M pThe element cij of the matrix product is obtained by multiplying each element in row i of matrix A for each element of column j of the matrix B and adding.<br /> <br />
15. 15. Product features matrix:<br /><ul><li>Associations:A ° (B ° C) = (A ° B) ° C
16. 16. Neutral element:A · I = AWhere I is the identity matrix of the same order as the matrix A.
17. 17. Not Commutative:A · B ≠ B * A
18. 18. Product distributive over addition:A ° (B + C) = A ° B + A × C.</li></li></ul><li>BIBLIOGRAPHY<br />http://www.ditutor.com<br />www.fisicanet.com<br />