1. Seismic Data Processing Code: ZGE 373/3 2013/2014 Dr. Amin E. Khalil

2. Contents: Complex Numbers Vectors Linear vector spaces Linear systems Matrices Determinants Eigenvalue problems Singular values Matrix inversion Series Taylor Fourier Delta Function Fourier integrals 3. Vector norm The lp- Norm for a vector x is defined as (p1):xlp n = xi i =11/ pp When P=2 it's called l2-norm. The norm here turn turns to Euclidean norm in the form:x Special case occur when p=l2= xT xxl= max xi 1 i n 4. Example Find the l1, l2, l3, l4 and l of the vector v ={1 2 3}Sol. l16l23.742l33.302l43.146l3.0 5. Matrix Norm 6. The 1-norm 7. The 1-norm 8. Infinity norm 9. Infinity norm 10. Infinity norm 11. The Eclidean Norm 12. The Euclidean norm 13. Determinants Definition of Determinant Determinant is a function which as an input accepts nxn matrix and out put is a real or a complex number that is called the determinant of the input matrix. One way to define determinant of an n x n matrix A is the following formula:Where the terms are summed over all permutations (i1 i2 .in). The sign is +ve for even permutation and -ve for odd ones. 14. Determinants The determinant of a square matrix A is a scalar number denoted det (A) or |A|, for example a det cb = ad bc d or a11 det a21 a31 = a11a22 a33a13 a22 a23 a32 a33 + a12 a23 a31 + a13 a21a32 a11a23 a32 a12 a21a33 a13 a22 a31 a12 15. Eigen values Starting from the fact that most vecors change their direction when multiplied by a matrix. However, there are some special vectors (called eigen vectors) that keep its direction when multiplied by a matrix. These follow the relation:Where A is an nxn matrix and x is eigenvector. The constant is called the eigenvalues. Eigenvalues are useful is studying dynamic problems. 16. Eigenvalues (example) 17. Matrix Inversion A square matrix is singular if det A=0. This usually indicates problems with the system (non-uniqueness, linear dependence, degeneracy ..)Matrix inverse using Gauss elimination method: In applying Gauss elimination we use the properties: 18. Matrix InversionRef. 19. Matrix Inversion 20. Matrix inversion 21. Thank youEnd of Lecture