1 / 46

An m  n matrix is an rectangular array of elements with m rows and n columns :

An m  n matrix is an rectangular array of elements with m rows and n columns :. Matrices. denotes the element in the i th row and j th column. Partitioning in submatrices. Matrices y vectores son fundamentales en el estudio formal de todas las ramas de la ingeniería.

miltond
Download Presentation

An m  n matrix is an rectangular array of elements with m rows and n columns :

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. An m  n matrix is an rectangular array of elements with m rows and n columns: Matrices denotes the element in the ith row and jth column

  2. Partitioning in submatrices

  3. Matrices y vectores son fundamentales en el estudio formal de todas las ramas de la ingeniería • Instrumentación • Diseño de circuitos • Comunicaciones • Microelectrónica

  4. Vectors A column vector is a matrix with n rows and 1 column A row vector is a matrix with 1 row and n columns

  5. Classification of matrices Square: m=n

  6. Symmetric: aji = aij

  7. Upper Triangular: aij = 0 when j < i

  8. Lower Triangular: aij = 0 when j >i

  9. Diagonal: aij = 0 when j  i

  10. Identity: aii = 1 aij = 0 when j  i

  11. Sum of matrices of the same dimension:

  12. Scalar multiplication B = kA • Dimensions: • Example

  13. Matrix multiplication C = AB Only possible if the number of columns of A is equal to the number of rows of B

  14. examples:

  15. Matrix multiplicationis a non-commutative operation (generally):

  16. Identity: aii = 1 aij = 0 when j  i

  17. Vector products: (u,v are column vectors) • Dot product or inner product • Outer product:

  18. Scalar product (of vectors) The product of a row vector a and a column vector b is a scalar a  b = a1b1 + ... + anbn

  19. Trace The trace of a nxn matrix A is given by:

  20. Properties of Matrix Operations • A+B = B+A • A+(B+C) = (A+B)+C • A(BC) = (AB)C • A(B+C) = AB+AC • (B+C)A = BA+CA • a(B+C) = aB+aC Commutative law for addition Associative law for addition Associative for multiplication Left distributive law Right distributive law Distributive law for scalar multiplication

  21. (a+b)C = aC+bC a(bC) = (ab)C a(BC) = (aB)C

  22. Transpose B = AT • Dimensions: • Formula: • Example

  23. Alternative notation used in some books B = AT B = A’ In this course we use the first one (B = AT )

  24. Transpose Matrix properties

  25. Symmetric matrix: AT = A • Skew-symmetric matrix: AT = -A

  26. Unitary matrix example :

  27. Symmetric Skew-symmetric Unitarymatrix

  28. Given any matrix A with real entries:

  29. Complex conjugate of matrices

  30. Alternative notation used in some books for Matrix Complex Conjugate In this notes we use the bar

  31. ComplexHermitian Example:

  32. Complex Hermitian Properties

  33. definitions

  34. examples: Hermitian: Skew-Hermitian Unitary

  35. Given any matrix A with complex entries:

  36. Exercises : (a) Find A such as: (b) Find A such as:

  37. Exercises

  38. Ejercicio: Simplificar

  39. Ejercicio: Simplificar

More Related