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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.

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

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  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

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