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Introduction to Matrices and Vectors: Concepts and Properties

This overview introduces matrices and vectors as fundamental elements in engineering disciplines such as instrumentation, circuit design, communications, and microelectronics. A matrix is defined as a rectangular array featuring m rows and n columns, with specific types classified by their properties, such as square, symmetric, and triangular matrices. The document covers matrix operations, including addition, scalar multiplication, and multiplication rules. It also discusses vector products, including dot and outer products, and highlights essential concepts such as transpose and complex conjugate. Exercises are provided to enhance understanding.

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Introduction to Matrices and Vectors: Concepts and Properties

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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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