Matrices An m n matrix is an rectangular
- Slides: 46
Matrices An m n matrix is an rectangular array of elements with m rows and n columns: denotes the element in the ith row and jth column
Partitioning in submatrices
Matrices y vectores son fundamentales en el estudio formal de todas las ramas de la ingeniería n Instrumentación n Diseño de circuitos n Comunicaciones n Microelectrónica
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
Classification of matrices Square: m=n
Symmetric: aji = aij
Upper Triangular: aij = 0 when j < i
Lower Triangular: aij = 0 when j >i
Diagonal: aij = 0 when j i
Identity: aii = 1 aij = 0 when j i
Sum of matrices of the same dimension:
Scalar multiplication B = k. A n Dimensions: n Example
Matrix multiplication C = AB Only possible if the number of columns of A is equal to the number of rows of B
examples:
Matrix multiplication is a non-commutative operation (generally) :
Identity: aii = 1 aij = 0 when j i
Vector products: (u, v are column vectors) n Dot product or inner product n Outer product:
Scalar product (of vectors) The product of a row vector a and a column vector b is a scalar a b = a 1 b 1 +. . . + anbn
Trace The trace of a nxn matrix A is given by:
Properties of Matrix Operations a) A+B = B+A b) A+(B+C) = (A+B)+C A(BC) = (AB)C A(B+C) = AB+AC (B+C)A = BA+CA a(B+C) = a. B+a. C c) d) e) f) Commutative law for addition Associative for multiplication Left distributive law Right distributive law Distributive law for scalar multiplication
j) k) l) (a+b)C = a. C+b. C a(b. C) = (ab)C a(BC) = (a. B)C
Transpose B = AT n Dimensions: n Formula: n Example
Alternative notation used in some books T A B= ’ B=A In this course we use the first one (B = AT )
Transpose Matrix properties
n Symmetric matrix: AT = A n Skew-symmetric matrix: AT = - A
n Unitary matrix example :
Symmetric Skew-symmetric Unitary matrix
n Given any matrix A with real entries:
Complex conjugate of matrices
Alternative notation used in some books for Matrix Complex Conjugate In this notes we use the bar
Complex Hermitian Example:
Complex Hermitian Properties
definitions
n examples: Hermitian: Skew-Hermitian Unitary
n Given any matrix A with complex entries:
Exercises : (a) Find A such as: (b) Find A such as:
Exercises
n Ejercicio: Simplificar
n Ejercicio: Simplificar
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