Matrices An m n matrix is an rectangular

Matrices y vectores son fundamentales en el estudio formal de todas las ramas de

Vectors A column vector is a matrix with n rows and 1 column A

Scalar multiplication B = k. A n Dimensions: n Example

Matrix multiplication C = AB Only possible if the number of columns of A

Matrix multiplication is a non-commutative operation (generally) :

Vector products: (u, v are column vectors) n Dot product or inner product n

Scalar product (of vectors) The product of a row vector a and a column

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) =

j) k) l) (a+b)C = a. C+b. C a(b. C) = (ab)C a(BC) =

Transpose B = AT n Dimensions: n Formula: n Example

Alternative notation used in some books T A B= ’ B=A In this course

n Symmetric matrix: AT = A n Skew-symmetric matrix: AT = - A

Alternative notation used in some books for Matrix Complex Conjugate In this notes we

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:

Slides: 46

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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