Matrices and Vector Concepts Introduction Vectors Binary Matrices

Matrices and Vector Concepts • Introduction • Vectors • Binary Matrices Operations • Unary Matrix Operations

Introduction • A matrix is a rectangular array of elements. • Row i of [A] has n elements and is • column j of [A] has m elements and is • The matrix [A] may be denoted by or

What are the special types of matrices? • Vector : A vector is a matrix that has only one row or one column § Row Vector: If a matrix [B] has one row, it is called a row vector. n is the dimension of the row vector § Column vector: If a matrix [C] has one column, it is called a column vector. m is the dimension of the vector.

What are the special types of matrices? • Submatrix: A submatrix of a matrix is obtained by deleting any collection of rows and/or columns, the remaining matrix is called a submatrix of [A]. • Square matrix: the number of rows m of a matrix is equal to the number of columns n. The entries are called the diagonal elements

What are the special types of matrices? • Upper triangular matrix: • Lower triangular matrix:

What are the special types of matrices? • Diagonal matrix: • Identity matrix: • Zero matrix:

Vectors : Introduction • A vector is a collection of numbers in a definite order • If it is a collection of n numbers, it is called a n -dimensional vector

Introduction

Introduction • A null vector is where all the components of the vector are zero. • A unit vector is defined as • If k is a scalar and is a n -dimensional vector, then

Linear Combination of Vectors

Linear Combination of Vectors • linearly independent

Linear Combination of Vectors

The rank of a set of vectors • From a set of n -dimensional vectors, the maximum number of linearly independent vectors in the set is called the rank of the set of vectors. Note that the rank of the vectors can never be greater than the vectors dimension.

The rank of a set of vectors • If a set of vectors contains the null vector, the set of vectors is linearly dependent. • If a set of vectors is linearly dependent, then at least one vector can be written as a linear combination of others.

Using vectors to write simultaneous linear equations

Binary Matrices Operations • Adding matrices • Subtracting matrices

Binary Matrices Operations • Matrices multiplication • The scalar product of a constant and a matrix

Binary Matrices Operations Rules

Binary Matrices Operations Rules • Beware: For multiplication is generally not true

Unary Matrix Operations • Transpose Matrix

Unary Matrix Operations • Transpose Matrix Also, note that • Skew-symmetric matrix

Unary Matrix Operations • Trace of a matrix

Determinant • The determinant of a square matrix is a single unique real number corresponding to a matrix and denotes as or • It is an amazing number containing rich information about a matrix such as inverse and linearly independent vector in matrix • For a 2× 2 matrix

Properties of Determinant 1. Determinant of nxn identity matrix is 1 2. The determinant change sign when two rows are exchanged 3. The determinant is a linear function of each row separately

Properties of Determinant 4. If two rows of A are equal, then det A = 0 5. Subtracting a multiple of one row from another row leaves det A unchanged 6. A matrix with a row of zeros has det A=0

Properties of Determinant 7. If A is a triangular then det A=a 11 a 22…ann=product of diagonal entries 8. If A is singular then det A =0, if A is invertible then det 9. The 10. The Question: can all row rules be used for columns as well? Why?

The determinant matrix: Cofactor