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Matrix and Vector Operations

Learning Objectives
• Lesson 1:
– Define matrices
– Distinguish vectors and matrices
– Identify square matrices
– Perform basic matrix operations
• Sum/subtraction
• Multiply matrices
Review of
Matrices
– Definition
– Types of
matrices
– Matrix operations
Matrix Definition

• Array of numbers that can be represented
in the following form:


• Where “n” represents the row number
• Where “m” represents the column
number
• Matrix A has dimensions (n x m)
 

Exercise 1

• Given the following matrices, identify the
dimensions of each matrix, what is the
value of the following elements a12, a21,
a32,a22

 

Review of
Matrices
– Definition
– Types of
matrices
– Matrix operations
Types of Matrices

• There are different types of
important matrices in linear algebra

• However, for the purpose of this
course we will focus on:
– Vectors
– Square matrices

 

Vectors

• Vectors are type of matrices that have
either one row or one column

• Matrices with one column (m=1) are
called column vectors, e.g.,

• Matrices with one row (n=1) are called
row vectors, e.g.,
{B} = [1 2 3]

Square matrices

• If m=n, we have a square matrix,
eg:

Matrix Operations

• Addition:
– Matrices can be added only if they have the
same dimensions
– To add matrices each element must be
added
– Steps for adding matrices:

• Identify the dimensions of the matrices
• If the dimensions are not the same the matrices
can’t be added
• If the dimensions are the same the matrices can
be added by adding each of the individual
elements

Example Adding Matrices

• Given A and B, calculate C= A+B

• Procedure:
– Dimensions of A: 2×2, Dimensions of B: 2×2
– Because both matrices have the same dimensions
they can be added
– Elements of C matrix:

 

Exercise 2

• Given the matrices, A, B, and C obtain matrix
D = A + B and E=B+C

 

Review of
Matrices
– Definition
– Types of
matrices
– Matrix operations
Matrix Operations

• Subtraction:
– Matrices can be subtracted only if they have
the same dimensions
– To subtract matrices each element must be
subtracted
– Steps for subtracting matrices:
• Identify the dimensions of the matrices
• If the dimensions are not the same the matrices
can’t be subtracted
• If the dimensions are the same the matrices can
be subtracted by subtracting each of the
individual elements
 

Example Subtracting Matrices

• Given A and B, calculate C= A-B

• Procedure:
– Dimensions of A: 2×2, Dimensions of B: 2×2
– Because both matrices have the same dimensions
they can be subtracted
– Elements of C matrix:

Exercise 3

• Given the matrices, A and B, obtain
matrix C = A - B

 

Review of
Matrices
– Definition
– Types of
matrices
– Matrix operations
Matrix Operations

• Multiplication of a matrix by a scalar:
– If “g” is a scalar, a matrix “A” can be multiply by “g”
– To do this, we need to multiply each element of
matrix “A” by the scalar “g”
– See procedure below:

 

Exercise 4

• Given the matrices A, B, C, D
• Perform the following calculations
– F=A- B+0.5D
– G=2(A+B+C)- D

Review of
Matrices
– Definition
– Types of
matrices
– Matrix operations
Matrix Operations
• Product of two matrices
– The number of columns in the first matrix(m1), must be equal to the number of rows
in the second matrix (n2). If the conditions
given above are not true, the two matrices
can’t be multiply
– The new matrix will have dimensions: n1m2
– Multiply each element of the row of the first matrix, by each element of the column of the second matrix and add them. This operation
will generate the new elements of the
product matrix
 

• Given Matrices A and B, calculate
C=A*B
 
Solution:
Dimensions of A:
Dimensions of B:
Because m1 is different to n2 the
product can’t be done

• Given Matrices A and B, calculate
C=A*B

Solution:
Dimensions of A:
Dimensions of B:
Because m1 is equal to n2 the product CAN be
done. The new matrix C will have dimensions 2x1

Solution Example Continues

 

Exercise 5

• Given the matrices X and Y, calculate
the matrix P = X Y

Review of
Matrices
– Definition
– Types of
matrices
– Matrix operations

Matrix Operations

• Operations of product of matrices
– The product of two matrices is not
commutative
AB ≠ BA

– The product of matrices is associative
( AB)C = A(BC)

– The product of matrices is distributive
A(B +C) = ( AB) + ( AC)
 

 

Exercise 6

• Given the matrices A, B, and C, and
the scalar g=2. Perform the following
computation:
D = g*A(B+C)

Review of
Matrices
– Definition
– Types of
matrices
– Matrix operations

Matrix Operations

• Transpose
– The transpose “AT” of a matrix “A”
transforms rows into columns
– Given a matrix “A”, the transpose
matrix “AT” is the matrix in which aij of
the transpose is aji of the original
matrix

 

Solution Example Continues

 

Exercise 7

• Given matrix A, calculate the transpose
of A

Summary

• Can you define a matrix?
• Can you identify a particular element in a
matrix?
• What are square matrices?
• What are vectors?
• Can you
– Add matrices
– Subtract matrices
– Multiply matrices (remember product properties)
– Calculate transpose of matrices