MATLAB Module 2: Vectors, Matrices, Array Operations

• EE 153: Introduction to MATLAB, Module 2 covers vectors, matrices, and array operations.

Learning Objectives

• Understand the basics of vectors and matrices.
• Learn matrix operations and array manipulation.
• Apply element-wise operations and matrix algebra in MATLAB.
• Use matrix operations for solving linear equations.
• Practice solving linear equations in power systems using matrix operations.

Array

• An array is a collection of data values organized into rows and columns, known by a single name.
• Arrays can be classified as vectors and matrices.
• A vector is a one-dimensional array.
• A matrix is a two-dimensional array.
• Scalars are treated as arrays by MATLAB (1 row and 1 column).
• Scalars are quantities represented just by their numerical value, such as 3, -5, 0.368.

Vectors

• A vector is an array of numbers or a list of scalar values in a one-dimensional array of numbers, which can be in a row or column.
• A = [5 6 7 8 9] is a 1 x 5 array with 5 elements, representing a row vector.
• Comma "," is used to separate each number for a row vector.
• >> VR = [6, 7, 8] results in VR = 6 7 8
• Semicolon ";" is used to separate each number for a column vector.
• >> VC = [5; 6; 7] results in VC as a column vector with elements 5, 6, and 7.

Vector Addressing

• Vector Addressing involves accessing a vector element in MATLAB with an integer index enclosed in parentheses.
• For example, from vector A, >> A(3) results in ans = 7, which is the 3rd element of vector A.
• Colon notation may be used to address a block of elements, with the syntax (start:increment:end).
• "start" is the starting index, "increment" is the amount to add to each successive index, and "end" is the ending index.
• (start: end) is used if the increment is 1.
• E.g., >> A(1:3) results in ans = 5 6 7, which are the 1st to 3rd elements of vector A.
• MATLAB index starts at 1.

Some Useful Commands

• x = start: end creates a row vector x starting with "start," counting by one, and ending at "end."
• length(x) returns the length of vector x.
• y = x' transposes vector x.
• dot(x, y) returns the scalar dot product of the vectors x and y.
• sum(x) returns the sum of all numbers in x.
• cross(x, y) returns the vector as the cross product of x and y.
• randi() uniformly distributes random integers.
• rand() generates a uniformly distributed array of random numbers between 0 and 1.

Matrices

• A matrix is a two-dimensional array.
• A matrix having i rows and j columns is called an i x j matrix.
• Matrix creation begins with "[" and ends with "]".
• Spaces or commas are used to separate elements in a row.
• A semicolon or enter is used to separate rows.
• Elements are arranged in rows (i) and columns (j).
• >> f = [ 1 2 3; 4 5 6] creates a matrix f.
• >> M = [4, 8, 9, 6; 9, 6, 9, 6; 3, 6, 9, 6] creates a matrix M.

Matrix Addressing

• The function matrixname(row, column) is used for matrix addressing.
• For example, >> f(2,3) returns the element in the 2nd row and 3rd column of matrix f.
• >> h(:,1) selects all rows in the first column of matrix h.
• A colon may be used in place of a row or column reference to select the entire row or column.
• >> a = [1, 2, 3; 4, 5, 6] defines matrix a in MATLAB.
• >> a(1,:) selects row 1 and every column in matrix a.
• >> a(1, 1:2) selects the first two elements in the first row in matrix a.
• >> a(:) changes matrix a into a column vector.

Special Matrices/Useful Commands

• zeros(M,N) creates an MxN matrix of zeros.
  • Example: x = zeros(1,3) results in x = 0 0 0
• ones(M,N) creates an MxN matrix of ones.
  • Example: x = ones(1,3) results in x = 1 1 1
• eig(x) finds the eigenvalues of x.
  • Example: x = [1,4;3,2], lambda = eig(x), results in lambda = -2 5

Matrix Operations

• Transpose: B = A'
• Identity Matrix: eye(n) returns an n x n identity matrix.
  • eye(m,n) returns an m x n matrix with ones on the main diagonal and zeros elsewhere.
• Addition and subtraction: C = A + B, C = A - B
• Scalar Multiplication: B = αA, where α is a scalar
• Matrix Multiplication: C = A*B
• Matrix Inverse: B = inv(A). A must be a square matrix; rank(A) returns the rank of the matrix A.
• Matrix Powers:
  • B = A.^2 squares each element in the matrix.
  • C = A*A computes A*A, and A must be a square matrix.
• Determinant: det(A), A must be a square matrix.
• A, B, C are defined as matrices, and m, n, α are scalars.

Operators (Element-wise)

• .* is the element-by-element multiplication operator.
• ./ is the element-by-element division operator.
• .^ is the element-by-element power operator.
• Example of element-by-element division:
  • If >> a = [1, 2, 12] and >> b = [1, 1, 2]
  • Then >> a ./ b results in: ans = 1 2 6
• Example of element-by-element multiplication:
  • If >> a = [1, 2, 3] and >> b = [0, 1, 2]
  • Then >> a .* b results in: ans = 0 2 6
• Example of element-by-element power:
  • If >> a = [1, 2, 3] and >> b = [0, 1, 2]
  • Then >> a .^ b results in: ans = 1 2 9

Practice 1

• Consider the vector: A = [4, 8, 15, 16, 23, 42]
• Write MATLAB commands to extract the first three elements of A using colon notation.
• Write commands to replace the fourth element of A with 100.
• Write commands to create a new vector B consisting of the last three elements of A.

Practice 2

• Create a 3×3 matrix M in MATLAB with rows: [1 2 3], [4 5 6], and [7 8 9].
• Extract the entire second row using proper matrix addressing.
• Compute a new matrix where each element of M is squared using the element-by-element power operator.
• Replace the third column of M with the vector [10; 11; 12].

Practice 3

• Solving Systems of linear equations, like the one below:
  • 3a + 2b - c = 10
  • -a + 3b + 2c = 5
  • a - b - c = -1
• Formulate the coefficient matrix and constant vector corresponding to the system.
• Solve the system using the matrix inverse approach.
• Solve the same system using MATLAB's matrix division operator.
• Verify that both methods yield the same solution.

Practice 4

• Modeling of the Kirchhoff's Voltage Law (KVL) states that the sum of all voltages around a closed loop in a circuit must be equal to zero.
• Write a set of linear equations for a circuit with m loops.
• Use symbolic math tool box