Slide 1

• Topic: Problem of Matrices - Function of Matrix
• Learning objectives:
  1. Understand the concept of a function of a matrix
  2. Learn how to solve problems related to functions of matrices

Slide 2

• Definition: Function of a Matrix
• A function of a matrix is a rule that assigns a unique matrix to every matrix in its domain
• The domain of the function is the set of matrices on which the function is defined

Slide 3

• Representation of Function of a Matrix
• Let A be a matrix and f(x) be a function of matrix A
• We can represent the function as f(A) = B, where B is the resultant matrix after applying the function to matrix A

Slide 4

• Types of Functions of Matrices
  • Scalar Function: A function that operates on each element of a matrix individually
  • Matrix Function: A function that operates on the entire matrix as a whole

Slide 5

• Example: Scalar Function
• Let A be a matrix and f(x) = 2x^2 + 3, where x is an element of matrix A
• Compute f(A) A = [1 2 3 4] Solution: f(A) = [2(1)^2 + 3 2(2)^2 + 3 2(3)^2 + 3 2(4)^2 + 3] = [5 11 21 35]

Slide 6

• Example: Matrix Function
• Let A be a matrix and f(A) = A^T, where A^T is the transpose of matrix A
• Compute f(A) A = [1 2 3 4] Solution: f(A) = A^T = [1 3 2 4]

Slide 7

• Properties of Function of a Matrix
  1. The function of a sum of matrices is equal to the sum of the functions of individual matrices
  2. The function of a scalar multiple of a matrix is equal to the scalar multiple of the function of the matrix

Slide 8

• Example: Properties of Function of a Matrix
• Let A and B be matrices, and f(x) be a function of matrices
• Properties:
  1. f(A + B) = f(A) + f(B)
  2. f(cA) = cf(A), where c is a scalar

Slide 9

• Example: Properties of Function of a Matrix (contd.)

• Let A and B be matrices, and f(x) = 2x^2 + 3, where x is an element of matrix A

• Compute f(A + B) and f(A) + f(B) `` A = [1 2 3 4]

  B = [5 6 7 8] Solution: f(A + B) = f([6 8 10 12]) = [75 99 143 179]

  f(A) + f(B) = [f(A)] + [f(B)] = [5 11 21 35] + [23 27 31 39] = [28 38 52 74] ``

Slide 10

• Summary
• Function of a matrix assigns a unique matrix to every matrix in its domain
• It can be a scalar function or a matrix function
• Some properties of function of a matrix include addition and scalar multiplication properties

11

• Problem 1:
  • Let A and B be matrices, and f(x) = x^2 - 3x, where x is an element of matrix A
  • Compute f(A + B)
• Solution:
  • f(A + B) = (A + B)^2 - 3(A + B) = A^2 + AB + BA + B^2 - 3A - 3B
• Problem 2:
  • Let A be a matrix and f(x) = x^3, where x is an element of matrix A
  • Compute f(2A)
• Solution:
  • f(2A) = (2A)^3 = 8A^3
• Problem 3:
  • Let A and B be matrices, and f(x) = x^2 + 5, where x is an element of matrix A^T
  • Compute f(A^T + B)
• Solution:
  • f(A^T + B) = (A^T + B)^2 + 5 = (A^T)^2 + 2A^TB + B^2 + 5
• Problem 4:
  • Let A and B be matrices, and f(x) = 3x + 2, where x is an element of matrix (A + B)
  • Compute f(A) + f(B)
• Solution:
  • f(A) + f(B) = (3A + 2) + (3B + 2) = 3A + 3B + 4
• Problem 5:
  • Let A be a matrix and f(x) = x^-1, where x is an element of matrix A
  • Compute f(A)
• Solution:
  • f(A) = A^-1

12

• Problem 1:
  • Let A be a matrix and f(x) = 2x^2 + 4x + 1, where x is an element of matrix A
  • Compute f(2A)
• Solution:
  • f(2A) = 2(2A)^2 + 4(2A) + 1 = 8A^2 + 8A + 1
• Problem 2:
  • Let A and B be matrices, and f(x) = 3x^2 - 2x - 1, where x is an element of matrix AB
  • Compute f(A) + f(B)
• Solution:
  • f(A) + f(B) = 3A^2 - 2A - 1 + 3B^2 - 2B - 1 = 3(A^2 + B^2) - 2(A + B) - 2
• Problem 3:
  • Let A and B be matrices, and f(x) = x^3 + 2, where x is an element of matrix (A - B)
  • Compute f(A) - f(B)
• Solution:
  • f(A) - f(B) = A^3 + 2 - (B^3 + 2) = A^3 - B^3
• Problem 4:
  • Let A be a matrix and f(x) = x^2 - 4, where x is an element of matrix A^2
  • Compute f(A) - f(A^T)
• Solution:
  • f(A) - f(A^T) = A^2 - 4 - (A^T)^2 - 4 = A^2 - A^2 - 8 = -8
• Problem 5:
  • Let A be a matrix and f(x) = x^-1 + 1, where x is an element of matrix A
  • Compute f(A)
• Solution:
  • f(A) = A^-1 + 1 = A^-1 + A^0 = A^-1 + I

Slide 21

• Problem 1:
  • Let A be a matrix and f(x) = sin(x), where x is an element of matrix A
  • Compute f(A)
• Solution:
  • To find f(A), we apply the function sin(x) to each element of matrix A
  • Thus, f(A) = [sin(a) sin(b) sin(c) sin(d)], where a, b, c, and d are the elements of matrix A
• Problem 2:
  • Let A be a matrix and f(x) = e^(2x), where x is an element of matrix A
  • Compute f(A)
• Solution:
  • To find f(A), we apply the function e^(2x) to each element of matrix A
  • Thus, f(A) = [e^(2a) e^(2b) e^(2c) e^(2d)], where a, b, c, and d are the elements of matrix A
• Problem 3:
  • Let A be a matrix and f(x) = log(x), where x is an element of matrix A
  • Compute f(A)
• Solution:
  • To find f(A), we apply the function log(x) to each element of matrix A
  • Since log(x) is not defined for negative elements, we need to ensure that all elements of matrix A are positive
• Problem 4:
  • Let A be a matrix and f(x) = 1/x, where x is an element of matrix A
  • Compute f(A)
• Solution:
  • To find f(A), we apply the function 1/x to each element of matrix A
  • Thus, f(A) = [1/a 1/b 1/c 1/d], where a, b, c, and d are the elements of matrix A
• Problem 5:
  • Let A be a matrix and f(x) = 1/(x^2), where x is an element of matrix A
  • Compute f(A)
• Solution:
  • To find f(A), we apply the function 1/(x^2) to each element of matrix A
  • Thus, f(A) = [1/(a^2) 1/(b^2) 1/(c^2) 1/(d^2)], where a, b, c, and d are the elements of matrix A

Slide 22

• Problem 1:
  • Given a matrix A, find another matrix B such that f(A + B) = f(A) + f(B), where f(x) = sin(x)
• Solution:
  • Let A and B be matrices
  • We need to find B such that sin(A + B) = sin(A) + sin(B)
  • There is no general method to find such a matrix B as it depends on the specific values of elements in matrices A and B
• Problem 2:
  • Given a matrix A, find another matrix B such that f(A + B) = f(A) + f(B), where f(x) = x^2
• Solution:
  • Let A and B be matrices
  • We need to find B such that (A + B)^2 = A^2 + B^2
  • Expanding both sides, solving for B, and rearranging the terms, we can find the matrix B
• Problem 3:
  • Given a matrix A, find another matrix B such that f(A + B) = f(A) + f(B), where f(x) = e^x
• Solution:
  • Let A and B be matrices
  • We need to find B such that e^(A + B) = e^A + e^B
  • There is no general method to find such a matrix B as it depends on the specific values of elements in matrices A and B
• Problem 4:
  • Given a matrix A, find another matrix B such that f(A + B) = f(A) + f(B), where f(x) = log(x)
• Solution:
  • Let A and B be matrices
  • We need to find B such that log(A + B) = log(A) + log(B)
  • There is no general method to find such a matrix B as it depends on the specific values of elements in matrices A and B
• Problem 5:
  • Given a matrix A, find another matrix B such that f(A + B) = f(A) + f(B), where f(x) = 1/x
• Solution:
  • Let A and B be matrices
  • We need to find B such that 1/(A + B) = 1/A + 1/B
  • There is no general method to find such a matrix B as it depends on the specific values of elements in matrices A and B

Slide 23

• Problem 1:
  • Given a matrix A, find another matrix B such that f(A) + f(B) = f(A + B), where f(x) = x^3
• Solution:
  • Let A and B be matrices
  • We need to find B such that A^3 + B^3 = (A + B)^3
  • Expanding both sides, solving for B, and rearranging the terms, we can find the matrix B
• Problem 2:
  • Given a matrix A, find another matrix B such that f(A) + f(B) = f(A + B), where f(x) = cos(x)
• Solution:
  • Let A and B be matrices
  • We need to find B such that cos(A) + cos(B) = cos(A + B)
  • There is no general method to find such a matrix B as it depends on the specific values of elements in matrices A and B
• Problem 3:
  • Given a matrix A, find another matrix B such that f(A) + f(B) = f(A + B), where f(x) = sqrt(x)
• Solution:
  • Let A and B be matrices
  • We need to find B such that sqrt(A) + sqrt(B) = sqrt(A + B)
  • There is no general method to find such a matrix B as it depends on the specific values of elements in matrices A and B
• Problem 4:
  • Given a matrix A, find another matrix B such that f(A) + f(B) = f(A + B), where f(x) = 1/x^2
• Solution:
  • Let A and B be matrices
  • We need to find B such that 1/(A^2) + 1/(B^2) = 1/((A + B)^2)
  • There is no general method to find such a matrix B as it depends on the specific values of elements in matrices A and B
• Problem 5:
  • Given a matrix A, find another matrix B such that f(A) + f(B) = f(A + B), where f(x) = 2^x
• Solution:
  • Let A and B be matrices
  • We need to find B such that 2^A + 2^B = 2^(A + B)
  • There is no general method to find such a matrix B as it depends on the specific values of elements in matrices A and B

Slide 24

• Problem 1:
  • Given a matrix A, find another matrix B such that f(A) - f(B) = f(A - B), where f(x) = sin(x)
• Solution:
  • Let A and B be matrices
  • We need to find B such that sin(A) - sin(B) = sin(A - B)
  • There is no general method to find such a matrix B as it depends on the specific values of elements in matrices A and B
• Problem 2:
  • Given a matrix A, find another matrix B such that f(A) - f(B) = f(A - B), where f(x) = x^2
• Solution:
  • Let A and B be matrices
  • We need to find B such that A^2 - B^2 = (A - B)^2
  • Expanding both sides, solving for B, and rearranging the terms, we can find the matrix B
• Problem 3:
  • Given a matrix A, find another matrix B such that f(A) - f(B) = f(A - B), where f(x) = e^x
• Solution:
  • Let A and B be matrices
  • We need to find B such that e^A - e^B = e^(A - B)
  • There is no general method to find such a matrix B as it depends on the specific values of elements in matrices A and B
• Problem 4:
  • Given a matrix A, find another matrix B such that f(A) - f(B) = f(A - B), where f(x) = log(x)
• Solution:
  • Let A and B be matrices
  • We need to find B such that log(A) - log(B) = log(A - B)
  • There is no general method to find such a matrix B as it depends on the specific values of elements in matrices A and B
• Problem 5:
  • Given a matrix A, find another matrix B such that f(A) - f(B) = f(A - B), where f(x) = 1/x
• Solution:
  • Let A and B be matrices
  • We need to find B such that 1/A - 1/B = 1/(A - B)
  • There is no general method to find such a matrix B as it depends on the specific values of elements in matrices A and B

Slide 25

• Problem 1:
  • Given a matrix A, find another matrix B such that f(A) - f(B) = f(A - B), where f(x) = x^3
• Solution:
  • Let A and B be matrices
  • We need to find B such that A^3 - B^3 = (A - B)^3
  • Expanding both sides, solving for B, and rearranging the terms, we can find the matrix B
• Problem 2:
  • Given a matrix A, find another matrix B such that f(A) - f(B) = f(A - B), where f(x) = cos(x)
• Solution:
  • Let A and B be matrices
  • We need to find B such that cos(A) - cos(B) = cos(A - B)
  • There is no general method to find such a matrix B as it depends on the specific values of elements in matrices A and B
• Problem 3: