Slide 1
Topic: Problem of Matrices - Function of Matrix
Learning objectives:
Understand the concept of a function of a matrix
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
The function of a sum of matrices is equal to the sum of the functions of individual matrices
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:
f(A + B) = f(A) + f(B)
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:
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:
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:
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Slide 1 Topic: Problem of Matrices - Function of Matrix Learning objectives: Understand the concept of a function of a matrix Learn how to solve problems related to functions of matrices