differential equation and linear algebra presentation

for b.tech cse students in 1st year

1. Product of two orthogonal matrices and its inverse, describe the mean of rotation Differential equation and linear algebra DISHA AGARWALLA (230301120281) KONCHADA BHARGAVI (230301120289) SHRABANEE ROUTRAY (230301120295) PRESENTED BY: B.TECH(CSE) SEC-F Guided by: Dr. Swarnalata Jena

2. CONTENTS: ORTHOGONAL MATRIX PRODUCT OF TWO ORTHOGONAL MATRIX INVERSE OF ORTHOGONAL MATRIX MEAN OF ROTATION

3. ORTHOGONAL MATRIX : We know that a square matrix has an equal number of rows and columns. A square matrix with real numbers or elements is said to be an orthogonal matrix, if its transpose is equal to its inverse matrix. DEFINITION: W hen the product of a square matrix and its transpose gives an identity matrix, then the square matrix is known as an orthogonal matrix . Suppose A is a square matrix with real elements and of n x n order and AT is the transpose of A. Then according to the definition, if, AT = A-1 is satisfied, then, A AT = I Where ‘I’ is the identity matrix, A-1 is the inverse of matrix A, and ‘n’ denotes the number of rows and columns

4. PROPERTIES: Orthogonal matrices preserve vector lengths, angles, and distances under multiplication. The transpose of an orthogonal matrix is its inverse. The determinant of an orthogonal matrix is either 1 or -1. Orthogonal matrices are used in various applications including rotation, reflection, and scaling transformations .

5. APPLICATIONS: Computer graphics: Orthogonal matrices are commonly used to represent 3D rotations and transformations in computer graphics and gaming. Signal processing: Orthogonal matrices play a crucial role in signal processing applications such as image compression and filtering. Quantum mechanics: In quantum mechanics, orthogonal matrices are used to represent unitary transformations that preserve probabilities.

6. ORTHOGONAL MATRIX :

7. PRODUCT OF TWO ORTHOGONAL MATRIX:

8. INVERSE OF ORTHOGONAL MATRIX: We know that the product of two orthogonal matrix is a orthogonal matrix then we find its inverse then its inverse is a orthogonal matrix as per the formula. A. = =  

9. THANK YOU!