## Orthogonal matrix in Discrete mathematicsA matrix will be known as the orthogonal matrix if the transpose of the given matrix and its inverse matrix are similar to each other. Now we will learn about the transpose of a matrix. A matrix will be known as the transpose matrix if we interchange the elements of rows or columns of the original matrix. We already learned that if the original matrix and its transpose are similar to each other, then that matrix will be known as the symmetric matrix. But in the case of an orthogonal matrix, the inverse of the matrix and transpose of the matrix will be the same. In this section, we will learn about the orthogonal matrix, its definition, its determinant, its inverse, its properties, its examples, and many more. ## What is an Orthogonal matrixIf there is a square matrix, then we can call it an orthogonal matrix if the transpose of this matrix and its inverse are similar to each other, i.e., A A Now we will multiply A on both sides of the above equation like this: AA As we have learned that AA Thus AA In the same way, we can also prove that A We will get the following equation from above two equations: AA Thus, an orthogonal matrix is used to contain two definitions which will be shown in the definition of an orthogonal matrix. ## Definition of Orthogonal matrixThere are two definitions of an orthogonal matrix. According to the first definition, if the transpose of the matrix and the inverse of the matrix are similar to each other, then that matrix will be known as the orthogonal matrix. According to the second definition, if we get the identity matrix as the multiplication of given matrix and its transpose matrix, then the given matrix will be known as the orthogonal matrix. Suppose there is a square matrix A, which contains the size n∗n. This matrix will be considered an orthogonal matrix if it contains the following conditions: A AA ## Note:- It is necessary that all the orthogonal matrices will be invertible. The transpose holds back the determinant. This is because we can say that in the case of an orthogonal matrix, the determinant will always be +1 or -1.
- It is necessary that all the orthogonal matrices will be square matrices, but it is not compulsory that all the square matrices will be orthogonal.
## Orthogonal matrix ExampleIn this example, we have a matrix A and its transpose matrix A The transpose matrix is described as follows: Now we will do the multiplication of both matrices in the following way: In the same way, we can also prove that AA ## How to find Orthogonal matrixThere are some steps that should be followed to determine whether the matrix is an orthogonal matrix or not. For this, we will assume a square matrix A. These steps are described as follows:
## Determinant of Orthogonal matrixIn case of the orthogonal matrix, if we calculate the determinant, then it will always be +1 or -1. Now we will prove it. For this, we will assume an orthogonal matrix A. With the help of its definition, we know that AA Now we will perform the determinant on both sides of the above equation and get the following: det(AA We have learned that we will get 1 if we determine the determinant of an identity matrix. Also, if there are two matrices, A and B, then det(AB) = detA ∗ detB. So det(A) ∗ det(A As we already know that det(A) = det(A det(A) ∗ det(A) = 1 [det(A)] det(A) = ±1 ## Inverse of Orthogonal matrixWith the help of definition of an orthogonal matrix, we know that any orthogonal matrix A, A AA If there are two matrices, A and B, then they will be inverse of each other if it satisfies the following condition: AB = BA = I .... (2) With the help of equations (1) and (2), we can see that B = A Thus it is proved that the inverse of an orthogonal matrix will be its transpose. ## Properties of Orthogonal matrixThere are various properties of an orthogonal matrix on the basis of its definitions, and some of them are described as follows: - The transpose and inverse of the orthogonal matrix are similar to each other, i.e., A
^{-1}= A^{T}. - If we multiply the given matrix A and its transpose matrix, then we will get an identity matrix as a result, i.e., AA
^{T}= A^{T}A = I. - If the diagonal matrix is used to contain the elements 1 or -1, then it will always be orthogonal. For example,
- A
^{T}is also orthogonal. If it contains the relation A^{-1}= A^{T}, in this case, A^{-1}will also be orthogonal. - The eigenvectors will be orthogonal, and eigenvalues of A will be ±1.
- If there is an identity matrix (I), then it will be orthogonal, i.e., I.I = I.I = I.
- The orthogonal matrices will be symmetric in nature.
- The determinant of orthogonal matrix will be det(A) =±1. The determinant of this matrix is not 0. That's why an orthogonal matrix will always be non-singular.
## Applications of Orthogonal matrixThere are various applications or uses in the orthogonal matrix, and some of them are described as follows: - We can perform multi-channel signal processing with the help of an orthogonal matrix.
- We can perform multivariate time series analysis with the help of an orthogonal matrix.
- We can use this matrix in many algorithms in linear algebra.
- We can use this matrix in QR decomposition.
## Important notes on Orthogonal matrixThere are some important points that we should know while learning the concept of an orthogonal matrix. These points are described as follows: - The matrix will be an orthogonal matrix if and only if the given matrix is a square matrix.
- If the inverse of the given matrix and its transpose matrix are similar to each other, then the given square matrix will be known as orthogonal.
- If there is an orthogonal matrix A, then the given matrix A and its transpose matrix A
^{T}will be inverses of each other. - If we calculate the determinant of an orthogonal matrix, then we will either get 1 or -1 as a result.
- All the identity matrices must be orthogonal matrices.
- In the orthogonal matrix, if we perform the dot product of any two rows or columns, then we will always get 0 as a result.
- There will always be a unit vector in any row or column of an orthogonal matrix.
## Examples of Orthogonal matrixThere are a lot of examples of orthogonal matrices, and some of them are described as follows:
From the question, we have a matrix A where The transpose of the above matrix is described as follows: Now we will multiply the matrix A and its transpose matrix A and get the following: We can see that we get an identity matrix as a result.
|A| = cos x∗cos x - sinx∗(-sin x) |A| = cos |A| = 1 The determinant of matrix A is 1.
When we do the transpose of matrix P, then the given matrix P and its transpose matrix will be similar to each other. As we have learned that there is a symmetric matrix if the given matrix and its transpose matrix are similar to each other. We also know that if there is a symmetric matrix, then it will be an orthogonal matrix.
We will get the inverse of this matrix if A∗A Now Similarly, we can also prove that A Hence it is proved that A is orthogonal. Then A
A = A For this, we will assume an orthogonal matrix A, which contains the following elements: (We can prove that this matrix is orthogonal by proving AA Now we will perform the transpose of the assumed matrix A and get the following: (We can prove it AA So we can say that the given matrix and its transpose matrix are not similar to each other, i.e., A ≠ A
AA Now we will add inverse on both sides of the above equation like this: (AA As we have learned that the inverse of an identity matrix will be itself. So I Now we will put this in the above equation and get the following: (AA
|A| = 4∗9 - (-9)∗7 |A| = 36 + 63 |A| = 99 ≠ 1.
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