Orthogonal Matrix

If a square matrix transpose is equal to its inverse, then it is known as an orthogonal matrix, where AT = A-1. We can use this definition to derive an important property of orthogonal matrices.
Proof:
Given: A^T = A^-1 
Multiply both sides by A, AA^T = AA^-1
Since AA^-1 = I, where I is the identity matrix, AA^T = I 
On multiplying both sides of the original equation by A, we get A^TA = A^-1A = I
So, AA^T = A^TA = I
This means that matrix A is only orthogonal if the product of the matrix and its transpose results in the identity matrix. This shows that a matrix can only be orthogonal if it produces an identity matrix when multiplied by its transpose.

Properties of an Orthogonal Matrix

Orthogonal matrices have structural and algebraic properties that define their characteristics. Some important properties of orthogonal matrices are listed below:

1. Inverse and transpose of the matrix are equal, i.e., A^-1 = A^T 
2. The identity matrix is the product of the orthogonal matrix and its transpose, i.e., AA^T = A^TA = I
3. Orthogonal matrices are always non-singular because their determinant is never zero. The determinant of an orthogonal matrix is always det(A) = 1.
4. An orthogonal matrix is diagonal only if the diagonal entries are either 1 or -1, and off-diagonal entries are zero.
5. Since the transpose and the inverse of an orthogonal matrix have the same defining conditions, they are also orthogonal.
6. Eigenvectors of an orthogonal matrix can be complex, but all of them have magnitude 1.
7. The identity matrix is orthogonal because I^T = I and I  I = I.
    

An orthogonal matrix is a square matrix whose product with its transpose results in the identity matrix. A matrix is also orthogonal if the transpose of the matrix and the inverse of the matrix are equal.
Let’s take a square matrix A having real elements in the n  n order. A^T is the transpose of matrix A. According to the definition, if A^T = A^-1, then A  A^T = I.

Determinant of Orthogonal Matrix

The determinant of an orthogonal matrix is 1. To prove so, let us consider an orthogonal matrix A.
Then by definition, AA^T = I
Taking determinants on both sides,
det(AA^T) = det(I)
The determinant of an identity matrix is 1. 

For an orthogonal matrix A, det(A) = 1:
Property of determinants, det(AB) = det(A)  det(B)
Since A is orthogonal, we know that A^T = A^-1
So, AA^T = I  det(AA^T) = det(I) = 1
Using the determinant property, 
det(AA^T) = det(A)  det(A^T)
Another property of determinants is det(A^T) = det(A)
Therefore, det(A)² = 1  det(A) =  1
So, for an orthogonal matrix A, 
det(A)² = 1 
and, det(A) = 1.

Inverse of Orthogonal Matrix

As defined, for any orthogonal matrix A, A^-1 = A^T. To prove this, we will use the other definition of orthogonal matrix, i.e., AA^T = A^TA = I 
Let this be (1)
Two matrices A and B are said to be each other’s inverses if
AB = BA = I
Let this be (2)
From (1) and (2), we get B = A^T.
B = A^T is equal to A^-1 = A^T because B is the inverse of A.
Hence, proved that the inverse of an orthogonal matrix is equal to its transpose.

Multiplicative Inverse of Orthogonal Matrices

The inverse of an orthogonal matrix is also orthogonal and is equal to the transpose of the original matrix. This shows that orthogonality is maintained during multiplication and inversion.

Orthogonal Matrix in Linear Algebra

The term “orthogonal” means perpendicular. Two vectors having a dot product of zero are considered orthogonal. In an orthogonal matrix, each row vector and column vector is a unit vector and perpendicular to every other row or column. 
Consider an orthogonal matrix:
                                                     
Check for the dot product of the first two rows, it should be zero.
Row 1: (12, 12)
Row 2: (-12, 12)
Their dot product: (12 -12) + (1212) = -12+ 12 = 0
We can see that the first two rows are orthogonal. Keep repeating the process for every two rows and columns. The dot product for each of them should be zero.

Now, let's find the magnitude of the first row: (12)2 + (12)2 = 12 + 12 = 1 =1
Similarly, the length of every row and column will be 1.
 

Orthogonal matrices are vital in many real-world applications due to their properties of preserving lengths, angles, and orthogonality. Some of these applications are listed below.

• 3D rotation in computer graphics
  Orthogonal matrices preserve shapes, angles, and sizes of objects, so they are used in 3D rotations of objects for realistic animations and simulation in graphics.

• Signal decomposition in audio and image processing
  In orthogonal matrices, each component remains independent, resulting in efficient filtering and compression. It is used in MP3, JPEG, and wireless communication systems.

• Dimensionality reduction in machine learning
  In algorithms like Principal Component Analysis (PCA), principal components are orthogonal vectors. They capture maximum variance without overlaps. This leads to better interpretation and visualization of the data.

• Attitude control in aerospace engineering
  Orthogonal matrices are used in attitude control systems of satellites, drones, and aircraft to maintain orientation without distortion.

• State transformations in quantum mechanics
  They preserve inner products and probabilities, ensuring physical realism, and hence are used in representing quantum states and transformations in real vector spaces