The matrix is said to be an orthogonal matrix if the product of a matrix and its transpose gives an identity value.  Before discussing it briefly, let us first know what matrices are. Matrix is a rectangular array of numbers which are arranged in rows and columns. Let us see an example of a 2×3 matrix;

\(\begin{bmatrix} 2 & 3 & 4\\ 4 & 5 & 6 \end{bmatrix}\)

In the above matrix, you can see there are two rows and 3 columns. The standard matrix format is given as:

\(\begin{bmatrix} a_{11}& a_{12} & a_{13} & ….a_{1n}\\ a_{21} & a_{22} & a_{23} & ….a_{2n}\\ . & . & .\\ . & . & .\\ . & . & .\\ a_{m1} & a_{m2} & a_{m3} & ….a_{mn} \end{bmatrix}\)

Where n is the number of columns and m is the number of rows, a_ij are its elements such that i=1,2,3,…n & j=1,2,3,…m.

If m=n, which means the number of rows and number of columns is equal, then the matrix is called a square matrix.

For example, \(\begin{bmatrix} 2 & 4 & 6\\ 1 & 3 & -5\\ -2 & 7 & 9 \end{bmatrix}\)

This is a square matrix, which has 3 rows and 3 columns.

There are a lot of concepts related to matrices. The different types of matrices are row matrix, column matrix, rectangular matrix, diagonal matrix, scalar matrix, zero or null matrix, unit or identity matrix, upper triangular matrix & lower triangular matrix. In linear algebra, the matrix and its properties play a vital role. In this article, a brief explanation of the orthogonal matrix is given with its definition and properties.

What do you mean by Orthogonal?

When we say two vectors are orthogonal, we mean that they are perpendicular or form a right angle. Now when we solve these vectors with the help of matrices, they produce a square matrix, whose number of rows and columns are equal. 

Orthogonal Matrix Definition

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. Or we can say, when 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 A^T is the transpose of A. Then according to the definition, if, A^T = A^-1 is satisfied, then,

A A^T = I

Where ‘I’ is the identity matrix, A^-1 is the inverse of matrix A, and ‘n’ denotes the number of rows and columns.

Note:

• All the orthogonal matrices are invertible. Since the transpose holds back determinant, therefore we can say, determinant of an orthogonal matrix is always equal to the -1 or +1.
• All orthogonal matrices are square matrices but not all square matrices are orthogonal.

Inverse of Orthogonal Matrix

The inverse of the orthogonal matrix is also orthogonal. It is matrix product of two matrices that are orthogonal to each other. 

If inverse of matrix is equal to its transpose, then it is a orthogonal matrix.

Unitary Matrix

A square matrix is called a unitary matrix if its conjugate transpose is also its inverse. 

A.A^T = I

So, basically, the unitary matrix is also an orthogonal matrix in linear algebra. 

Orthogonal Matrix Properties

• We can get the orthogonal matrix if the given matrix should be a square matrix.
• The orthogonal matrix has all real elements in it.
• All identity matrices are orthogonal matrices.
• The product of two orthogonal matrices is also an orthogonal matrix.
• The collection of the orthogonal matrix of order n x n, in a group, is called an orthogonal group and is denoted by ‘O’.
• The transpose of the orthogonal matrix is also orthogonal. Thus, if matrix A is orthogonal, then is A^T is also an orthogonal matrix.
• In the same way, the inverse of the orthogonal matrix, which is A^-1 is also an orthogonal matrix.
• The determinant of the orthogonal matrix has a value of ±1.
• It is symmetric in nature
• If the matrix is orthogonal, then its transpose and inverse are equal
• The eigenvalues of the orthogonal matrix also have a value of ±1, and its eigenvectors would also be orthogonal and real.

Determinant of Orthogonal Matrix

The number which is associated with the matrix is the determinant of a matrix. The determinant of a square matrix is represented inside vertical bars. Let Q be a square matrix having real elements and P is the determinant, then,

Q = \(\begin{bmatrix} a_{1} & a_{2} \\ b_{1} & b_{2} & \end{bmatrix}\)

And |Q| =\(\begin{vmatrix} a_{1} & a_{2} \\ b_{1} & b_{2}\end{vmatrix}\)

|Q| = a₁.b₂ – a₂.b₁

If Q is an orthogonal matrix, then,

|Q| = ±1

Therefore, the value of determinant for orthogonal matrix will be either +1 or -1.

Dot Product of Orthogonal Matrix

When we learn in Linear Algebra, if two vectors are orthogonal, then the dot product of the two will be equal to zero. Or we can say, if the dot product of two vectors is zero, then they are orthogonal. Also, if the magnitude of the two vectors is equal to one, then they are called orthonormal. 

To check, we can take any two columns or any two rows of the orthogonal matrix, to find they are orthonormal and perpendicular to each other. Since the transpose of an orthogonal matrix is an orthogonal matrix itself. 

Solved Examples

Let us see an example of the orthogonal matrix.

Q.1: Determine if A is an orthogonal matrix. 

\(A=\left[\begin{array}{cc} -1 & 0 \\ 0 & 1 \end{array}\right]\)

Solution: To find if A is orthogonal, multiply the matrix by its transpose to get Identity matrix.

Given, 

\(A=\left[\begin{array}{cc} -1 & 0 \\ 0 & 1 \end{array}\right]\)

Transpose of A,

\(A^{T}=\left[\begin{array}{cc} -1 & 0 \\ 0 & 1 \end{array}\right]\)

Now multiply A and A^T

\(\text { A } A^{T}=\left[\begin{array}{cc} (-1)(-1) & (0)(0) \\ (0)(0) & (1)(1) \end{array}\right]=\left[\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}\right]\)

Since, we have got the identity matrix at the end, therefore the given matrix is orthogonal.

Q.2: Prove Q = \(\begin{bmatrix} cosZ & sinZ \\ -sinZ & cosZ\\ \end{bmatrix}\) is orthogonal matrix.

Solution:

Given, Q = \(\begin{bmatrix} cosZ & sinZ \\ -sinZ & cosZ\\ \end{bmatrix}\)

So, Q^T = \(\begin{bmatrix} cosZ & -sinZ \\ sinZ & cosZ\\ \end{bmatrix}\) ….(1)

Now, we have to prove Q^T = Q^-1

Now let us find Q^-1.

Q^-1 = \(\frac{Adj(Q)}{|Q|}\)

Q^-1 = \(\frac{\begin{bmatrix} cosZ & -sinZ\\ sinZ & cosZ \end{bmatrix}}{cos^2Z + sin^2 Z}\)

Q^-1 = \(\frac{\begin{bmatrix} cosZ & -sinZ\\ sinZ & cosZ \end{bmatrix}}{1}\)

Q^-1 = \(\begin{bmatrix} cosZ & -sinZ \\ sinZ & cosZ\\ \end{bmatrix}\) …(2)

Now, compare (1) and (2), we get Q^T = Q^-1

Therefore, Q is an orthogonal matrix