Orthogonal Matrix

Matrix is a very important and useful topic of mathematics. One important type of matrix is the orthogonal matrix. When the product of one matrix with its transpose matrix gives the identity matrix value, then that matrix is termed Orthogonal Matrix. These matrices are useful in science for many vector related applications. This article will explain the Orthogonal matrix and related formulae in an easy way.  Let us begin it.

Orthogonal Matrix Definition

In mathematics, Matrix is a rectangular array, consisting of numbers, expressions, and symbols arranged in various rows and columns. If n is the number of columns and m is the number of rows, then its order will be m × n. Also, if m=n, then a number of rows and the number of columns will be equal, and such a matrix will be termed as a square matrix.

 

Source: en.wikipedia.org

A square matrix with real numbers or values is termed as an orthogonal matrix if its transpose is equal to the inverse matrix of it. In other words, the product of a square orthogonal matrix and its transpose will always give an identity matrix.

Suppose A is the square matrix with real values, of order n × n. Also, let \(A^{T}\) is the transpose matrix of A. Then according to the definition:

If, \(A^{T} = A^{-1}\) condition  is satisfied, then

A \(\times A^{T}\) = I

Where ‘I’ is the identity matrix of the order n × n. \(A^{-1}\) is the inverse of matrix A and ‘n’ denotes the number of rows and columns.

Then we will call A as the orthogonal matrix.

Orthonormal Matrix:

The orthonormal matrix is a special type of orthogonal matrix. A set of vectors will be orthonormal if the set is orthogonal as well as the inner product of every vector in the set with itself is always 1. Orthonormal is actually a shorter way to say orthogonal and every vector in the set as a unit vector.

Unitary Matrix:

This matrix is having its inverse and transpose, whose corresponding elements are the pairs of the conjugate complex values. Therefore, for real matrices, unitary is the same as the orthogonal matrix.

Example

One popular example is:

\(\begin{bmatrix} cos \theta & sin \theta \\ -sin \theta & cos \theta \\ \end{bmatrix}\)

Orthogonal Matrix Properties:

• The orthogonal matrix is always a symmetric matrix.
• All identity matrices are hence the orthogonal matrix.
• The product of two orthogonal matrices will also be an orthogonal matrix.
• The transpose of the orthogonal matrix will also be an orthogonal matrix.
• The determinant of the orthogonal matrix will always be +1 or -1.
• The eigenvalues of the orthogonal matrix will always be \(\pm{1}\).

How to find an orthogonal matrix?

Let given square matrix is A. To check for its orthogonality steps are:

1. Find the determinant of A. If, it is 1 then, matrix A may be the orthogonal matrix.
2. Find the inverse matrix of A i.e. \(A^{-1}\) as well transpose of A i.e. \(A^{T}\).
3. If \(A^{T} \times A^{-1}\) = I , then A will be orthogonal matrix, otherwise not.
4. Here I is the identity matrix of the same order.

Orthogonal matrix calculator is using the above-said procedure. This calculator is also useful to get inverse of a matrix.

Frequently Asked Questions

Q.1: Let A = \(\begin{bmatrix} 4 & 7\\ -9 & 9 \end{bmatrix}\). Whether A is an orthogonal matrix or not?

Solution: Since \(\left | A \right | = 99 \neq 1.\)

Thus matrix A is not possible to be an orthogonal matrix.

Q.2: What are Orthogonal Matrices?

Solution: Orthogonal matrices are the square matrices, which after multiplication with its transpose matrix will give an identity matrix.

Q.3: What will be the value of determinant for the Orthogonal Matrix?

Solution: The value of the determinant of an orthogonal matrix must be \(\pm{1}\).