Orthogonal Matrix

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 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, aij 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 their properties play a vital role. In this article, a brief explanation of the orthogonal matrix is given with its definition and properties.

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 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.

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 an orthogonal matrix.
  • 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 AT 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.
  • The eigenvalues of the orthogonal matrix also have a value as ±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.

Orthogonal Matrix Example

Let us see an example of the orthogonal matrix.

Example:

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, QT = \(\begin{bmatrix} cosZ & -sinZ \\ sinZ & cosZ\\ \end{bmatrix}\) ….(1)

Now, we have to prove QT = 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 QT = Q-1

Therefore, Q is an orthogonal matrix

Frequently Asked Questions

What are Orthogonal Matrices?

Orthogonal matrices are square matrices which, when multiplied with its transpose matrix results in an identity matrix. So, for an orthogonal matrix, A•AT = I

How to Know if a Matrix is Orthogonal?

To check if a given matrix is orthogonal, first find the transpose of that matrix. Then, multiply the given matrix with the transpose. Now, if the product is an identity matrix, the given matrix is orthogonal, otherwise, not.

What is the Value of Determinant for an Orthogonal Matrix?

The value of the determinant of an orthogonal matrix is always ±1.

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