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Orthogonal Matrix - Example, Definition and Formula

 

Matrices have a vast array of concepts. A matrix is a collection of numbers represented in the form of rows and columns. There are numerous types of matrices like the row matrix, column matrix, rectangular matrix, diagonal matrix, zero or null matrix, identity matrix, upper and lower triangular matrix, etc. The matrix and its properties play an essential role in linear algebra. We will learn a special kind of matrix known as an orthogonal matrix in this article.

Initially, let us understand the term Orthogonal. When two vectors are orthogonal, it means they are perpendicular or create a right angle with each other. When we solve these vectors using matrices and related formulas, we get a square matrix. To understand an orthogonal matrix, let us first understand what a transpose of a matrix is. If in a matrix, say A, we interchange the positions of the data in the rows and columns and change the row into a column and vice versa, the resultant matrix is known as the transpose of the original matrix denoted by A’.

Now that we know what the transpose of a matrix is let us learn about the orthogonal matrix. A matrix is referred to as an orthogonal matrix if the multiplication of any matrix and its transpose yields an identity matrix.

We have earlier clarified the fact that a square matrix has rows and columns in equal numbers. Thus, an orthogonal matrix consists of real numbers or elements, and the inverse of that matrix is equal to its transpose. An orthogonal matrix can also be defined as a square matrix whose product and transpose gives an identity matrix.

Suppose K is a square matrix with elements belonging to real numbers, and the order of the square matrix is a x a; the transpose of the matrix will be K’ or KT. According to the concepts and theories mentioned above, K.K’ = I. Here ‘I’ is an identity matrix. Identity Matrix: A square matrix with 1s on the main diagonal and 0s everywhere else is an identity matrix.

Some properties of the orthogonal matrix are listed below:

  • An orthogonal matrix is its inverse which implies that all orthogonal matrices are invertible. When we inverse an orthogonal matrix, we always get a matrix that is also orthogonal. We can conclude that the orthogonal matrix is a product of two different orthogonal matrices.
  • The determinant of an orthogonal matrix is always either +1 or -1. The orthogonal matrix's eigenvalues are also 1, and its eigenvectors are both orthogonal and real.
  • All orthogonal matrices are square matrices, but all square matrices are not necessarily orthogonal.
  • In linear algebra, a unitary matrix is considered an orthogonal matrix because the product of a unitary matrix and its transpose is always I (Identity matrix). This also proves the fact that all identity matrices are orthogonal.
  • An orthogonal matrix is the product of two orthogonal matrices, and it has a symmetrical design.
  • If a matrix is orthogonal, its transpose and inverse are both identical.
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