Linear Algebra tutorial: Orthogonal Matrix (original) (raw)
A matrix is orthogonal if the transpose is equal to its inverse , that is
.
Since computing matrix inverse is rather difficult while computing matrix transpose is straightforward, orthogonal matrix make difficult operation easier. Orthogonal matrix is important in many applications because of its properties.
Example:
Is matrix
an orthogonal matrix?
Answer:
To test whether a matrix is an orthogonal matrix, we multiply the matrix to its transpose. If the result is an identity matrix, then the input matrix is an orthogonal matrix.
Thus, matrix
is an orthogonal matrix.
To create random orthogonal matrix as in the interactive program below, I created random symmetric matrix and compute the modal matrix from concatenation of the Eigen vectors .
The interactive program below is designed to answers the question whether the given input matrix is an orthogonal matrix. When you click Random Example button, it will create random input matrix to provide you with many examples of both orthogonal and non-orthogonal matrices. You can also try to input your own matrix to test whether it is an orthogonal matrix or not.
Properties
Some important properties of orthogonal matrix are
See also : Singular Value Decomposition , orthogonal vector , spectral decomposition
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