A matrix is considered orthogonal if its columns and rows are all orthogonal (perpendicular) to each other. This means that the dot product of any two columns or rows is equal to 0, indicating that they are at right angles to each other.
To determine if a matrix is orthogonal, you can calculate the dot product of its columns or rows. If the dot product is 0 for all pairs, then the matrix is orthogonal. Another way is to check if the inverse of the matrix is equal to its transpose. If this is true, then the matrix is orthogonal.
No, a non-square matrix cannot be orthogonal. For a matrix to be orthogonal, it must have the same number of rows and columns, meaning it must be a square matrix. This is because the dot product can only be calculated for two vectors of the same dimension.
There are a few methods for proving that a matrix is orthogonal. One way is to calculate the dot product of its columns or rows and show that it is equal to 0 for all pairs. Another way is to show that the inverse of the matrix is equal to its transpose. You can also use the Gram-Schmidt process to orthogonalize the columns or rows of the matrix.
Yes, all orthogonal matrices are also orthonormal. In addition to being orthogonal (having perpendicular columns and rows), an orthonormal matrix also has columns and rows that are unit vectors (length of 1). This means that the dot product of any two columns or rows is not only equal to 0, but also equal to 1.