Semi-orthogonal matrix
In linear algebra, a semi-orthogonal matrix is a non-square matrix with real entries where: if the number of columns exceeds the number of rows, then the rows are orthonormal vectors; but if the number of rows exceeds the number of columns, then the columns are orthonormal vectors.
Properties
Let be an semi-orthogonal matrix.- Either
- A semi-orthogonal matrix is an isometry. This means that it preserves the norm either in row space, or column space.
- A semi-orthogonal matrix always has full rank.
- A square matrix is semi-orthogonal if and only if it is an orthogonal matrix.
- A real matrix is semi-orthogonal if and only if its non-zero singular values are all equal to 1.
- A semi-orthogonal matrix A is semi-unitary and either left-invertible or right-invertible.
Examples
Tall matrix (sub-isometry)
Consider the matrix whose columns are orthonormal:Here, its columns are orthonormal. Therefore, it is semi-orthogonal, which is confirmed by:
Short matrix
Consider the matrix whose rows are orthonormal:Here, its rows are orthonormal. Therefore, it is semi-orthogonal, which is confirmed by:
Non-example
The following matrix has orthogonal, but not orthonormal, columns and is therefore not semi-orthogonal:The calculation confirms this:
Proofs
Preservation of Norm
If a matrix is tall or square, its semi-orthogonality implies. For any vector, preserves its norm:If a matrix is short, it preserves the norm of vectors in its Row and [column spaces|row space].
Justification for Full Rank
If, then the columns of are linearly independent, so the rank of must be.If, then the rows of are linearly independent, so the rank of must be.
In both cases, the matrix has full rank.