Do not test whether matrix columns are normalized:

This indicates the columns are orthogonal but do not have unit norm:

With the default value of the setting, True, the columns are tested:

This is equivalent to testing that is an identity matrix:

Note that while the columns are orthogonal, the rows are not:

Consider the following matrix m:

UnitaryMatrixQ gives True when applied to m when Normalized is set to False:

UnitaryMatrixQ gives False with the default setting:

The rows and columns of m are complex-orthogonal to each other but do not have unit norm:

Dividing m by the norm of the entries gives a matrix for which UnitaryMatrixQ always gives True:

The following matrix satisfies OrthogonalMatrixQ with the setting NormalizedFalse:

Using the default setting of Normalized gives False:

The columns of m are orthogonal but not normalized:

The rows are neither normalized nor orthogonal:

Normalize the columns:

Now is a true orthogonal matrix:

Use NormalizedFalse to avoid having to normalize symbolic matrices:

Test whether the similarity matrix of JordanDecomposition can be made unitary:

The matrix is not strictly unitary, but can be made unitary by normalizing its columns:

Since the input matrix is unitarily equivalent to a diagonal matrix, it must be normal:

If a predicate gives True with NormalizedTrue, it will also give True with the setting False:

If a matrix has more columns than rows, NormalizedTrue tests if the rows are normalized:

The columns need not be normalized:

If a matrix has more rows than columns, NormalizedTrue tests if the columns are normalized:

The rows need not be normalized: