is an option that determines whether to test if matrix columns or rows are normalized.
- Functions such as OrthogonalMatrixQ and UnitaryMatrixQ test whether the columns or rows of a matrix contains normalized vectors.
- If a matrix is square or has more rows than columns, NormalizedTrue indicates that the columns should be normalized.
- If a matrix has more columns than rows, NormalizedTrue indicates that the rows must be normalized.
Examplesopen allclose all
Basic Examples (1)
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 NormalizedFalse:
Using the default setting of Normalized gives False:
The columns of m are orthogonal but not normalized:
The rows are neither normalized nor orthogonal:
Now is a true orthogonal matrix:
Use NormalizedFalse 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:
Properties & Relations (3)
If a predicate gives True with NormalizedTrue, it will also give True with the setting False:
If a matrix has more columns than rows, NormalizedTrue tests if the rows are normalized:
The columns need not be normalized:
If a matrix has more rows than columns, NormalizedTrue tests if the columns are normalized:
Wolfram Research (2014), Normalized, Wolfram Language function, https://reference.wolfram.com/language/ref/Normalized.html.
Wolfram Language. 2014. "Normalized." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/Normalized.html.
Wolfram Language. (2014). Normalized. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Normalized.html