Normalized

Normalized

is an option that determines whether to test if matrix columns or rows are normalized.

Details

  • 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.

Examples

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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 TemplateBox[{m}, Transpose].m is an identity matrix:

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

Scope  (3)

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:

Normalize the columns:

Now is a true orthogonal matrix:

Use NormalizedFalse to avoid having to normalize symbolic matrices:

Applications  (1)

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:

The rows need not be normalized:

Wolfram Research (2014), Normalized, Wolfram Language function, https://reference.wolfram.com/language/ref/Normalized.html.

Text

Wolfram Research (2014), Normalized, Wolfram Language function, https://reference.wolfram.com/language/ref/Normalized.html.

CMS

Wolfram Language. 2014. "Normalized." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/Normalized.html.

APA

Wolfram Language. (2014). Normalized. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Normalized.html

BibTeX

@misc{reference.wolfram_2022_normalized, author="Wolfram Research", title="{Normalized}", year="2014", howpublished="\url{https://reference.wolfram.com/language/ref/Normalized.html}", note=[Accessed: 19-August-2022 ]}

BibLaTeX

@online{reference.wolfram_2022_normalized, organization={Wolfram Research}, title={Normalized}, year={2014}, url={https://reference.wolfram.com/language/ref/Normalized.html}, note=[Accessed: 19-August-2022 ]}