OrthogonalMatrix

OrthogonalMatrix[omat]

converts the orthogonal matrix omat to a structured array.

Details and Options

  • Orthogonal matrices, when represented as structured arrays, allow for a convenient specification and more efficient operations, including Inverse and LinearSolve.
  • Matrix decompositions that use orthogonal matrices include QR, Hessenberg, Schur and singular value decompositions.
  • For an orthogonal matrix , the column vectors are orthonormal so that v_i.v_j=TemplateBox[{{i, ,, j}}, KroneckerDeltaSeq].
  • A square orthogonal matrix is a matrix whose transpose is equal to its inverse; that is, it satisfies the relation TemplateBox[{V}, Transpose]=TemplateBox[{V}, Inverse].
  • The inverse of a square orthogonal matrix is also orthogonal.
  • A matrix of dimensions p×q is orthogonal if pq and TemplateBox[{V}, Transpose]V is the q×q identity matrix, or pq and VTemplateBox[{V}, Transpose] is the p×p identity matrix.
  • Orthogonal matrices are closed under matrix multiplication, so is again an orthogonal matrix.
  • For an OrthogonalMatrix sa, the following properties "prop" can be accessed as sa["prop"]:
  • "Matrix"orthogonal matrix, represented as a full array
    "Properties"list of supported properties
    "Structure"type of structured array
    "StructuredData"internal data stored by the structured array
    "StructuredAlgorithms"list of functions with special methods for the structured array
    "Summary"summary information, represented as a Dataset
  • Normal[OrthogonalMatrix[]] gives the orthogonal matrix as an ordinary matrix.
  • OrthogonalMatrix[,TargetStructure->struct] returns the orthogonal matrix in the format specified by struct. Possible settings include:
  • Automaticautomatically choose the representation returned
    "Dense"represent the matrix as a dense matrix
    "Structured"represent the matrix as a structured array
  • OrthogonalMatrix[,TargetStructureAutomatic] is equivalent to OrthogonalMatrix[,TargetStructure"Structured"].

Examples

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Basic Examples  (1)

Construct an orthogonal matrix:

Show the elements:

Normal can convert an OrthogonalMatrix to its ordinary representation:

Scope  (5)

Matrices corresponding to isometric transformations (reflections and rotations) are orthogonal:

Real reflection matrices are both orthogonal and symmetric:

A complex orthogonal matrix:

The matrix is orthogonal, but not unitary:

A rectangular orthogonal matrix:

Its transpose is also orthogonal:

OrthogonalMatrix objects include properties that give information about the matrix:


The "Summary" property gives a brief summary of information about the matrix:

The "StructuredAlgorithms" property lists the functions that have structured algorithms:

Options  (1)

TargetStructure  (1)

Return the orthogonal matrix as a dense matrix:

Return the orthogonal matrix as a structured array:

Applications  (5)

Matrices drawn from CircularRealMatrixDistribution are orthogonal:

A rotation matrix:

Orthogonal matrices preserve the standard inner product on TemplateBox[{}, Reals]^n. In other words, if is orthogonal and and are vectors, then :

This means the angles between the vectors are unchanged:

Since the norm is derived from the inner product, norms are preserved as well:

Any orthogonal matrix represents a rotation and/or reflection. If the matrix has determinant , it is a pure rotation. If it the determinant is , the matrix includes a reflection. Consider the following matrix:

It has determinant :

Thus, it is a pure rotation; the Cartesian unit vectors and maintain their relative positions:

The following matrix is orthogonal but has determinant :

Thus, it includes a reflection; the Cartesian unit vectors and reverse their relative positions:

Orthogonal matrices play an important role in many matrix decompositions:

The matrix is always orthogonal for any nonzero real vector :

is called a Householder reflection; as a reflection, its determinant is :

It represents a reflection through a plane perpendicular to , sending to :

Any vector perpendicular to is unchanged by :

In matrix computations, is used to set to zero selected components of a given column vector :

Properties & Relations  (1)

The transpose of an OrthogonalMatrix is equivalent to the inverse of the original matrix:

Wolfram Research (2023), OrthogonalMatrix, Wolfram Language function, https://reference.wolfram.com/language/ref/OrthogonalMatrix.html.

Text

Wolfram Research (2023), OrthogonalMatrix, Wolfram Language function, https://reference.wolfram.com/language/ref/OrthogonalMatrix.html.

CMS

Wolfram Language. 2023. "OrthogonalMatrix." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/OrthogonalMatrix.html.

APA

Wolfram Language. (2023). OrthogonalMatrix. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/OrthogonalMatrix.html

BibTeX

@misc{reference.wolfram_2023_orthogonalmatrix, author="Wolfram Research", title="{OrthogonalMatrix}", year="2023", howpublished="\url{https://reference.wolfram.com/language/ref/OrthogonalMatrix.html}", note=[Accessed: 13-April-2024 ]}

BibLaTeX

@online{reference.wolfram_2023_orthogonalmatrix, organization={Wolfram Research}, title={OrthogonalMatrix}, year={2023}, url={https://reference.wolfram.com/language/ref/OrthogonalMatrix.html}, note=[Accessed: 13-April-2024 ]}