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BlockUpperTriangularMatrix
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BlockUpperTriangularMatrix

[Experimental]

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represents the block upper triangular matrix umat as a structured array.

Details and Options

  • Block upper triangular matrices, when represented as structured arrays, allow for efficient storage and more efficient operations, including Det and LinearSolve.
  • A block upper triangular matrix generalizes an upper triangular matrix, where the scalar elements in an upper triangular matrix that are on or above the diagonal are replaced by matrices of appropriate dimensions.
  • For a BlockUpperTriangularMatrix sa, the following properties "prop" can be accessed as sa["prop"]:
  • "Matrix"block upper triangular matrix, represented as a full array
    "BlockSizes"sizes of the diagonal blocks
    "RowPermutation"permutation of the rows, represented as a permutation list
    "ColumnPermutation"permutation of the columns, represented as a permutation list
    "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[BlockUpperTriangularMatrix[…]] gives the block upper triangular matrix as an ordinary list.
  • BlockUpperTriangularMatrix[…,TargetStructure->struct] returns the block upper triangular matrix in the format specified by struct. Possible settings include:
  • Automaticautomatically choose the representation returned
    "Dense"represent the matrix as a dense matrix
    "Sparse"represent the matrix as a sparse array
    "Structured"represent the matrix as a structured array
  • BlockUpperTriangularMatrix[…,TargetStructureAutomatic] is equivalent to BlockUpperTriangularMatrix[…,TargetStructure"Structured"].

Examples

open allclose all

Basic Examples  (2)Summary of the most common use cases

Construct a block upper triangular matrix:

In[1]:=1
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https://wolfram.com/xid/0do80oezu9tb9vyjm-fqwn3m
Out[1]=1

Show the elements:

In[2]:=2
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https://wolfram.com/xid/0do80oezu9tb9vyjm-ssx5x

Normal can convert a BlockUpperTriangularMatrix to its ordinary representation:

In[3]:=3
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https://wolfram.com/xid/0do80oezu9tb9vyjm-fgkucx
Out[3]=3

Construct a block upper triangular matrix with symbolic entries:

In[1]:=1
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https://wolfram.com/xid/0do80oezu9tb9vyjm-fmk8zc
Out[1]=1

Show the elements:

In[2]:=2
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https://wolfram.com/xid/0do80oezu9tb9vyjm-kvwuo4

Get the determinant:

In[3]:=3
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https://wolfram.com/xid/0do80oezu9tb9vyjm-10uirg
Out[3]=3

Scope  (4)Survey of the scope of standard use cases

BlockUpperTriangularMatrix objects include properties that give information about the array:

In[1]:=1
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https://wolfram.com/xid/0do80oezu9tb9vyjm-b621dk
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0do80oezu9tb9vyjm-llpipt
Out[2]=2

The "BlockSizes" property gives the dimensions of the diagonal blocks:

In[3]:=3
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https://wolfram.com/xid/0do80oezu9tb9vyjm-na801u
Out[3]=3

The "RowPermutation" property encodes row permutations done to the original matrix:

In[4]:=4
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https://wolfram.com/xid/0do80oezu9tb9vyjm-d6k021
Out[4]=4

The "ColumnPermutation" property encodes column permutations done to the original matrix:

In[5]:=5
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https://wolfram.com/xid/0do80oezu9tb9vyjm-eiir7r
Out[5]=5

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

In[6]:=6
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https://wolfram.com/xid/0do80oezu9tb9vyjm-2t1hkh
Out[6]=6

The "StructuredAlgorithms" property lists the functions that use the structure of the representation:

In[7]:=7
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https://wolfram.com/xid/0do80oezu9tb9vyjm-33g6sl
Out[7]=7

Structured algorithms are typically faster:

In[1]:=1
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https://wolfram.com/xid/0do80oezu9tb9vyjm-j4zpk3

Compute the determinant:

In[2]:=2
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https://wolfram.com/xid/0do80oezu9tb9vyjm-7pxqt
Out[2]=2
In[3]:=3
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https://wolfram.com/xid/0do80oezu9tb9vyjm-ho3i1l
Out[3]=3

Compute the eigenvalues:

In[4]:=4
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https://wolfram.com/xid/0do80oezu9tb9vyjm-c304or
Out[4]=4
In[5]:=5
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https://wolfram.com/xid/0do80oezu9tb9vyjm-mxky2
Out[5]=5

When appropriate, structured algorithms return another BlockUpperTriangularMatrix object:

In[1]:=1
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https://wolfram.com/xid/0do80oezu9tb9vyjm-otpl4i
Out[1]=1

Transposing bu gives a block lower triangular matrix:

In[2]:=2
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https://wolfram.com/xid/0do80oezu9tb9vyjm-hz2rsy
Out[2]=2

The product is no longer a block triangular matrix:

In[3]:=3
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https://wolfram.com/xid/0do80oezu9tb9vyjm-0pfzn8
Out[3]=3

Elements in BlockUpperTriangularMatrix are coerced to the precision of the nonzero elements of the input.

Exact matrix:

In[1]:=1
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https://wolfram.com/xid/0do80oezu9tb9vyjm-ilelcc

Machine-number matrix:

In[2]:=2
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https://wolfram.com/xid/0do80oezu9tb9vyjm-medfd3

Arbitrary-precision number matrix:

In[3]:=3
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https://wolfram.com/xid/0do80oezu9tb9vyjm-f053i

Generalizations & Extensions  (1)Generalized and extended use cases

Represent a rectangular block upper triangular matrix:

In[1]:=1
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https://wolfram.com/xid/0do80oezu9tb9vyjm-bifwcj
Out[1]=1

Show the sizes of the diagonal blocks:

In[2]:=2
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https://wolfram.com/xid/0do80oezu9tb9vyjm-myf1cu
Out[2]=2

Options  (1)Common values & functionality for each option

TargetStructure  (1)

Return the block upper triangular matrix as a dense matrix:

In[1]:=1
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https://wolfram.com/xid/0do80oezu9tb9vyjm-e3t5fj
Out[1]=1

Return the block upper triangular matrix as a structured array:

In[2]:=2
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https://wolfram.com/xid/0do80oezu9tb9vyjm-numczz
Out[2]=2

Return the block upper triangular matrix as a sparse array:

In[3]:=3
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https://wolfram.com/xid/0do80oezu9tb9vyjm-jnxo7
Out[3]=3

Applications  (1)Sample problems that can be solved with this function

The Kronecker product of an upper triangular matrix and a general square matrix is a block upper triangular matrix:

In[1]:=1
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https://wolfram.com/xid/0do80oezu9tb9vyjm-da36rs
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0do80oezu9tb9vyjm-fe7i8n

Properties & Relations  (2)Properties of the function, and connections to other functions

Upper triangular matrices are treated as block upper triangular matrices with 1×1 diagonal blocks:

In[1]:=1
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https://wolfram.com/xid/0do80oezu9tb9vyjm-cf4l1p
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0do80oezu9tb9vyjm-ebnlj
Out[2]=2

If a given matrix cannot be transformed into a block triangular form, BlockUpperTriangularMatrix returns the matrix itself:

In[1]:=1
✖
https://wolfram.com/xid/0do80oezu9tb9vyjm-em3ltz
Out[1]=1
Wolfram Research (2022), BlockUpperTriangularMatrix, Wolfram Language function, https://reference.wolfram.com/language/ref/BlockUpperTriangularMatrix.html (updated 2023).
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Wolfram Research (2022), BlockUpperTriangularMatrix, Wolfram Language function, https://reference.wolfram.com/language/ref/BlockUpperTriangularMatrix.html (updated 2023).

Text

Wolfram Research (2022), BlockUpperTriangularMatrix, Wolfram Language function, https://reference.wolfram.com/language/ref/BlockUpperTriangularMatrix.html (updated 2023).

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Wolfram Research (2022), BlockUpperTriangularMatrix, Wolfram Language function, https://reference.wolfram.com/language/ref/BlockUpperTriangularMatrix.html (updated 2023).

CMS

Wolfram Language. 2022. "BlockUpperTriangularMatrix." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2023. https://reference.wolfram.com/language/ref/BlockUpperTriangularMatrix.html.

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Wolfram Language. 2022. "BlockUpperTriangularMatrix." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2023. https://reference.wolfram.com/language/ref/BlockUpperTriangularMatrix.html.

APA

Wolfram Language. (2022). BlockUpperTriangularMatrix. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/BlockUpperTriangularMatrix.html

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Wolfram Language. (2022). BlockUpperTriangularMatrix. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/BlockUpperTriangularMatrix.html

BibTeX

@misc{reference.wolfram_2025_blockuppertriangularmatrix, author="Wolfram Research", title="{BlockUpperTriangularMatrix}", year="2023", howpublished="\url{https://reference.wolfram.com/language/ref/BlockUpperTriangularMatrix.html}", note=[Accessed: 24-May-2025 ]}

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@misc{reference.wolfram_2025_blockuppertriangularmatrix, author="Wolfram Research", title="{BlockUpperTriangularMatrix}", year="2023", howpublished="\url{https://reference.wolfram.com/language/ref/BlockUpperTriangularMatrix.html}", note=[Accessed: 24-May-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_blockuppertriangularmatrix, organization={Wolfram Research}, title={BlockUpperTriangularMatrix}, year={2023}, url={https://reference.wolfram.com/language/ref/BlockUpperTriangularMatrix.html}, note=[Accessed: 24-May-2025 ]}

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@online{reference.wolfram_2025_blockuppertriangularmatrix, organization={Wolfram Research}, title={BlockUpperTriangularMatrix}, year={2023}, url={https://reference.wolfram.com/language/ref/BlockUpperTriangularMatrix.html}, note=[Accessed: 24-May-2025 ]}