DiagonalMatrix

DiagonalMatrix[list]

gives a matrix with the elements of list on the leading diagonal, and zero elsewhere.

DiagonalMatrix[list,k]

gives a matrix with the elements of list on the k^(th) diagonal.

DiagonalMatrix[list,k,n]

pads with zeros to create an n×n matrix.

Details and Options

  • A diagonal matrix can be used to scale the rows of a matrix by multiplying from the left, or scale the columns by multiplying from the right.
  • In the diagonal matrix given by DiagonalMatrix[{d1,…,dn}], the entries are given by D〚i,j〛=d_iTemplateBox[{{i, ,, j}}, KroneckerDeltaSeq]; that is, for the i^(th) leading diagonal entry and 0 otherwise.
  • DiagonalMatrix[list] is a symmetric matrix, and is thus equal to its transpose.
  • For positive k, DiagonalMatrix[{d1,…,ds},k] puts the elements k positions above the leading diagonal, while DiagonalMatrix[{d1,…,ds},-k] puts the elements k positions below the leading diagonal of a matrix of size (TemplateBox[{k}, Abs]+s)x(TemplateBox[{k}, Abs]+s). Different values of k lead to different matrix dimensions.
  • DiagonalMatrix[list] is equivalent to DiagonalMatrix[list,0].
  • DiagonalMatrix[list,k,n] always creates an n×n matrix, even if this requires dropping elements of list. »
  • The determinant of a diagonal matrix is the product of the leading diagonal entries if , and 0 otherwise.
  • DiagonalMatrix[list,k,{m,n}] creates an m×n matrix.
  • DiagonalMatrix[SparseArray[…],…] gives a SparseArray object.
  • DiagonalMatrix[…,TargetStructure->struct] returns the diagonal 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
  • With DiagonalMatrix[…,TargetStructureAutomatic], a dense matrix is returned if the number of matrix entries is less than a preset threshold, and a structured array is returned otherwise.
  • Diagonal matrices, when represented as structured arrays, allow for efficient storage and more efficient operations, including Det, Inverse and LinearSolve.
  • Operations that are accelerated for DiagonalMatrix include:
  • Dettime
    Dottime
    Inversetime
    LinearSolvetime
  • For a structured DiagonalMatrix sa, the following properties "prop" can be accessed as sa["prop"]:
  • "Diagonal"list of entries on the k^(th) diagonal
    "Offset"the offset k
    "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[DiagonalMatrix[…]] gives the diagonal matrix as an ordinary matrix.

Examples

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Basic Examples  (1)

Construct a diagonal matrix:

A superdiagonal matrix:

A subdiagonal matrix:

Scope  (6)

The elements in DiagonalMatrix are chosen to match the elements of the vector:

Exact number entries:

Machine-number entries:

Arbitrary-precision number entries:

Pad with zeros to make a larger square matrix:

Make a square matrix with the specified dimension:

Rectangular diagonal matrices:

Construct a sparse diagonal matrix using the option setting TargetStructure"Sparse":

The sparse representation saves a significant amount of memory for larger matrices:

Generate a structured diagonal matrix using the option setting TargetStructure"Structured":

The representation and computation are efficient for the structured array:

The normal representation is dramatically bigger and slower:

The inverse needs a large amount of storage:

DiagonalMatrix objects include properties that give information about the array:

"Diagonal" gives the diagonal entries:

"Offset" gives the offset, which is zero, positive or negative if the diagonal elements are respectively on, above or below the main diagonal:

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

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

Options  (3)

TargetStructure  (3)

Return the diagonal matrix as a dense matrix:

Return the diagonal matrix as a structured array:

Return the diagonal matrix as a sparse array:

With the setting TargetStructureAutomatic, a dense matrix is returned for small dimensions:

For large dimensions, a structured representation is returned:

The dense representation uses a lot of memory for large lists:

The sparse representation typically uses less memory:

The structured representation uses even less memory:

Applications  (5)

Express a matrix as the sum of its diagonal and off-diagonal parts:

Verify the similarity of a matrix to the diagonal matrix of its eigenvalues:

Define a Jordan matrix:

Construct a 5×5 tridiagonal matrix:

This can also be done using Band:

Extract the diagonal from a diagonal rectangular matrix:

Reconstruct the original matrix from the diagonal:

Properties & Relations  (9)

IdentityMatrix is a special case of DiagonalMatrix:

Convert IdentityMatrix to a structured DiagonalMatrix:

DiagonalMatrix creates matrices that are DiagonalMatrixQ:

DiagonalMatrix[l] is UpperTriangularMatrixQ and LowerTriangularMatrixQ:

Several simple properties hold for diagonal matrices:

Inverse, MatrixExp and MatrixPower commute with DiagonalMatrix:

Det and Tr have commuting relations:

The permanent of the product of a diagonal matrix and a general matrix is equal to the permanent of the general matrix multiplied by the diagonal elements of the diagonal matrix:

Diagonal of DiagonalMatrix gives the original vector:

This is true even if the vector is a SparseArray object:

Matrices with only subdiagonals or superdiagonals are always nilpotent:

The size of the matrix generated by DiagonalMatrix[list,k] equals Length[list]+Abs[k]:

Band can be used to construct diagonals equivalent to DiagonalMatrix[list,k]:

They will be SameQ with the setting TargetStructure"Sparse":

Wolfram Research (1988), DiagonalMatrix, Wolfram Language function, https://reference.wolfram.com/language/ref/DiagonalMatrix.html (updated 2023).

Text

Wolfram Research (1988), DiagonalMatrix, Wolfram Language function, https://reference.wolfram.com/language/ref/DiagonalMatrix.html (updated 2023).

CMS

Wolfram Language. 1988. "DiagonalMatrix." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2023. https://reference.wolfram.com/language/ref/DiagonalMatrix.html.

APA

Wolfram Language. (1988). DiagonalMatrix. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/DiagonalMatrix.html

BibTeX

@misc{reference.wolfram_2023_diagonalmatrix, author="Wolfram Research", title="{DiagonalMatrix}", year="2023", howpublished="\url{https://reference.wolfram.com/language/ref/DiagonalMatrix.html}", note=[Accessed: 22-September-2023 ]}

BibLaTeX

@online{reference.wolfram_2023_diagonalmatrix, organization={Wolfram Research}, title={DiagonalMatrix}, year={2023}, url={https://reference.wolfram.com/language/ref/DiagonalMatrix.html}, note=[Accessed: 22-September-2023 ]}