BUILT-IN MATHEMATICA SYMBOL

DiagonalMatrix

DiagonalMatrix[list]
gives a matrix with the elements of list on the leading diagonal, and

elsewhere.

DiagonalMatrix

gives a matrix with the elements of list on the k

diagonal.

DiagonalMatrix

pads with

s to create an n×n matrix.

MORE INFORMATION

For positive k, DiagonalMatrix puts the elements k positions above the main diagonal. DiagonalMatrix puts the elements k positions below.

DiagonalMatrix fills the k diagonal of a square matrix with the elements from list. Different values of k lead to different matrix dimensions.

DiagonalMatrix always creates an n×n matrix, even if this requires dropping elements of list. »

DiagonalMatrix creates an m×n matrix.

DiagonalMatrix[SparseArray[...], ...] gives a SparseArray object.

EXAMPLES

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Basic Examples (1)

Construct a diagonal matrix:

A superdiagonal matrix:

A subdiagonal matrix:

Construct a diagonal matrix:

In[1]:=

Out[1]//MatrixForm=

A superdiagonal matrix:

In[2]:=

Out[2]//MatrixForm=

A subdiagonal matrix:

In[3]:=

Out[3]//MatrixForm=

Scope (4)

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

Exact number entries:

Machine-number entries:

Arbitrary-precision number entries:

When the vector is a SparseArray object, DiagonalMatrix will give a SparseArray object:

Pad with zeros to make a larger square matrix:

Make a square matrix with the specified dimension:

Rectangular diagonal matrices:

Applications (4)

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:

Properties & Relations (6)

IdentityMatrix is a special case of DiagonalMatrix:

Several simple properties hold for diagonal matrices:

Inverse, MatrixExp, and MatrixPower commute with DiagonalMatrix:

Det and Tr have commuting relations:

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

equals Length[list]+Abs[k]:

Band can be used to construct diagonals equivalent to DiagonalMatrix

:

They will be SameQ if the vector is a SparseArray: