Construct a diagonal matrix:

A superdiagonal matrix:

A subdiagonal matrix:

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:

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:

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":