Construct a 3×3 identity matrix:

Visualize the matrix:

Create a 2×3 identity matrix:

A square identity matrix:

A rectangular identity matrix:

Compute the rank of an identity matrix:

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

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

Generate a structured identity 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:

IdentityMatrix objects include properties that give information about the array:

"WorkingPrecision" gives the precision of the matrix entries:

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

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

Use IdentityMatrix to quickly define the standard basis on :

The variables , and can now be used as the standard basis variables:

Compute the characteristic polynomial using IdentityMatrix:

Compare with a direct computation using CharacteristicPolynomial:

Form the augmented matrix that combines a matrix m with the identity matrix:

Row reduction of the augmented matrix gives an identity matrix augmented with Inverse[m]:

Verify that the right half of r truly is Inverse[m]:

The determinant of a square identity matrix is always 1:

For an nm matrix ℳ, Tr[ℳ]==Min[n,m]:

A square identity matrix is its own inverse and its own transpose:

The scalar multiple of an identity matrix is a diagonal matrix:

The , entry of any identity matrix is given by KroneckerDelta[i,j]:

The row or column of IdentityMatrix[n] is UnitVector[n,i]:

For IdentityMatrix[{n,m}], the rows are instead UnitVector[m,i] for i≤Min[n,m]:

IdentityMatrix can be converted to a structured DiagonalMatrix:

Use DiagonalMatrix for general diagonal matrices:

For an invertible n×n matrix m, Inverse[m].m==m.Inverse[m]==IdentityMatrix[n]:

For an n×m matrix a, a.PseudoInverse[a]==IdentityMatrix[n]:

The pseudoinverse of an identity matrix is its transpose:

For a nonsingular n×n matrix m, MatrixPower[m,0]==IdentityMatrix[n]:

The WorkingPrecision option is equivalent to creating the matrix, then applying N:

If IdentityMatrix is a square matrix, it can be converted to a PermutationMatrix:

Get the cycle representation of the identity permutation: