40. Identity Matrix

The identity matrix is the diagonal matrix with on every diagonal entry and elsewhere.

In index notation: (the Kronecker delta).

40.1. Why it’s “the identity”

It’s the multiplicative identity for matrix multiplication:

(provided shapes are compatible).

Equivalently, the identity linear transformation , , has matrix .

40.2. Key properties

• Determinant:
• Trace:
• Rank: (full)
• Inverse:
• Transpose:
• Eigenvalues: with multiplicity
• Powers: for all

40.3. Where it shows up

• Defining the matrix inverse:
• Eigenvalue equation: (eigenvalues)
• Block-matrix algebra
• Characterization of orthogonal matrices: