40. Identity Matrix

The identity matrix ${𝐼}_{𝑛}$ is the $𝑛\times 𝑛$ diagonal matrix with $1$ on every diagonal entry and $0$ elsewhere.

In index notation: ${\left({𝐼}_{𝑛}\right)}_{𝑖𝑗}={𝛿}_{𝑖𝑗}$ (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 $𝐼:{ℝ}^{𝑛}\to {ℝ}^{𝑛}$, $\overrightarrow{𝑥}\mapsto \overrightarrow{𝑥}$, has matrix ${𝐼}_{𝑛}$.

40.2. Key properties

• Determinant: $det\left({𝐼}_{𝑛}\right)=1$
• Trace: $\text{tr}\left({𝐼}_{𝑛}\right)=𝑛$
• Rank: $\text{rank}\left({𝐼}_{𝑛}\right)=𝑛$ (full)
• Inverse: ${𝐼}_{𝑛}^{-1}={𝐼}_{𝑛}$
• Transpose: ${𝐼}_{𝑛}^{𝑇}={𝐼}_{𝑛}$
• Eigenvalues: $1$ with multiplicity $𝑛$
• Powers: ${𝐼}_{𝑛}^{𝑘}={𝐼}_{𝑛}$ for all $𝑘$

40.3. Where it shows up

• Defining the matrix inverse: $𝐴{𝐴}^{-1}={𝐴}^{-1}𝐴=𝐼$
• Eigenvalue equation: $det\left(𝐴-𝜆𝐼\right)=0$ (eigenvalues)
• Block-matrix algebra
• Characterization of orthogonal matrices: ${𝑄}^{𝑇}𝑄=𝐼$