Trace of Unit Matrix

Theorem

Let $\mathbf I_n$ be the unit matrix of order $n$.

Then:

$\map \tr {\mathbf I_n} = n$

where $\map \tr {\mathbf I_n}$ denotes the trace of $\mathbf I_n$.

Proof

By definition:

$\mathbf I_n := \sqbrk a_n: a_{i j} = \delta_{i j}$

That is: each of the elements on the main diagonal is equal to $1$.

There are $n$ such elements.

Hence the result.

$\blacksquare$

Sources

• 1980: A.J.M. Spencer: Continuum Mechanics ... (previous) ... (next): $2.1$: Matrices: $(2.8)$