Trace of Unit Matrix
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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: $(2.8)$