Trace of Sum of Matrices is Sum of Traces

Theorem

Let $\mathbf A = \sqbrk a_n$ and $\mathbf B = \sqbrk b_n$ be square matrices of order $n$.

let $\mathbf A + \mathbf B$ denote the matrix entrywise sum of $\mathbf A$ and $\mathbf B$.

Then:

$\map \tr {\mathbf A + \mathbf B} = \map \tr {\mathbf A} + \map \tr {\mathbf B}$

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

Proof

\(\ds \map \tr {\mathbf A} + \map \tr {\mathbf B}\)

\(=\)

\(\ds \sum_{k \mathop = 1}^n a_{kk} + \sum_{k \mathop = 1}^n b_{kk}\)

Definition of Trace of Matrix

\(\ds \)

\(=\)

\(\ds \sum_{k \mathop = 1}^n \paren {a_{kk} + b_{kk} }\)

Sum of Summations equals Summation of Sum

\(\ds \)

\(=\)

\(\ds \map \tr {\mathbf A + \mathbf B}\)

Definition of Matrix Entrywise Addition, Definition of Trace of Matrix

$\blacksquare$

Sources

• 1998: Richard Kaye and Robert Wilson: Linear Algebra ... (previous) ... (next): Part $\text I$: Matrices and vector spaces: $1$ Matrices: $1.6$ Determinant and trace