ThmDex – An index of mathematical definitions, results, and conjectures.

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Result R5544 on D5946: Real matrix trace
Subresult of R5545: Trace of conjugate transpose equals conjugate of trace

Real matrix trace is preserved by transposing

Formulation 0

Let

$A \in \mathbb{R}^{N \times N}$ be a D6160: Real square matrix such that

(i)	$A^T$ is a D5841: Real matrix transpose for $A$

Then

$\begin{equation} \text{Trace} A^T = \text{Trace} A \end{equation}$

Also known as

Real matrix trace is invariant under transposing

Proofs

Proof 0

Let

$A \in \mathbb{R}^{N \times N}$ be a D6160: Real square matrix such that

(i)	$A^T$ is a D5841: Real matrix transpose for $A$

Clear.

$\square$