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I'm looking into the derivation of Gauss-Newton method.

Why $\frac 12(F_k+J_k\Delta x)^T(F_k+J_k\Delta x)$ become $\frac 12F_k^TF_k+F_k^TJ_k\Delta x+\frac12\Delta x^TJ_k^TJ_k\Delta x$?

Is $F_k^TJ_k$ equal $J_k^TF_k$?

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Mulitply the factors as if they were scalars but with the additional observation that (A + B)^T = A^T + B^T and (A B)^T = B^T A^T. If you work through the algebra you'll find that:

$\frac 12(F_k+J_k\Delta x)^T(F_k+J_k\Delta x)$ = $\frac 12F_k^TF_k+F_k^TJ_k\Delta x+\frac12\Delta x^TJ_k^TJ_k\Delta x$?

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