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Background: Electric-quadrupole ($E2$) strengths relate to the underlyingquadrupole deformation of a nucleus and present a challenge for many nucleartheories. Mirror nuclei in the vicinity of the line of $N=Z$ represent aconvenient laboratory for testing deficiencies in such models, making use ofthe isospin-symmetry of the systems. Purpose: Uncertainties associated withliterature $E2$ strengths in \textsuperscript{23}Mg are some of the largest in$T_z=\left|\frac{1}{2}\right|$ nuclei in the $sd$-shell. The purpose of thepresent work is to improve the precision with which these values are known, toenable better comparison with theoretical models. Methods: Coulomb-excitationmeasurements of $^{23}$Mg and $^{23}$Na were performed at the TRIUMF-ISACfacility using the TIGRESS spectrometer. They were used to determine the $E2$matrix elements of mixed $E2$/$M1$ transitions. Results: Reduced $E2$transition strengths, $B(E2)$, were extracted for \textsuperscript{23}Mg and\textsuperscript{23}Na. Their precision was improved by factors ofapproximately six for both isotopes, while agreeing within uncertainties withprevious measurements. Conclusions: A comparison was made with both shell-modeland {\it ab initio} valence-space in-medium similarity renormalization groupcalculations. Valence-space in-medium similarity-renormalization-groupcalculations were found to underpredict the absolute $E2$ strength - inagreement with previous studies.

Coulomb excitation of the $\left|T_z\right|=\frac{1}{2}$, $A=23$ mirror pair