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\begin{document}$$f(x)=x^TAx+2a^Tx+c$$\end{document} and h(x)=xTBx+2bTx+d\documentclass[12pt]{minimal}
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\begin{document}$$h(x)=x^TBx+2b^Tx+d$$\end{document} be two quadratic functions having symmetric matrices A\documentclass[12pt]{minimal}
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\begin{document}$$A$$\end{document} and B\documentclass[12pt]{minimal}
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\begin{document}$$B$$\end{document}. The S-lemma with equality asks when the unsolvability of the system f(x)<0,h(x)=0\documentclass[12pt]{minimal}
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\begin{document}$$f(x)<0, h(x)=0$$\end{document} implies the existence of a real number μ\documentclass[12pt]{minimal}
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\begin{document}$$\mu $$\end{document} such that f(x)+μh(x)≥0,∀x∈Rn\documentclass[12pt]{minimal}
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\begin{document}$$f(x) + \mu h(x)\ge 0, ~\forall x\in \mathbb {R}^n$$\end{document}. The problem is much harder than the inequality version which asserts that, under Slater condition, f(x)<0,h(x)≤0\documentclass[12pt]{minimal}
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\begin{document}$$f(x)<0, h(x)\le 0$$\end{document} is unsolvable if and only if f(x)+μh(x)≥0,∀x∈Rn\documentclass[12pt]{minimal}
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\begin{document}$$f(x) + \mu h(x)\ge 0, ~\forall x\in \mathbb {R}^n$$\end{document} for some μ≥0\documentclass[12pt]{minimal}
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\begin{document}$$\mu \ge 0$$\end{document}. In this paper, we show that the S-lemma with equality does not hold only when the matrix A\documentclass[12pt]{minimal}
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\begin{document}$$A$$\end{document} has exactly one negative eigenvalue and h(x)\documentclass[12pt]{minimal}
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\begin{document}$$h(x)$$\end{document} is a non-constant linear function (B=0,b≠0\documentclass[12pt]{minimal}
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\begin{document}$$B=0, b\not =0$$\end{document}). As an application, we can globally solve inf{f(x):h(x)=0}\documentclass[12pt]{minimal}
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\begin{document}$$\inf \{f(x): h(x)=0\}$$\end{document} as well as the two-sided generalized trust region subproblem inf{f(x):l≤h(x)≤u}\documentclass[12pt]{minimal}
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\begin{document}$$\inf \{f(x): l\le h(x)\le u\}$$\end{document} without any condition. Moreover, the convexity of the joint numerical range {(f(x),h1(x),…,hp(x)):x∈Rn}\documentclass[12pt]{minimal}
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\begin{document}$$\{(f(x), h_1(x),\ldots , h_p(x)):x\in \mathbb R^n\}$$\end{document} where f\documentclass[12pt]{minimal}
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\begin{document}$$f$$\end{document} is a (possibly non-convex) quadratic function and h1(x),…,hp(x)\documentclass[12pt]{minimal}
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\begin{document}$$h_1(x),\ldots ,h_p(x)$$\end{document} are affine functions can be characterized using the newly developed S-lemma with equality.
