Ketaksamaan Cauchy-Schwarz dan Determinan Gram

Di ruang hasil kali dalam (X,\langle\cdot,\cdot\rangle), ketaksamaan Cauchy-Schwarz

\langle x,y\rangle^2 \le \langle x,x\rangle \langle y,y\rangle

ekuivalen dengan ketaksamaan

\left| \begin{array}{cc} \langle x,x\rangle & \langle x,y\rangle\\ \langle y,x\rangle & \langle y,y\rangle \end{array} \right| \ge 0

untuk setiap x,y\in X.

Nah, di ruang hasil kali dalam-2, sila periksa bahwa ketaksamaan Cauchy-Schwarz

\left| \begin{array}{cc} \langle x,y\rangle & \langle x,z\rangle\\ \langle z,y\rangle & \langle z,z\rangle \end{array} \right|^2 \le \left| \begin{array}{cc} \langle x,x\rangle & \langle x,z\rangle\\ \langle z,x\rangle & \langle z,z\rangle \end{array} \right| \, \left| \begin{array}{cc} \langle y,y\rangle & \langle y,z\rangle\\ \langle z,y\rangle & \langle z,z\rangle \end{array} \right|

ekuivalen dengan ketaksamaan

\left| \begin{array}{ccc} \langle x,x\rangle & \langle x,y\rangle & \langle x,z\rangle\\ \langle y,x\rangle & \langle y,y\rangle & \langle y,z\rangle\\  \langle z,x\rangle & \langle z,y\rangle & \langle z,z\rangle \end{array}\right| \ge 0

untuk setiap x,y,z\in X.

Determinan \left| \begin{array}{ccc} \langle x,x\rangle & \langle x,y\rangle & \langle x,z\rangle\\ \langle y,x\rangle & \langle y,y\rangle & \langle y,z\rangle\\  \langle z,x\rangle & \langle z,y\rangle & \langle z,z\rangle \end{array}\right| dikenal sebagai determinan Gram (berukuran 3 \times 3). Secara geometris, akar dari determinan ini menyatakan volume paralelpipedium yang direntang oleh x,y,z\in X.

*

Bandung, 22-09-2018

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