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็งฉ

  • \(\text{rank}(AB)\leq \min\{\text{rank}(A),\text{rank}(B)\}\)

  • ไน˜็งฏ็Ÿฉ้˜ต็š„็งฉไธไผš่ถ…่ฟ‡ๅ› ๅญ็Ÿฉ้˜ต็š„็งฉ

    • ๅ–็ญ‰ๅทๆ—ถไน˜็งฏ็Ÿฉ้˜ต็š„ๅˆ—ๅ‘้‡็ป„ๅ’Œๅ…ถไธญไธ€ไธช็Ÿฉ้˜ต็š„ๅˆ—ๅ‘้‡็ป„็ญ‰ไปท

  • \(\text{rank}(A+B)\leq \text{rank}(A)+\text{rank}(B)\)

  • ๅ’Œ็Ÿฉ้˜ต็š„็งฉไธไผš่ถ…่ฟ‡ไธคไธช็Ÿฉ้˜ต็š„็งฉไน‹ๅ’Œ

  • \(\text{Sylvestorไธ็ญ‰ๅผ}: \text{rank}(A)+\text{rank}(B) \leq \text{rank}(AB)+n\)

  • ไธคไธช็Ÿฉ้˜ต็š„็งฉไน‹ๅ’Œไธไผš่ถ…่ฟ‡ไน˜็งฏ็Ÿฉ้˜ต็š„็งฉๅŠ ไธŠ็Ÿฉ้˜ต็š„้˜ถๆ•ฐ

    • \(\implies \text{rank}(A)+\text{rank}(B)-n \leq \text{rank}(AB) \leq \min\{\text{rank}(A),\text{rank}(B)\}\)

  • \(\text{Frobeniusไธ็ญ‰ๅผ}: \text{rank}(ABC)\geq \text{rank}(AB)+\text{rank}(BC)-n\)

    • ่ฟ™้‡Œ \(B\) ๅ– \(E\) ๅฐฑๅพ—ๅˆฐ \(\text{Sylvestorไธ็ญ‰ๅผ}\)

  • \(\text{rank}(A^T)=\text{rank}(A) = \text{rank}(A^TA)\)

  • \(\text{rank}(A-ABA) = \text{rank}(A)+\text{rank}(I-BA)-n\)

    • \(\left.\left[\begin{array}{cc}A&\\&E_n-BA\end{array}\right.\right]\to\left[\begin{array}{cc}A&O\\\\BA&E_n-BA\end{array}\right]\to\left[\begin{array}{cc}A&A\\\\BA&E_n\end{array}\right]\to\left[\begin{array}{cc}A-ABA&O\\\\O&E_n\end{array}\right]\)

  • \(A^2=A\iff \text{rank}(A)+\text{rank}(I-A)=n\)

  • \(A^2=E\iff \text{rank}(A+I)+\text{rank}(A-I)=n\)