Menu Close

we-consider-that-application-n-1-det-M-n-R-R-A-det-A-1-verify-that-H-M-n-R-and-t-R-if-A-I-n-det-A-tH-1-t-Tr-H-t-2-suppose-that-A-GL-n-R-prouve-that-the-d




Question Number 133482 by AbderrahimMaths last updated on 22/Feb/21
    we consider that application n≥1    det : M_n (R)→R                        A det(A)  1−verify that ∀H∈M_n (R) and t∈R   if A=I_n ⇒det(A+tH)=1+t.Tr(H)+○(t)  2−suppose that: A∈GL_n (R)   prouve that the differntial of det in A is given by:     H Tr[(com(A))^T H]   Tr: trace of matrix  (com(A))^T : transpose of the comatrix
$$\:\:\:\:{we}\:{consider}\:{that}\:{application}\:{n}\geqslant\mathrm{1} \\ $$$$\:\:{det}\::\:{M}_{{n}} \left(\mathbb{R}\right)\rightarrow\mathbb{R} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{A} {det}\left({A}\right) \\ $$$$\mathrm{1}−{verify}\:{that}\:\forall{H}\in{M}_{{n}} \left(\mathbb{R}\right)\:{and}\:{t}\in\mathbb{R} \\ $$$$\:{if}\:{A}={I}_{{n}} \Rightarrow{det}\left({A}+{tH}\right)=\mathrm{1}+{t}.{Tr}\left({H}\right)+\circ\left({t}\right) \\ $$$$\mathrm{2}−{suppose}\:{that}:\:{A}\in{GL}_{{n}} \left(\mathbb{R}\right) \\ $$$$\:{prouve}\:{that}\:{the}\:{differntial}\:{of}\:{det}\:{in}\:{A}\:{is}\:{given}\:{by}: \\ $$$$\:\:\:{H} {Tr}\left[\left({com}\left({A}\right)\right)^{{T}} {H}\right] \\ $$$$\:{Tr}:\:{trace}\:{of}\:{matrix} \\ $$$$\left({com}\left({A}\right)\right)^{{T}} :\:{transpose}\:{of}\:{the}\:{comatrix} \\ $$
Commented by AbderrahimMaths last updated on 25/Feb/21
•j
$$\bullet{j} \\ $$

Leave a Reply

Your email address will not be published. Required fields are marked *