Soit
une matrice orthogonale.
Démontrer que :
![\[\sum_{1 \leq i,j \leq n} |a_{ij}| \leq n\sqrt{n}.\]](https://exercicesmath.fr/wp-content/ql-cache/quicklatex.com-876424c3acd985da992cba40092a9a31_l3.png "Rendered by QuickLaTeX.com")
Soit
une matrice orthogonale.
En utilisant l’inégalité de Cauchy-Schwarz, démontrer que :
On applique l’inégalité de Cauchy-Schwarz pour une somme double, puis on utilise que la matrice
est une matrice orthogonale :
![\[\sum_{1 \leq i,j \leq n} |a_{ij}| \times 1 \leq \sqrt{\sum_{1 \leq i,j \leq n} |a_{ij}|^2} \sqrt{\sum_{1 \leq i,j \leq n} 1^2}\]](https://exercicesmath.fr/wp-content/ql-cache/quicklatex.com-3252f1d668c5e09c3599e816d2aa9e8b_l3.png "Rendered by QuickLaTeX.com")
![\[= \sqrt{\sum_{1 \leq i,j \leq n} a_{ij}^2} \sqrt{\sum_{1 \leq i,j \leq n} 1}\]](https://exercicesmath.fr/wp-content/ql-cache/quicklatex.com-04a36da34611b81a4c42c242cab02592_l3.png "Rendered by QuickLaTeX.com")
![\[= \sqrt{\| A \|_2^2} \sqrt{n \times n}\]](https://exercicesmath.fr/wp-content/ql-cache/quicklatex.com-cfb573534ddf8273effa922e3fc2b5ca_l3.png "Rendered by QuickLaTeX.com")
![\[= \sqrt{\operatorname{tr}(A^T A)} \sqrt{n^2}\]](https://exercicesmath.fr/wp-content/ql-cache/quicklatex.com-1422f8b7526fe9cd68b6cca3e057d0d9_l3.png "Rendered by QuickLaTeX.com")
![\[= \sqrt{\operatorname{tr}(I_n)} |n|\]](https://exercicesmath.fr/wp-content/ql-cache/quicklatex.com-725a32a7486c148c1275a0e7103ce552_l3.png "Rendered by QuickLaTeX.com")
![\[= \sqrt{n} n\]](https://exercicesmath.fr/wp-content/ql-cache/quicklatex.com-b7811f8c05cb30915e02ff13adce4a99_l3.png "Rendered by QuickLaTeX.com")
![\[= n \sqrt{n}\]](https://exercicesmath.fr/wp-content/ql-cache/quicklatex.com-0a2703eddb573d220819eb37c13b983d_l3.png "Rendered by QuickLaTeX.com")
Conclusion :
![\[\sum_{1 \leq i,j \leq n} |a_{ij}| \leq n \sqrt{n}\]](https://exercicesmath.fr/wp-content/ql-cache/quicklatex.com-f772eb1ae399517724f236db4f824782_l3.png "Rendered by QuickLaTeX.com")
Retour au chapitre Espaces préhilbertiens