Matematyka
Ewelka23p
2017-06-24 15:59:24
Obliczyć wartości własne macierzy, oraz wektory własne: (1wiersz)[1,-1,0] (2wiersz)[-1,2,0] (3wiersz)[0,0,3]
Odpowiedź
rene714
2017-06-24 22:56:04

[latex]A=left[egin{array}{ccc}1&-1&0\-1&2&0\0&0&3end{array} ight][/latex] [latex]A-lambdacdot{I}=left[egin{array}{ccc}1-lambda&-1&0\-1&2-lambda&0\0&0&3-lambdaend{array} ight][/latex] [latex]det(A-lambdacdot{I})=(1-lambda)(2-lambda)(3-lambda)-(3-lambda)=[/latex] [latex]=(3-lambda)[(1-lambda)(2-lambda)-1]=(3-lambda)(lambda^2-3lambda+1)[/latex] Równanie charakterystyczne macierzy: [latex](3-lambda)(lambda^2-3lambda+1)=0[/latex] [latex]3-lambda=0qquadlorqquad{lambda}^2-3lambda+1=0[/latex] [latex]lambda_1=3qquadlambda_2=cfrac{3-sqrt{5}}{2}qquadlambda_3=cfrac{3+sqrt{5}}{2}[/latex] [latex]lambda_1,lambda_2,lambda_3[/latex] - wartości własne. [latex]left[egin{array}{ccc}1-lambda&-1&0\-1&2-lambda&0\0&0&3-lambdaend{array} ight]cdotleft[egin{array}{c}u\v\tend{array} ight]=left[egin{array}{c}0\0\0end{array} ight][/latex] [latex]egin{cases}(1-lambda)u-v=0\-u+(2-lambda)v=0\(3-lambda)t=0end{cases}[/latex] Dla [latex]lambda_1=3[/latex] mamy: [latex]egin{cases}-2u-v=0\-u-v=0\0=0end{cases}[/latex] [latex]v=2uRightarrow u=0Rightarrow v=0[/latex] [latex][u,v,t]=[0,0,t],; tinmathbb{R}[/latex] - wektory własne dla [latex]lambda_1[/latex] Dla [latex]lambda_2=cfrac{3-sqrt{5}}{2}[/latex] mamy: [latex]egin{cases}cfrac{sqrt{5}-1}{2}u-v=0\-u+cfrac{sqrt{5}+1}{2}v=0\cfrac{sqrt{5}+3}{2}t=0end{cases}[/latex] [latex]egin{cases}(sqrt{5}-1)u-2v=0\-2u+(sqrt{5}+1)v=0\t=0end{cases}[/latex] [latex]v=cfrac{sqrt{5}-1}{2}uRightarrow-2u+cfrac{5-1}{2}u=0[/latex] [latex]uinmathbb{R}[/latex] [latex][u,v,t]=left[u,cfrac{sqrt{5}-1}{2}u,0 ight],;uinmathbb{R}[/latex] - wektory własne dla [latex]lambda_2[/latex] Dla [latex]lambda_3=cfrac{3+sqrt{5}}{2}[/latex] mamy: [latex]egin{cases}cfrac{-sqrt{5}-1}{2}u-v=0\-u+cfrac{1-sqrt{5}}{2}v=0\cfrac{3-sqrt{5}}{2}t=0end{cases}[/latex] [latex]egin{cases}(-sqrt{5}-1)u-2v=0\-2u+(1-sqrt{5})v=0\t=0end{cases}[/latex] [latex]v=cfrac{-sqrt{5}-1}{2}uRightarrow-2u+cfrac{5-1}{2}u=0[/latex] [latex]uinmathbb{R}[/latex] [latex][u,v,t]=left[u,cfrac{-sqrt{5}-1}{2}u,0 ight],;uinmathbb{R}[/latex] - wektory własne dla [latex]lambda_3[/latex]

Dodaj swoją odpowiedź