Hi,
I have got the following matrix. I have found the eigen values but in some eq x, y & z terms are vanishing, so how to find the value of eigen vector??? Also why we have to do normalization??
\(\displaystyle \mbox{Given: }\, A\, =\, \left[\begin{array}{ccc}1&1&0\\1&1&0\\0&0&1\end{array} \right]\)
A−λI=0:
. . . . .⎣⎢⎡110110001⎦⎥⎤−⎣⎢⎡λ000λ000λ⎦⎥⎤=0
. . . . .⎣⎢⎡1−λ1011−λ0001−λ⎦⎥⎤=0
. . . . .(1−λ)∣∣∣∣∣1−λ001−λ∣∣∣∣∣−1∣∣∣∣∣1001−λ∣∣∣∣∣+0=0
. . . . .(1−λ)(1−λ)2−(1−λ)=0
Taking the common factor of (1−λ) out front:
. . . . .(1−λ)[(1−λ)2−1]=0
First eigen value λ1=1
Now consider:
. . . . .[(1−λ)2−1]=0
. . . . .1−2λ+λ2−1=0
. . . . .−2λ+λ2=0
. . . . .λ(λ−2)=0
. . . . .\(\displaystyle \lambda_2\, =\, 0\, \mbox{ and }\, \lambda_3\, =\, 2\)
For λ1=1,Ax=λ⋅x⋅I
. . . . .⎣⎢⎡110110001⎦⎥⎤⎣⎢⎡xyz⎦⎥⎤=1⋅⎣⎢⎡xyz⎦⎥⎤
. . . . .\(\displaystyle \begin{array}{lcr}x\, +\, y\, =\, x\, &\,&\mbox{eqn}\, (1)\\x\, +\, y\, =\, y\, &\,&\mbox{eqn}\, (2)\\z\, =\, z\, &\,&\mbox{eqn}\, (3)\end{array}\)
Above x vanishes in eq(1), y vanishes in eq(2) & z vanishes in eq(3).
What is the value of μ1?????
For λ2=0:
. . . . .⎣⎢⎡110110001⎦⎥⎤⋅⎣⎢⎡xyz⎦⎥⎤=0⋅⎣⎢⎡xyz⎦⎥⎤
. . . . .\(\displaystyle \begin{array}{lcr}x\, +\, y\, =\, 0&\,&\mbox{eqn}\, (4)\\x\, +\, y\, =\, 0&\,&\mbox{eqn}\, (5)\\z\, =\, 0&\,&\mbox{eqn}\, (6)\end{array}\)
In my view, it should be:
. . . . .μ=⎣⎢⎡−1−10⎦⎥⎤
but teacher has got different answer. In addition to this he has done normalization, please guide me what is the need for normalization
For λ=2:
. . . . .⎣⎢⎡110110001⎦⎥⎤⎣⎢⎡xyz⎦⎥⎤=2⋅⎣⎢⎡xyz⎦⎥⎤
. . . . .\(\displaystyle \begin{array}{lcr}x\, +\, y\, =\, 2x&\,&\mbox{eqn}\, (7)\\x\, =\, -y&\,&\,\\x\, +\, y\, =\, 2y&\,&\mbox{eqn}\, (8)\\y\, =\, x&\,&\,\\z\, =\, 2z&\,&\mbox{eqn}\, (9)\end{array}\)
Therefore:
. . . . .μ=⎣⎢⎡−110⎦⎥⎤
Why we have to normalize the vector?
Somebody please guide me.
Zulfi.
I have got the following matrix. I have found the eigen values but in some eq x, y & z terms are vanishing, so how to find the value of eigen vector??? Also why we have to do normalization??
\(\displaystyle \mbox{Given: }\, A\, =\, \left[\begin{array}{ccc}1&1&0\\1&1&0\\0&0&1\end{array} \right]\)
A−λI=0:
. . . . .⎣⎢⎡110110001⎦⎥⎤−⎣⎢⎡λ000λ000λ⎦⎥⎤=0
. . . . .⎣⎢⎡1−λ1011−λ0001−λ⎦⎥⎤=0
. . . . .(1−λ)∣∣∣∣∣1−λ001−λ∣∣∣∣∣−1∣∣∣∣∣1001−λ∣∣∣∣∣+0=0
. . . . .(1−λ)(1−λ)2−(1−λ)=0
Taking the common factor of (1−λ) out front:
. . . . .(1−λ)[(1−λ)2−1]=0
First eigen value λ1=1
Now consider:
. . . . .[(1−λ)2−1]=0
. . . . .1−2λ+λ2−1=0
. . . . .−2λ+λ2=0
. . . . .λ(λ−2)=0
. . . . .\(\displaystyle \lambda_2\, =\, 0\, \mbox{ and }\, \lambda_3\, =\, 2\)
For λ1=1,Ax=λ⋅x⋅I
. . . . .⎣⎢⎡110110001⎦⎥⎤⎣⎢⎡xyz⎦⎥⎤=1⋅⎣⎢⎡xyz⎦⎥⎤
. . . . .\(\displaystyle \begin{array}{lcr}x\, +\, y\, =\, x\, &\,&\mbox{eqn}\, (1)\\x\, +\, y\, =\, y\, &\,&\mbox{eqn}\, (2)\\z\, =\, z\, &\,&\mbox{eqn}\, (3)\end{array}\)
Above x vanishes in eq(1), y vanishes in eq(2) & z vanishes in eq(3).
What is the value of μ1?????
For λ2=0:
. . . . .⎣⎢⎡110110001⎦⎥⎤⋅⎣⎢⎡xyz⎦⎥⎤=0⋅⎣⎢⎡xyz⎦⎥⎤
. . . . .\(\displaystyle \begin{array}{lcr}x\, +\, y\, =\, 0&\,&\mbox{eqn}\, (4)\\x\, +\, y\, =\, 0&\,&\mbox{eqn}\, (5)\\z\, =\, 0&\,&\mbox{eqn}\, (6)\end{array}\)
In my view, it should be:
. . . . .μ=⎣⎢⎡−1−10⎦⎥⎤
but teacher has got different answer. In addition to this he has done normalization, please guide me what is the need for normalization
For λ=2:
. . . . .⎣⎢⎡110110001⎦⎥⎤⎣⎢⎡xyz⎦⎥⎤=2⋅⎣⎢⎡xyz⎦⎥⎤
. . . . .\(\displaystyle \begin{array}{lcr}x\, +\, y\, =\, 2x&\,&\mbox{eqn}\, (7)\\x\, =\, -y&\,&\,\\x\, +\, y\, =\, 2y&\,&\mbox{eqn}\, (8)\\y\, =\, x&\,&\,\\z\, =\, 2z&\,&\mbox{eqn}\, (9)\end{array}\)
Therefore:
. . . . .μ=⎣⎢⎡−110⎦⎥⎤
Why we have to normalize the vector?
Somebody please guide me.
Zulfi.
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