#Please help me get this answer for matrices

53 messages · Page 1 of 1 (latest)

copper badger
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Im not sure how to get the required eigenvectors

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after reducing the matrix A to
| 1 -1 0 |
| 0 0 0 |
| 0 0 0 |

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for A + 4*I

fading mantle
copper badger
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let me show my working

fading mantle
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good idea....

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Yeah your RREF for A+4I is correct, so just solve (A+4I)x=0

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clear that y and z are free, and x=y

copper badger
fading mantle
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so a vector in ker(A+4I) is [x,y,z]=[y,y,z]=y[1,1,0]+z[0,0,1]

copper badger
fading mantle
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[y,y,z]=y[1,1,0]+z[0,0,1]

copper badger
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I have no idea how they got this solution

image.png
fading mantle
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just undo the linear combination.

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you take 2 linearly independent vectors from the -4-eigenspace

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you're given A is diagonalizable, hence the arithmetic and geometric multiplicities must be equal, so since -4 appears twice, the -4-eigenspace is a 2 dimensional subspace

copper badger
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right

fading mantle
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so any basis for the -4-eigenspace will fill the last 2 columns

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they just picked not obvious ones ig.

copper badger
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wait wht

fading mantle
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what?

copper badger
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I am still a bit stuck on the "undo the linear combination"

fading mantle
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do you agree [y,y,z]=y[1,1,0]+z[0,0,1]?

copper badger
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yes

fading mantle
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so then you're not stuck

copper badger
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but how does z have a value when

fading mantle
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z is any real number

copper badger
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[1,-1,0] = x - y?

fading mantle
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since y and z were free variables

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x and y are scalars, so that equation is complete nonsense

copper badger
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right so x = y and then [y,y,z] = y[1,1,0] + z[0,0,1] then how did they procede?

fading mantle
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so ker(A+4I)=span([1,1,0],[0,0,1]), and clearly the 2 vectors are linearly independent

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so they're a basis for the -4-eigenspace

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so you can put those as the columns..

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or, as I said, you can take any basis for for the -4-eigenspace

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they, again as I already said, took a non-obvious basis.

copper badger
fading mantle
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Yeah

copper badger
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could z be any value?

fading mantle
copper badger
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so they just

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happen to choose those numbers

fading mantle
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again, as I said

copper badger
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literally

fading mantle
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they just chose a non-obvious basis

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read what I say please.

copper badger
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got it.

copper badger
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so another valid eigen vector could be like [-2,-2, 5] ?

fading mantle
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yes since that's in the span

copper badger
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right