Linear Algebra, solution to homogeneous equation

  • Thread starter srfriggen
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  • #1
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Homework Statement



The problem has to do with diagonalizing a square matrix, but the part I'm stuck on is this:

Bx=0, where B is the matrix with rows [000], [0,-4, 0], and [-3, 0, -4].

After performing rref on the augmented matrix Bl0, I get rows [1,0,4/3,0], [0,1,0,0], [0000].

I am trying now to solve for column vector x=[x1,x2,x3]^transposed.



Homework Equations





The Attempt at a Solution





I'm stuck. From every other problem like this I would set x3 = r, where r is an arbitrary constant. x2 has a leading entry and I set that equal to zero, which seems to be the only thing I'm getting correct. I know the answer should be [-4,0,3]^transposed, so x1=-4, x2=0, x3=3 but I can't seem to algebraically get there.

Can someone let me know what I'm missing? Thank you!
 

Answers and Replies

  • #2
301
4

Homework Statement



The problem has to do with diagonalizing a square matrix, but the part I'm stuck on is this:

Bx=0, where B is the matrix with rows [000], [0,-4, 0], and [-3, 0, -4].

After performing rref on the augmented matrix Bl0, I get rows [1,0,4/3,0], [0,1,0,0], [0000].

I am trying now to solve for column vector x=[x1,x2,x3]^transposed.



Homework Equations





The Attempt at a Solution





I'm stuck. From every other problem like this I would set x3 = r, where r is an arbitrary constant. x2 has a leading entry and I set that equal to zero, which seems to be the only thing I'm getting correct. I know the answer should be [-4,0,3]^transposed, so x1=-4, x2=0, x3=3 but I can't seem to algebraically get there.

Can someone let me know what I'm missing? Thank you!





Ok, I now see that what works to get the answer is using the equation, x1+(3/4)r = 0, where r is an arbitrary constant. Setting r = 4, I have x1=-3, x2 = 0, and x3 = 4.

So I've figured out how to get to the solution, but I just don't understand why I should have used the equation x1 + (3/4)r = 0.

I'm gonna keep plugging away at the intuition but if someone can help explain it to me I would be grateful. I don't like knowing how to do something without understanding what I'm doing.
 
  • #3
301
4
Disregard, I've got it.

I realize that the rref form of B is equal to B*, and B*x yields the correct solutions.
 
  • #5
301
4

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