Find the necessary and sufficient conditions on the real numbers a,b,c

In summary, if you want to diagonalize a matrix, you need to find the necessary and sufficient conditions on the real numbers a,b,c for the matrix.
  • #1
trap101
342
0
Find the necessary and sufficient conditions on the real numbers a,b,c for the matrix:
\begin{bmatrix} 1 & a & b\\ 0 & 1 & c \\ 0 & 0 & 2 \end{bmatrix} to be diagonalizable.

Attempt: Now for this one I also solved for the eigenvlues which were: λ1 = 1, λ2 = 1, λ3 = 2

So the problematic eigenvalues will be the one of multiplicity 2, i.e λ = 1.

So this means I'd have to obtain two linearly independent eigenvectors for λ = 1.

I tried solving and got to this matrix: \begin{bmatrix} 0 & a & b \\ 0&0&c \\ 0&0 & 1 \end{bmatrix}

But I won't be able to find two linearly independent eigenvectors from setting any of the variables equal to anything...I don't think. What's the next step?[/QUOTE]
 
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  • #2


Take your matrix down to reduce row echelon form.

From that point, you should be able to figure what one of the variables should be.
 
  • #3


Do you mean just reduce the variables: a , b, c? i.e: divide out a by 1/a, etc and row reduce those?
 
  • #4


trap101 said:
Do you mean just reduce the variables: a , b, c? i.e: divide out a by 1/a, etc and row reduce those?


Start with the 2nd and 3rd row first. Reduces those and then see what you can do to the top.
 
  • #5


\begin{bmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{bmatrix} a row of zeros which means one free variable, or would this mean that a, c = 0 and b could be any value?
 
  • #6


trap101 said:
\begin{bmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{bmatrix} a row of zeros which means one free variable, or would this mean that a, c = 0 and b could be any value?

Instead of dividing out the 1st row by a, leave it as a. You will still have a = 0.

Why is c = 0? the second row means x_3 = 0.
 
  • #7


trap101 said:
\begin{bmatrix} 0 & a & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{bmatrix} a row of zeros which means one free variable, or would this mean that a, c = 0 and b could be any value?

So then a = 0, x2 is a free variable, and x3 = 0. Now this leaves me with still only one free variable...actually...x1 is a free variable as well isn't it?...ahhhhhh, if that's the case I see what your getting at. But I didn't really state any conditions, all I did was solve a reduced matrix.

Thanks
 
  • #8


trap101 said:
So then a = 0, x2 is a free variable, and x3 = 0. Now this leaves me with still only one free variable...actually...x1 is a free variable as well isn't it?...ahhhhhh, if that's the case I see what your getting at. But I didn't really state any conditions, all I did was solve a reduced matrix.

Thanks

What you know now is a must be 0. The question is can b and c be anything?
 
  • #9


Well if the same process would have to be applied to find any matrix that is diagonalizable, it must mean that b, c can be any real number because they will be eliminated through row reduction.
 

1. What does it mean to find necessary and sufficient conditions?

Finding necessary and sufficient conditions means identifying the specific criteria or requirements that must be met in order for a particular statement or problem to be true or valid. These conditions are both necessary and sufficient, meaning that they are the minimum requirements for the statement or problem to hold true.

2. Why is it important to find necessary and sufficient conditions?

Identifying necessary and sufficient conditions is important because it helps us to understand the underlying principles and factors that determine the validity of a statement or the solution to a problem. It also allows us to make more accurate and precise conclusions based on these conditions.

3. How do you determine necessary and sufficient conditions?

To determine necessary and sufficient conditions, one must carefully analyze the statement or problem and identify any key factors or constraints that must be met. This may involve using logical reasoning, mathematical equations, or other analytical methods to establish the specific conditions that must be satisfied.

4. Can necessary and sufficient conditions be proven?

Yes, necessary and sufficient conditions can be proven using mathematical or logical methods. Once the conditions have been identified, they can be tested and verified to demonstrate their validity. However, it is important to note that these conditions may vary depending on the context and assumptions of the problem.

5. How do necessary and sufficient conditions differ from each other?

Necessary conditions are requirements that must be met in order for a statement or problem to be true, while sufficient conditions are conditions that, if met, guarantee the truth of the statement or solution to the problem. In other words, necessary conditions are essential for the statement to hold true, while sufficient conditions are enough to make it true.

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