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Further Maths question

Hello,

I was wondering if someone knew about why if a matrix transformation is singular, the transformation has only 1 invariant line. A question I came across said that it being singular means there cannot be a second invariant line. Thank you!
Reply 1
Original post by Justagirl378
Hello,

I was wondering if someone knew about why if a matrix transformation is singular, the transformation has only 1 invariant line. A question I came across said that it being singular means there cannot be a second invariant line. Thank you!

Assuming it's a 2*2 nonzero matrix. What do you understand about singularity and invariant lines?
Original post by mqb2766
Assuming it's a 2*2 nonzero matrix. What do you understand about singularity and invariant lines?

So singularity is when the determinant is zero so the matrix does not have an inverse. But I’m not sure about the link between singularity and invariant lines.
Reply 3
Another interpretation of singularity is that there must exist some nonzero x such that
Ax = 0

If there was two (different) invariant lines, would this be possible?
Original post by mqb2766
Another interpretation of singularity is that there must exist some nonzero x such that
Ax = 0

If there was two (different) invariant lines, would this be possible?

Is A a matrix? If there were two different invariant lines then I guess it wouldn’t equal zero anymore so there can only be 1 invariant line- that’s quite helpful, thank you!
Reply 5
Original post by Justagirl378
Is A a matrix? If there were two different invariant lines then I guess it wouldn’t equal zero anymore so there can only be 1 invariant line- that’s quite helpful, thank you!

Yes A is the matrix and x a vector. If you had two invariant lines, you could imagine they were axes. Each non zero vector x could be represented in terms of these two axes and multiplying x by A would produce another non zero vector. So two invariant lines would mean A was nonsingular.
Original post by mqb2766
Yes A is the matrix and x a vector. If you had two invariant lines, you could imagine they were axes. Each non zero vector x could be represented in terms of these two axes and multiplying x by A would produce another non zero vector. So two invariant lines would mean A was nonsingular.

Right! that’s really helpful, I think I understand that now- thank you!!
Reply 7
Original post by mqb2766
Yes A is the matrix and x a vector. If you had two invariant lines, you could imagine they were axes. Each non zero vector x could be represented in terms of these two axes and multiplying x by A would produce another non zero vector. So two invariant lines would mean A was nonsingular.

I know that A is a singular 2x2 matrix, and there are a whole line of vectors that fit x, such that Ax = 0. There's also another line of vectors that are parallel to the new two basis vectors after the transformation, and this line is invariant but does not fit the description of x. So, you're saying if there were two invariant lines like axes, and after they have been transformed by the matrix A, they wouldn't collapse to the origin and so would every other non-zero vector x? Therefore, it contradicts the singularity of A because these non-zero vectors do collapse to the origin, because there are no singular matrices A that have two invariant lines that behave like axes in the first place? So, the whole line of vectors that fit the description of x, cannot be invariant, because they also do not behave like axes and collapse to the origin instead?

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