I have a vector $v(0,1,1,)$ in $\mathbb{R}^3$ and its being tranformed by a matrix $A$ to a new vector $v(1,3,2)$. How do I know if those 2 vectors are moving in the same direction, and also without calculating the determinant of A how do I know if the determinant is positive or negative? My understanding is if the vector changes direction about an axis then the determinant is negative, otherwise it's positive?
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You aren't able to tell the sign of the determinant just from one input-output pair. I can construct matrices with a positive and negative determinant that transforms that vector in that way. You're right about the vector changing the direction. But if two vectors change directions, the determinant is positive (double negative). Maybe I'm wrong, but you need at least 3 input-output pairs to determine whether the sign of the determinant is positive or negative, and even then those 3 inputs have to be linearly independent...I wonder if I'm right on that. – Spencer Kraisler Mar 27 '20 at 20:34
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I know the matrix A and it is linearly independent but how can you tell the sign without computing the determinant and knowing the matrix? – mathcomp guy Mar 27 '20 at 20:37
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I'd be a bit careful on the terminology you're using. A matrix can't be linearly independent, but I'm guessing you're talking about the column vectors (right?). Here's a promising lead: https://math.stackexchange.com/questions/900553/is-there-an-easy-way-to-find-the-sign-of-the-determinant-of-an-orthogonal-matrix – Spencer Kraisler Mar 27 '20 at 20:45
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yes the columns are linearly independent in the sense that the vectors x,y,z span R^3 and the matrix is not orthogonal – mathcomp guy Mar 27 '20 at 20:51
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What do you mean by "(if those) 2 vectors are moving in the same direction"? Please be more precise. – Ben Grossmann Mar 27 '20 at 21:07
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My question is if you graph both these vectors their direction to my understanding does not change about an axis, yet the determinant of A is negative and I'm unclear as to why. My understanding is if the vectors do not change directions about an axis then the determinant is positive? – mathcomp guy Mar 27 '20 at 21:11
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@mathcompguy You're continuing to use bizarre terminology. What does it mean for a vector to "change about an axis"? – Ben Grossmann Mar 27 '20 at 21:13
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I meant it does not change 90 degrees about an axis in which case the determinant of the matrix transform would be negative to my understanding – mathcomp guy Mar 27 '20 at 21:18
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In order for me (and for most other people, I suspect) to understand your understanding of negative determinants, you will have to explain it in a way that doesn't use the phrase "change about an axis". Or, you'll have to explain what "change about an axis" means (in a way that doesn't use the phrase that you're trying to define) – Ben Grossmann Mar 27 '20 at 21:21
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For example in $R^2$ if you have the point (2,3) and it shifts 90 degrees about the y-axis to (-2,3) you multiply the (2,3) vector by matrix (-1,0 | 0,1) whose determinant is -1. That is what I'm simply trying to understand with respect to the sign of determinant and graphing vectors – mathcomp guy Mar 27 '20 at 21:29
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If $A$ is a $3 \times 3$ matrix, then the following statements are equivalent:
- $\det(A)$ is negative,
- The columns of $A$ form a negatively oriented basis,
- The columns of $A$ satisfy the left-hand rule (as opposed to the right-hand rule).
Ben Grossmann
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In response to a comment on the question: it is true that if $A$ has a negative determinant, then there exists a vector $v$ for which $Av = -kv$ for some number $k>0$. However, the reverse is not true (for instance, since the rotation by $180^\circ$ has positive determinant). – Ben Grossmann Mar 27 '20 at 21:14