Positive Definite Quadratic Form Puns

I have a 3x3 matrix that contains 3 time-varying variables. It turns out that the rref of the matrix is the identity matrix, independent of the time-varying variables.

According to https://www.math.utah.edu/~zwick/Classes/Fall2012_2270/Lectures/Lecture33_with_Examples.pdf , a matrix is positive when the pivots are positive. The pivots of an identity matrix are obviously positive; does this mean that my matrix is positive definite?

I'm working through Artin's Algebra, and while he introduces forms that are not positive definite, I haven't yet seen any examples of where the extra generality is actually used.

I have the following exercise I want to solve: https://imgur.com/a/0ucXhSW

We already solved this at the seminar a few months ago, and I have it in my notebook yet I forgot how to do it and I'm struggling to understand what my professor did.

I know how to solve a).

I got ax^2 - 4xy + 2y^2 - 2yz + z^2 .

I am trying to solve b). and yet I find nothing like my professor did by Googling. I assume "canonical form" is the same thing as "diagonalized form". All of the methods I found include eigenvalues, orthonormal vectors, and other concepts half of which I don't even know what they mean. Yet he wants it to be solved "by the Gauss method" and I assume he will do similar stuff at the exam.

What he did was manipulate the form algebraically by factoring and completing squares until he got to:

2(y - x - (1/2)z)^2 + 3/2(z - (2/3)x)^2 + (a - 8/3)x^2

Then he replaced each thing in the three parenthesis with 3 other letters and that is the canonical form. What are the steps to factoring and completing the squares in order to write that as a sum of squares?

He replaced x with m (he actually used other letters but I'm not gonna use subscripts here to save time). Therefore the form that I just wrote becomes:

2n^2 + (3/2)p^2 + (a-(8/3))m^2 , where

m = x
p  = z - (2/3)x
n  = y - x - (1/2)z

This is the second part I don't understand. How did he find out what n and p are equal to in terms of x, y, z from the first algebraic form?

Before going to point c). he wrote the matrix that changes the basis from the first form to the canonical form, or something similar to that (I don't think what I just said was mathematically correct).

  1     0      0 
 -1     1    -1/2
-2/3    0      1

At c). he found the inverse of the above matrix. He then multiplied the inverse matrix with each of the standard basis vectors of R^3 and wrote the three. This thing I understand to do.

At d). he wrote that the quadratic form Q(x) = 2n^2 + (3/2)p^2 + (a-(8/3))m^2 is strictly positive for a > 8/3 , semipositive for a = 8/3 and undefined for a < 8/3. Why is it undefined for when m gets a negative coefficient?

This was the whole solution in my notebook. I'm here literally having the solution to an exercise and not understanding it lol. This is because I don't know all the theory behind it and I'm cramming linear algebra in the last month of the exam. Before solving an exercise I try to Google the theory behind it but this time I didn't find anything

Hello, I'm trying to run a simulation and have been receiving this error and I was wondering what I needed to do to correct this. The simulation itself is fine as I have ran it with other materials so I know it is something with the custom materials I'm adding, but I'm not sure how to correct this. Thank you.

Hey all, need help with this question badly, don't even know where to start. I understand roots, parabolas, etc. and I listen in class, but this assignment has this question that I've been working on all weekend and I don't know how to answer this, I have tried solving for a, y, different forms of equations.

Thanks much.

Determine the quadratic equation in standard form (y=ax2+bx+c) given:

Zeroes at -3 and 7 and an optimal value of 4; using your equation, find the y value, if x = -5

I am giving a presentation, would it be ok if the bullet points in the powerpoint (which are very short) remove the le/la/les? For example,

Fraude électorale

Croissance de l’idéologie révolutionnaire et fondation du FLN

Merci à l'avance.

I am struggling to set up a vector minimization equation and trying to figure out what my model, "A" is.

I know A must be a positive definite matrix which means x^(T)Ax > 0 for all x =/= 0.

What is x? What is the benefit of having/assuming a positive definite matrix? Where else are these useful? Wikipedia asserts that the matrix is also symmetric, why?

Hi, I hope this question fits here.

I have a question regarding the SVD and Eigen decomposition of a symmetric, positive semi-definite matrix. As far as I know, both decompositions should be the same, but I don't understand why. Can you guys help me understand this/find me a good explanation online?

For f(x) = x(squared) - 2X -3

I can’t seem to make the X and Y value list as easy for graphing. I found the vertex coordinates but I am not sure how to plug in more X values to get Y values for the domain.

My vertex is (1,-4) and my X values I need to test are -1,0,2,3 (two values surrounding each “side)