Linear Algebra, Quadratic Forms, Change of Variable (concept)

In summary, the conversation discusses a change of variable for a quadratic form with no cross-product term. The eigenvalues for the problem are 1 and 11, and the order in which D is constructed affects the coefficients of the quadratic form. Both answers presented in the conversation are acceptable, but the convention is to put the smaller eigenvalue first. The change of basis is to one composed of the eigenvectors of the matrix A, and taking the eigenvectors corresponding to 1 or 11 as the first basis vector will result in the two different forms of D.
  • #1
calvert11
32
0

Homework Statement



Make a change of variable that transforms the quadratic form with no cross-product term:

9x1^2 - 8x1x2 = 3x2^2

Homework Equations



A = PDP^-1
Q = y^TDy

The Attempt at a Solution



I know the answer. This is a question regarding concept.

The eigenvalues for this problem are 1 and 11. The order in which I construct D affect the coefficients of the quadratic form following a change of variable.

Consider constructing D as either

1 0
0 11

or

11 0
0 1

The two resulting quadratic forms would have their coefficients switched.
Basically, I'm asking, is this ok?

Would both answers be acceptable?
 
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  • #2
yes they are both ok, though the convention is usually to put the smaller eigenvalue first

moving the the diagonal coordinate system is change of basis to one composed of the eignevectors of the matrix A .

If you take the eigenvector corresponding to 1 as your first basis vector, and make it the first column of P then you will end up with the first form of D.

Alternately if you take the eigenvector corresponding to 11 as your first basis vector, and make it the first column of P then you will end up with the 2nd form of D.
 

FAQ: Linear Algebra, Quadratic Forms, Change of Variable (concept)

1. What is linear algebra and why is it important?

Linear algebra is a branch of mathematics that deals with linear equations and their representations in vector spaces. It is used in various fields such as physics, engineering, economics, and computer graphics. It allows us to solve complex systems of equations and understand the relationships between different variables in a mathematical model.

2. What is a quadratic form and how is it related to linear algebra?

A quadratic form is a mathematical expression that involves quadratic terms of variables. It can be represented in the form of a matrix, making it closely related to linear algebra. Quadratic forms are used to study the properties of quadratic equations and to solve optimization problems.

3. What is the concept of change of variable in linear algebra?

Change of variable is a technique used in linear algebra to transform a system of equations into a more convenient form for solving. It involves substituting new variables for the existing ones in a way that simplifies the equations or reveals underlying patterns. This concept is particularly useful in solving differential equations and performing transformations in geometry.

4. How is linear algebra used in data analysis and machine learning?

Linear algebra is a fundamental tool in data analysis and machine learning. It is used to represent and manipulate large datasets, perform statistical analysis, and develop predictive models. Techniques such as matrix factorization, least squares, and eigenvalue decomposition are commonly used in data analysis and machine learning algorithms.

5. What are some real-world applications of linear algebra, quadratic forms, and change of variable?

Linear algebra, quadratic forms, and change of variable have numerous real-world applications in various fields. They are used in image and signal processing, computer graphics, robotics, finance, and optimization problems in engineering and physics. They also play a crucial role in developing algorithms for machine learning, data analysis, and artificial intelligence.

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