Basic linear algebra help. Converting equation to matrix form

In summary, the equation q=x_1 - 6x_2 + 3x_1^2 + 5x_1 x_2 can be expressed in matrix form as 1/2x^T Q*x+c^T x, where Q is a 2x2 matrix and c is a 2x1 column vector. This representation is necessary for solving the problem using Gaussian elimination.
  • #1
DyslexicHobo
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Homework Statement


Express the equation [tex]q=x_1 - 6x_2 + 3x_1^2 + 5x_1 x_2[/tex] in the matrix form [tex]1/2x^T Q*x+c^T x[/tex]

Homework Equations


The only mention of a matrix c that I could find in my book is in the section of Gaussian elimination:

[tex]c= \frac {a_{ik}}{a_{kk}}[/tex]

But I don't feel like this has anything to do with the solution form I'm trying to find.

The Attempt at a Solution


I'm not sure where to begin really. I feel like this should be a very simple problem, but I'm not sure where to start. I tried defining x = [x1 x2] and Q = [q1 q2] but I'm not sure what the "c" matrix is supposed to be.

As some background, this is taken from the review portion of my Finite Elements book.
 
Last edited:
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  • #2
Q will be a 2x2 matrix, while c will be a 2x1 column vector - they have to be of that form to match the multiplication and as q is a scalar
 

FAQ: Basic linear algebra help. Converting equation to matrix form

1. How do I convert a linear equation to matrix form?

To convert a linear equation to matrix form, you need to rearrange the equation so that all the variables are on one side and the constants are on the other side. Then, you can write the coefficients of each variable as a row in a matrix and the constants as a column vector. The resulting matrix equation will be in the form Ax = b, where A is the coefficient matrix, x is the variable vector, and b is the constant vector.

2. Why is it helpful to convert equations to matrix form?

Converting equations to matrix form allows us to use the powerful tools and techniques of linear algebra to solve them. It also makes it easier to manipulate and solve systems of equations with multiple variables and equations.

3. What are the basic operations in linear algebra?

The basic operations in linear algebra include addition, subtraction, scalar multiplication, matrix multiplication, and vector operations such as dot product and cross product. These operations are used to manipulate and solve equations in matrix form.

4. How do I solve a system of linear equations using matrices?

To solve a system of linear equations using matrices, you can use techniques such as Gaussian elimination, Gauss-Jordan elimination, or Cramer's rule. These methods involve manipulating the coefficient matrix and constant vector to solve for the variable vector x.

5. Can I use matrices to represent and solve other types of equations?

Yes, matrices can be used to represent and solve systems of nonlinear equations as well. However, the methods for solving these equations may be more complex and involve techniques such as Newton's method or gradient descent. Matrices can also be used to represent and solve differential equations, optimization problems, and many other mathematical models.

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