Quadratic forms of symmetric matrices

In summary: You can see this if you take the derivative of the quadratic form with respect to any of the coordinate variables and plot it on a coordinate plane- it will be a straight line in the direction of the coordinate variable.
  • #1
mathusers
47
0
hi i just wanted a quick explanation of what a symmetric matrix is and what they mean by the quadratic form by the standard basis?

(1)
for example why is this a symmetric matrix

[1 3]
[3 2]

and what is the quadratic form of the matrix by the standard basis?

(2)
also how would i go about figuring out the quadratic form corresponding to the matrix by the standard basis for
[ 0 1 1]
[ 1 3 5]
[ 1 5 0]
 
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  • #2
The first one is a symmetric matrix because it IS symmetric! aij= aji.

You get the quadratic form of an n by n matrix by multiplying the row vector [x1, x2, ..., xn] times the matrix times the column vector [x1, x2, ..., xn]T.

You get the quadratic form by multiplying the matrices
[tex][x y]\left[\begin{array}{cc}1 & 3 \\ 3 & 1\end{array}\right]\left[\begin{array}{c}x \\ y\end{array}\right]= x^3+ 6xy+ y^2[/tex]

Similarly, you get the quadratic form for a 3 by 3 matrix by multiplying
[tex][x y z]\left[\begin{array}{ccc}0 & 1 & 1 \\ 1 & 3 & 5 \\ 1& 5 & 0\end{array}\right]\left[\begin{array}{c}x \\ y \\ z\end{array}\right]= 2y^2+ 2xy + 2xz+ 10yz[/tex]

Notice that given ANY matrix, doing that gives a quadratic form. Going the other way, there are many matrices corresponding to a given quadratic form- but only one symmetric matrix.
 
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  • #3
Draw a line from corner to corner, tilt your head to the left a little bit and check the elements on the left and right...
 
  • #4
How do you know which elements in the quadratic equation go into which spots in the matrix .. for a two by two it seems easy as [1 goes with x^2 1 at the bottom right goes with y^2 and the two 3's are from 3xy +3xy


but i don't get the 3 x 3 matrices from that equation :frown:
 
  • #5
Oh, now you are going the other way- from the quadratic form to the symmetric matrix!

If we have, for example, [itex]x^2- 4xy+ y^2+ 5xz+ 2yz+ z^2[/itex], I would notice first the coefficients of [itex]x^2[/itex], [itex]y^2[/itex], [itex]z^2[/itex]. They will be the diagonal elements. (In whatever order I choose to put x, y, and z in the vector- if it in that order, they would be top left, center, bottom right).

To find the other numbers, look at the coefficient of xy: -4. Since x and y are the "first" and "second" in order (I just choose them that way) I would put that coefficient in the "first row, second column" and "second row, first column", dividing it equally, -2 in each, between them in order that the matrix be symmetric.

The coefficient of xz (first and third variables in my order) is 5. Put 5/2 in the "first row, third column" and 5/2 in the "third row, first column".

Finally, the coefficient of yz (second and third variables) is 2. Put 1 in the "second row, third column" and 1 in the "third row, second column".

[tex]\left[\begin{array}{ccc} x & y & z\end{array}\right]\left[\begin{array}{ccc}1 & -2 & \frac{5}{2} \\ -2 & 1 & 1 \\\frac{5}{2} & 1 & 1\end{array}\right]\left[\begin{array}{ccc} x \\ y \\ z\end{array}\right]= x^2- 4xy+ y^2+ 5xz+ 2yz+ z^2[/tex]
 
  • #6
Alright thanks a lot makes sense :)
 
  • #7
And, since the matrix is symmetric, it is diagonalizable. There exist a new "basis" (i.e. new coordinate system) in which the matrix is diagonal. Those give the "principle directions" for the surface define by the quadratic form.
 

1. What is a quadratic form of a symmetric matrix?

A quadratic form of a symmetric matrix is a mathematical expression that involves the elements of the matrix raised to the second power, along with linear and constant terms. It can be written in the form of xTAx, where x is a vector and A is a symmetric matrix.

2. What is the significance of symmetric matrices in quadratic forms?

Symmetric matrices are important in quadratic forms because they have special properties that make it easier to analyze and solve equations involving quadratic forms. For example, a symmetric matrix has real eigenvalues and can be diagonalized, which simplifies calculations and allows for easier interpretation of the results.

3. How are quadratic forms of symmetric matrices used in real-world applications?

Quadratic forms of symmetric matrices have numerous applications in fields such as physics, engineering, and statistics. They can be used to model and analyze physical systems, design optimal structures, and perform statistical tests for data analysis.

4. What is the relationship between positive definite matrices and positive definite quadratic forms?

A positive definite matrix is a symmetric matrix where all of its eigenvalues are positive. Similarly, a positive definite quadratic form is a quadratic form that always produces a positive value for any non-zero vector x. Thus, a positive definite matrix will always produce a positive definite quadratic form.

5. How do you determine the type of a quadratic form of a symmetric matrix?

The type of a quadratic form of a symmetric matrix depends on the sign of its eigenvalues. If all eigenvalues are positive, the quadratic form is positive definite. If all eigenvalues are negative, it is negative definite. If there are both positive and negative eigenvalues, it is indefinite. Finally, if there are zero eigenvalues, it is semi-definite.

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