Question

The matrix of the guadratic form \(q=4 x^{2}-2 y^{2}+2^{2}-2 x y+6 x\) is

Short answer

The matrix of the quadratic form is \( \begin{bmatrix} 4 & -1 & 0 \\ -1 & -2 & 0 \\ 0 & 0 & 2 \end{bmatrix} \).

Step-by-step solution

Identify the quadratic form

The given quadratic form is \( q = 4x^2 - 2y^2 + 2^2 - 2xy + 6x \). Here, it appears the term \(2^2\) might be a typo, as constants do not fit into a quadratic form transformation. Let's assume it is meant to be \(2z^2\) instead.

Rewrite the quadratic form

Assume the typo is addressed: the quadratic form is \( q = 4x^2 - 2y^2 + 2z^2 - 2xy + 6x \). Identify quadratic terms (\(x^2, y^2, z^2\)), cross-product terms (\(xy\)), and linear terms (\(x\)).

Express the quadratic form in matrix notation

The quadratic form can be expressed in matrix notation as \( q = \mathbf{x}^T \mathbf{A} \mathbf{x} + \mathbf{b}^T \mathbf{x} + c \), where \( \mathbf{x} = \begin{bmatrix} x \ y \ z \end{bmatrix} \). \(\mathbf{A}\) is the symmetric matrix of quadratic coefficients, \(\mathbf{b}\) is the vector for linear terms, and \(c\) is a constant.

Define the elements of the matrix

For the given quadratic form, \(\mathbf{A}\) consists of coefficients:- \([1,1] = 4\) for \(x^2\)- \([2,2] = -2\) for \(y^2\)- \([3,3] = 2\) for \(z^2\)- \([1,2] = [2,1] = -1\) for \(xy\) (since coefficient is -2)- \([1,3], [2,3], [3,1], [3,2]\) are zero, since there are no \(xz\) or \(yz\) terms.

Construct the matrix

The matrix \( \mathbf{A} \) for the quadratic part is \[\mathbf{A} = \begin{bmatrix}4 & -1 & 0 \-1 & -2 & 0 \0 & 0 & 2 \\end{bmatrix}\]

Consider linear and constant terms (if needed)

For many purposes, only the symmetric matrix \(\mathbf{A}\) is needed to describe the quadratic form. If linear or constant terms are necessary, they are dealt with separately. In this case, focusing on \(\mathbf{A}\) satisfies the problem statement requiring the matrix of the quadratic form.

Key concepts

Symmetric Matrix

A symmetric matrix is fundamental when working with quadratic forms. Its main characteristic is that it is identical to its transpose.
In mathematical terms, if \(\mathbf{A} = \begin{bmatrix}a_{11} & a_{12} & a_{13} \a_{21} & a_{22} & a_{23} \a_{31} & a_{32} & a_{33} \end{bmatrix}\)then \(\mathbf{A}^T = \begin{bmatrix}a_{11} & a_{21} & a_{31} \a_{12} & a_{22} & a_{32} \a_{13} & a_{23} & a_{33} \end{bmatrix}\)

• All corresponding off-diagonal entries are equal: \(a_{ij} = a_{ji}\).
• The main diagonal \(a_{11}, a_{22}, a_{33},...\) contains values from the quadratic terms.

Symmetric matrices simplify many mathematical calculations and make systems predictable. They are especially nice to work with in calculus and linear algebra.

Quadratic Form

A quadratic form is an expression that involves variables squared, mixed with linear terms and possibly cross-product terms. For example, our quadratic form \(q = 4x^2 - 2y^2 + 2z^2 - 2xy + 6x\).

• Quadratic terms: These are terms where variables are squared, like \(x^2, y^2, \text{and} z^2\).
• Linear terms: These are first-degree terms in the variables, like \(6x\).

The representation of a quadratic form in a matrix equation is \( q = \mathbf{x}^T \mathbf{A} \mathbf{x} + \mathbf{b}^T \mathbf{x} + c \). Here, \(\mathbf{x}\) is a column vector of variables, \(\mathbf{A}\) is the symmetric matrix, and \(\mathbf{b}\) represents linear coefficients.

Linear Terms

In a quadratic form, linear terms are the simple first-order variables. They manifest as coefficients multiplied by a single variable, like \(6x\) in our example. Linear terms do not include products of different variables or squares.

• These appear in matrix representation as part of the vector \(\mathbf{b}\).
• They affect the position but not the shape of the quadratic graph when graphed.

For our quadratic form, only the \(x\) variable has a linear term; there were no linear terms involving \(y\) or \(z\).While often separate, linear terms can impact the overall interpretation of the function.

Cross-Product Terms

Cross-product terms occur when two different variables are multiplied, like \(xy\) in the quadratic form. These terms drive the interaction between separate variables.

• For the quadratic form \(q = 4x^2 - 2y^2 + 2z^2 - 2xy + 6x\), the cross-product term is \(-2xy\).
• In the symmetric matrix \(\mathbf{A}\), these terms are off-diagonal elements, here represented as \(a_{12}\) and \(a_{21} = -1\), deriving from \(-2xy\).

Cross-product terms shape the interaction between variables, affecting how the quadratic form behaves, influencing contours or curves where variables exhibit joint variation.