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Discovery guide 41.1 Discovery guide

Discovery 41.1.1.

In each of the following, you are given an \(n \times n\) matrix \(A\text{.}\) Using the matrix to create a pairing
\begin{equation*} \inprod{\uvec{u}}{\uvec{v}} = \utrans{\uvec{v}} A \uvec{u} \end{equation*}
between column vectors in \(\R^n\text{,}\) obtain a description for the multivariable function
\begin{equation*} q(x_1,x_2,\dotsc,x_n) = q(\uvec{x}) = \inprod{\uvec{x}}{\uvec{x}} \end{equation*}
as a formula in the input coordinate variables \(x_1,x_2,\dotsc,x_n\text{.}\)
Warning. The pairings in this discovery activity are not necessarily inner products on \(\R^n\) — see Proposition 36.6.9 and Theorem 36.6.10.

(a)

\(\displaystyle A = \bidentmattwo \text{,}\) \(q_A(\uvec{x}) = \utrans{\uvec{x}} A \uvec{x} = \fillinmath{XXXXXXXXXXXXXXXXXXXX} \text{.}\)

(b)

\(\displaystyle A = \begin{abmatrix}{cr} 2 \amp 0 \\ 0 \amp -3 \end{abmatrix} \) \(q_A(\uvec{x}) = \utrans{\uvec{x}} A \uvec{x} = \fillinmath{XXXXXXXXXXXXXXXXXXXX} \text{.}\)

(c)

\(\displaystyle A = \begin{abmatrix}{rrr} 1 \amp -2 \amp 0 \\ 0 \amp 2 \amp 0 \\ 0 \amp 0 \amp -3 \end{abmatrix} \text{.}\) \(q_A(\uvec{x}) = \utrans{\uvec{x}} A \uvec{x} = \fillinmath{XXXXXXXXXXXXXXXXXXXXXXXXXXXXXX} \text{.}\)

(d)

\(\displaystyle A = \begin{abmatrix}{rrr} 1 \amp -1 \amp 0 \\ -1 \amp 2 \amp 0 \\ 4 \amp 0 \amp -3 \end{abmatrix} \text{.}\) \(q_A(\uvec{x}) = \utrans{\uvec{x}} A \uvec{x} = \fillinmath{XXXXXXXXXXXXXXXXXXXXXXXXXXXXXX} \text{.}\)

Discovery 41.1.2. Patterns.

Let’s analyze the patterns of Discovery 41.1.1.

(a)

A multivariable function \(q(x_1,x_2,\dotsc,x_n) \) of the kind explored in Discovery 41.1.1 is called a quadratic form.
Can you see why from the example formulas you computed?

(b)

What is the pattern of how the coefficients in each formula from Discovery 41.1.1 relate to the entries in the corresponding matrix?

(c)

Make some example quadratic polynomials for yourself, and then for each example determine a matrix \(A\) so that \(\utrans{\uvec{x}} A \uvec{x}\) gives you back your quadratic polynomial. Can you determine a symmetric matrix \(A\) that represents your quadratic polynomial?
Make sure to mix it up! (Don’t just use “diagonal” quadratic polynomials.)

Discovery 41.1.3. Shapes.

A level set \(q(\uvec{x}) = c\) of a quadratic form creates a quadric curve/surface/hypersurface in \(\R^n\text{.}\)
In each of the following, determine the type of curve in \(\R^2\) or the surface in \(\R^3\) defined by setting \(q(\uvec{x}) = 1\text{.}\)

(a)

\(q(x_1,x_2) = x_1^2 + x_2^2 \text{.}\)

(b)

\(q(x_1,x_2) = x_1^2 + 2 x_2^2 \text{.}\)

(c)

\(q(x_1,x_2) = x_1^2 - x_2^2 \text{.}\)

(d)

\(q(x_1,x_2,x_3) = x_1^2 + x_2^2 + x_3^2 \text{.}\)

(e)

\(q(x_1,x_2,x_3) = x_1^2 + 2 x_2^2 + x_3^2 \text{.}\)

Discovery 41.1.4. Change of variables.

Suppose \(A\) is a symmetric real matrix and \(q_A(\uvec{x}) = \utrans{\uvec{x}} A \uvec{x}\) is the associated quadratic form.
Symmetric matrices are orthogonally diagonalizable, so there exists orthogonal \(P\) so that \(D = \utrans{P} A P\) is diagonal, with the eigenvalues \(\lambda_1,\lambda_2,\dotsc,\lambda_n\) of \(A\) down the diagonal.

(a)

Using the change of variables \(\uvec{x} = P \uvec{w}\text{,}\) express \(q_A(\uvec{x})\) in terms of \(\uvec{w}\text{:}\)
\begin{equation*} q_A(\uvec{x}) = \utrans{\uvec{x}} A \uvec{x} = \utrans{\uvec{w}} \boxed{\phantom{D}} \uvec{w} \text{.} \end{equation*}

(b)

Based on your answer to Task (a), write out a quadratic polynomial for \(q_A(\uvec{x})\) in terms of the new variables \(w_1,w_2,\dotsc,w_n\) instead of in terms of \(x_1,x_2,\dotsc,x_n\text{.}\)

(c)

What is the point of this activity?

Discovery 41.1.5. Put it all together.

Consider the quadratic form \(q_A(\uvec{x})\) for symmetric matrix
\begin{equation*} A = \begin{abmatrix}{rr} 13 \amp -5 \\ -5 \amp 13 \end{abmatrix} \text{.} \end{equation*}

(a)

Write out the quadratic polynomial for \(q_A(\uvec{x}) = \utrans{\uvec{x}} A \uvec{x}\text{.}\)

(b)

The eigenvalues of \(A\) are \(\lambda_1 = 8\) and \(\lambda_2 = 18\text{.}\) Determine an orthogonal transition matrix \(P\) so that \(D = \utrans{P} A P\) is diagonal.

(c)

As in Task (a) of Discovery 41.1.4, use change of variables \(\uvec{x} = P \uvec{w}\) to express \(q_A(\uvec{x})\) as a quadratic polynomial in terms of new variables \(w,z\) (where \(\uvec{w} = (w,z)\)).

(d)

Let \(q_D(\uvec{w})\) represent the new quadratic polynomial in \(w,z\) from Task (c).
Sketch the level curve \(q_D(w,z) = 72\) on a set of \(wz\)-axes.

(e)

On a set of \(xy\)-axes, overlay a set of principal axes for \(A\text{:}\) use the orthonormal columns of your transition matrix \(P\) to determine a new set of orthogonal \(wz\)-axes overlaid on top of a set of \(xy\)-axes.
Sketch the level curve \(q_A(x,y) = 72\) on these axes by transferring your previous sketch of \(q_D(w,z)\) from your standalone set of \(wz\)-axes to your new \(wz\)-axes superimposed on the set of \(xy\)-axes.
Hint.
When transferring your sketch from one set of \(wz\)-axes to the other, remember that the columns of \(P\) are an orthonormal set. So each column vector in \(P\) represents one unit along its corresponding axis.
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