DLA Discovery guide

Print preview

Highlight workspace

Discovery guide 41.1 Discovery guide

Discovery 41.1.

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 41.2. Patterns.

Let’s analyze the patterns of Discovery 41.1.

🔗

(a)

A multivariable function \(q(x_1,x_2,\dotsc,x_n) \) of the kind explored in Discovery 41.1 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 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 41.3. 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 41.4. 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 41.5. 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 41.4, 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.

🔗

Prev Top Next