Question

In Exercises 21 and 22, matrices are \(n \times n\) and vectors are in \({{\bf{R}}^{\bf{n}}}\) . Mark each statement True or false. Justify each answer.

22. a. The expression \({\left\| {\bf{x}} \right\|^{\bf{2}}}\) is not a quadratic form.

b. If \(A\) is symmetric and \(P\) is an orthogonal matrix, then be the change of the variable \({\rm{x}} = P{\rm{y}}\) transforms \({x^T}Ax\) into a quadratic form with no cross product term.

c. If \(A\)is a\({\bf{2 \times 2}}\)symmetric matrix, then the set of\(x\)such that\({x^T}Ax = c\) corresponds to either a circle, an ellipse or a hyperbola.

d. An indefinite quadratic form is neither positive semidefinite nor negative semidefinite.

e.If \(A\) is symmetric and the quadratic form \({x^T}Ax\) has only negative values for \({\bf{x}} \ne {\bf{0}}\) then the eigenvalues of \(A\) are all positive.

Short answer

1. The given statement is False.
2. The given statement is False.
3. The given statement is False.
4. The given statement is True.
5. The given statement is False.

Step-by-step solution

(a) Step 1: Show that the expression \({\left\| {\bf{x}} \right\|^{\bf{2}}}\) is not a quadratic form

Since the identity matrix\(I\)is symmetric so,

\(\begin{aligned}{}Q({\bf{x}}) = {{\bf{x}}^T}I{\bf{x}}\\ = {{\bf{x}}^T}{\bf{x}}\\ = {\left\| {\bf{x}} \right\|^2}\end{aligned}\)

The above conclusion proves that an expression\({\left\| {\bf{x}} \right\|^2}\)is a quadratic form.

Hence the statement is false.

(b) Step 2: Show that if \(A\) is symmetric and \(P\) is an orthogonal matrix, then be the change of the variable \(x = Py\) transforms \({{\bf{x}}^{\bf{T}}}{\bf{Ax}}\) into a quadratic form with no cross-product term

By the principal Axes theorem, there is an orthogonal changeof the variable \(x = Py\) which transforms \({{\bf{x}}^T}A{\bf{x}}\) into a quadratic form \({y^T}Dy\) with no cross-product term.

Hence the statement is False.

(c) Step 3: Show that if \(A\) is a \({\bf{2 \times 2}}\) symmetric matrix, then the set of \({\bf{x}}\) such that \({x^T}Ax = c\) corresponds to either a circle, an ellipse or a hyperbola

From a geometric view of Principle axes, if\(A\)is\(2 \times 2\)a symmetric matrix then the set of\({\rm{x}}\)such that\({{\bf{x}}^T}A{\bf{x}} = c\)(constant) corresponds to a circle, ellipse, a hyperbola, two intersecting lines, a single point or contains no points at all. Hence, it can take any of the forms from a circle, ellipse, a hyperbola, two intersecting lines, a single point or contains no points at all.

Hence the statement is false.

(d) Step 4: Show that an indefinite quadratic form is neither positive semidefinite nor negative semidefinite

A quadratic form\(Q({\bf{x}})\)is indefinite then neither both positive and negative values. So, thequadratic form\(Q\)is indefinite form is neither positive semidefinite nor negative semidefinite.

Therefore, the statement is True.

(e) Step 5: Show that if \(A\) is symmetric and the quadratic form \({x^T}Ax\) has only negative values for \({\bf{x}} \ne {\bf{0}}\) then the eigenvalues of \(A\) are all positive.

If\(Q(x) = {x^T}Ax < 0,Q(x) = {x^T}Ax < 0\),for all\(x \ne 0\),then\(Q\)is negative definite. The\(Q\)is negative definite if and only if the eigenvalues of are all negative.

Hence the statement is false.