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.

21. a. The matrix of a quadratic form is a symmetric matrix.

b. A quadratic form has no cross-product terms if and only if the matrix of the quadratic form is a diagonal matrix.

c. The principal axes of a quadratic form \({x^T}Ax\) are eigenvectors of \(A\).

d. A positive definite quadratic form\(Q\)satisfies\(Q\left( x \right){\bf{ > 0}}\)for all\({\bf{x}}\)in\({{\bf{R}}^{\bf{n}}}\).

e.If the eigenvalues of a symmetric matrix \(A\) are all positive, then the quadratic form \({x^T}Ax\) is positive definite.

f. A Cholesky factorization of a symmetric matrix \(A\) has the form \(A = {R^T}R\) for an upper triangular matrix \(R\) with positive diagonal entries

Short answer

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

Step-by-step solution

(a) Step 1: Show that the matrix of a quadratic form is a symmetric matrix

As per the definition of quadratic form definition: A quadratic form on\({\mathbb{R}^n}\)is a function\(Q\)defined on\({\mathbb{R}^n}\)whose value at a vector\({\bf{x}}\)in\({\mathbb{R}^n}\)can be computed by an expression of the form\(Q({\bf{x}}) = {{\bf{x}}^T}A{\bf{x}}\)where\(A\)is a\(n \times n\)symmetric matrix. The matrix\(A\)is called the matrix of the quadratic form.

Thus, the matrix of the quadratic form is always symmetric. Hence the statement is true.

(b) Step 2: Show that a quadratic form has no cross-product terms if and only if the matrix of the quadratic form is a diagonal matrix

Let \(A\) be a diagonal matrix.

\(\begin{aligned}{}Q({\bf{x}}) &= {{\bf{x}}^T}A{\bf{x}},\\ &= {a_{11}}x_1^2 + {a_{22}}x_2^2 + ... + {a_{mn}}x_n^2\end{aligned}\)

It can be seen that the quadratic form of matrix A has no cross-product terms.

Hence the statement is true

(c) Step 3: Show that the principal axes of a quadratic form \({x^T}Ax\) are eigenvectors of \(A\)

By the principal axis’s theorem, the principal axes of a quadratic form \(Q({\bf{x}}) = {{\bf{x}}^T}A{\bf{x}},\) are the eigenvectors of the matrix therefore the statement is true.

(d) Step 4: Show that a positive definite quadratic form \(Q\) satisfies \(Q\left( x \right){\bf{ > 0}}\) for all \({\bf{x}}\) in \({{\bf{R}}^{\bf{n}}}\).

If the eigenvalues of a symmetric matrix\(A\)are all positive, then the quadratic form\(Q({\bf{x}}) = {{\bf{x}}^T}A{\bf{x}},\)is positive definite.When\({\rm{x}} = 0\), the value of \(Q\left( {\rm{x}} \right) = 0\). This implies that for all values of \({\rm{x}}\), \(Q\left( {\rm{x}} \right)\) is not greater than 0.

Therefore, the statement is False.

(e) Step 5: Show that If the eigenvalues of a symmetric matrix \(A\) are all positive, then the quadratic form \({x^T}Ax\) is positive definite

Since thesymmetric matrix\(A\) is all positive then the\(Q({\bf{x}}) = {{\bf{x}}^T}A{\bf{x}},\)is positive definite.

Hence the statement is true.

(f) Step 6: Show that a Cholesky factorization of a symmetric matrix \(A\) has the form \(A = {R^T}R\) for an upper triangular matrix \(R\) with positive diagonal entries

Asymmetric matrix\(A\)are positive definite if\(A\)can be a factor in the form\(A = {R^T}R\).

Such a Cholesky factorization is possible if and only if\(A\)is positive definite.

Therefore, the statement is true