Skew-Symmetric Matrix Property and Quadratic Forms

We are given a skew-symmetric matrix $A$ with real entries and a real vector $x$ such that $x^T x = 1$. We need to find the value of the quantity $q = x^T A x$.

Properties of Skew-Symmetric Matrices

• A matrix $A$ is skew-symmetric if its transpose equals its negative: $A^T = -A$.
• For a real skew-symmetric matrix, all diagonal elements are zero. This is because $a_{ii} = -a_{ii}$, implying $2a_{ii} = 0$, so $a_{ii} = 0$.

Evaluating $x^T A x$

Let the quantity be $q = x^T A x$. Since $x$ and $A$ contain real entries, $q$ must be a scalar (a single real number). Consider the transpose of $q$:

1. $q^T = (x^T A x)^T$
2. Using the property $(ABC)^T = C^T B^T A^T$, we get: $q^T = x^T A^T (x^T)^T$
3. Since $(x^T)^T = x$, we have: $q^T = x^T A^T x$
4. Given that $A$ is skew-symmetric, $A^T = -A$. Substituting this: $q^T = x^T (-A) x$
5. $q^T = - (x^T A x)$
6. Since $q = x^T A x$, we have: $q^T = -q$

Because $q$ is a scalar, its transpose $q^T$ is equal to itself, $q^T = q$. Therefore, we have the equation:

$q = -q$

Adding $q$ to both sides gives:

$2q = 0$

Dividing by 2, we find:

$q = 0$

Thus, the quantity $x^T A x$ evaluates to 0.

The condition $x^T x = 1$ ensures $x$ is a unit vector but doesn't change the result derived from the skew-symmetric property of $A$.