Finding Eigenvalues for a Matrix with p² + q² = 1

The question asks us to find the eigenvalues of a specific 2x2 matrix where the elements are related by the condition $p^2 + q^2 = 1$. Eigenvalues are special scalar values associated with a linear transformation (represented by a matrix).

Understanding Eigenvalues

For a square matrix $A$, an eigenvalue $\lambda$ is a scalar such that there exists a non-zero vector $v$ (called an eigenvector) satisfying the equation $Av = \lambda v$. This can be rewritten as $(A - \lambda I)v = 0$, where $I$ is the identity matrix. For a non-trivial solution (i.e., $v \neq 0$), the matrix $(A - \lambda I)$ must be singular, meaning its determinant is zero: $det(A - \lambda I) = 0$. This equation is known as the characteristic equation.

Applying the Characteristic Equation

Let the given matrix be $A = \begin{bmatrix} p & q \\ q & -p \end{bmatrix}$. We need to find the values of $\lambda$ that satisfy $det(A - \lambda I) = 0$. The identity matrix $I$ is $\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$. So, $A - \lambda I = \begin{bmatrix} p & q \\ q & -p \end{bmatrix} - \lambda \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} = \begin{bmatrix} p - \lambda & q \\ q & -p - \lambda \end{bmatrix}$.

Calculating the Determinant

Now, we calculate the determinant of $(A - \lambda I)$: $det \left( \begin{bmatrix} p - \lambda & q \\ q & -p - \lambda \end{bmatrix} \right) = (p - \lambda)(-p - \lambda) - (q)(q)$ Expanding this expression: $= -(p - \lambda)(p + \lambda) - q^2$ Using the difference of squares formula $(a-b)(a+b) = a^2 - b^2$: $= -(p^2 - \lambda^2) - q^2$ $= -p^2 + \lambda^2 - q^2$ Rearranging the terms: $= \lambda^2 - (p^2 + q^2)$

Solving for Eigenvalues

The characteristic equation is therefore: $\lambda^2 - (p^2 + q^2) = 0$ We are given the condition that $p$ and $q$ are real numbers such that $p^2 + q^2 = 1$. Substituting this condition into the characteristic equation: $\lambda^2 - 1 = 0$ Solving for $\lambda$: $\lambda^2 = 1$ $\lambda = \pm \sqrt{1}$ $\lambda = 1$ or $\lambda = -1$ Thus, the eigenvalues of the matrix are 1 and -1.

Conclusion

The eigenvalues are determined by solving the characteristic equation derived from the matrix. Given the specific condition $p^2 + q^2 = 1$, the eigenvalues simplify to 1 and -1.