Let A=( a b c d ). (a) Write down...

Let $A=\left(\begin{array}{ll}a & b \\ c & d\end{array}\right)$. (a) Write down necessary and sufficient conditions on the entries $a, b, c, d$ that ensures that $A$ has only real eigenvalues. (b) Verify that all symmetric $2 \times 2$ matrices satisfy your conditions. (c) Write down a non-symmetric matrix that satisfies your conditions.

Let $A=\left(\begin{array}{ll}a & b \\ c & d\end{array}\right)$.
(a) Write down necessary and sufficient conditions on the entries $a, b, c, d$ that ensures that $A$ has only real eigenvalues.
(b) Verify that all symmetric $2 \times 2$ matrices satisfy your conditions.
(c) Write down a non-symmetric matrix that satisfies your conditions.

Key Concepts

Eigenvalues

Eigenvalues are scalars associated with a square matrix that satisfy the equation A*v = ?*v for some nonzero vector v. They are found by solving the characteristic equation det(A - ?I) = 0, which for a 2×2 matrix leads to a quadratic polynomial in ?. The nature of the eigenvalues, whether real or complex, is determined by the coefficients of this polynomial.

Characteristic Polynomial and Discriminant

For a 2×2 matrix, the characteristic polynomial is given by ?² - (a+d)? + (ad-bc). The discriminant of this quadratic, (a+d)² - 4(ad-bc), must be non-negative for the eigenvalues to be real. Therefore, the condition for having only real eigenvalues is that (a+d)² - 4(ad-bc) ? 0.

Symmetric Matrices

Symmetric matrices (where the off-diagonal entries are equal) inherently have real eigenvalues because they are self-adjoint. This property is a consequence of the spectral theorem, which guarantees that every symmetric matrix can be diagonalized by an orthogonal matrix, and as a result, all its eigenvalues are real. In the context of a 2×2 symmetric matrix, the necessary discriminant condition is automatically satisfied.

Non-Symmetric Matrices with Real Eigenvalues

Although symmetry guarantees real eigenvalues, it is not a necessary condition. A non-symmetric matrix can still have real eigenvalues if the discriminant of its characteristic polynomial is non-negative. This involves carefully choosing the entries so that the inequality (a+d)² - 4(ad-bc) ? 0 holds, ensuring the roots of the quadratic (characteristic equation) are real even though the matrix is not symmetric.

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