Discussion

For a homogeneous Fredholm integral equation with symmetric kernel :

\[ x(t) = \lambda \int_{a}^{b} k(t, s)\, x(s)\, ds \]

consider the following statements.

(I) The eigen values are not real.
(II) The eigen functions corresponding to distinct eigen values are orthogonal on \([a, b]\). Then :

Answer: B

Only (II) is true.

For a real symmetric kernel \(k(t, s) = k(s, t)\), the Fredholm theory parallels that of a symmetric matrix:

The eigenvalues are real. Multiply \(x = \lambda K x\) by \(\bar{x}\) and integrate; symmetry makes \(\int\!\int k\, x \bar{x}\) real, so \(\lambda\) is real. Statement (I), which says they are not real, is false.
Eigenfunctions for distinct eigenvalues are orthogonal: if \(x_{1} = \lambda_{1} K x_{1}\) and \(x_{2} = \lambda_{2} K x_{2}\), symmetry gives \((\lambda_{1} - \lambda_{2})\int x_{1} x_{2} = 0\), so \(\int_{a}^{b} x_{1} x_{2} = 0\). Statement (II) is true.

So 'only (I)', 'both' and 'both false' are ruled out.

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