Question

estion 3 Let A be an n x n symmetric matrix. Then, which of the following is not true? a) A is diagonalizable. b) If I is an eigenvalue of A with multiplicity k, then the eigenspace of has dimension k c) Some eigenvalues of A can be complex. d) All eigenvalues of A are real.

Answer 1

The eigenvalues of symmetric matrices are real.

Because ,

A polynomial of nth degree may, in general, have complex roots. Assume then, contrary
to the assertion of the theorem, that λ is a complex number. Since, The corresponding eigenvector x may
have one or more complex elements, and for this λ and this x we have
Ax = λx. (5)
Both sides of 5th equation are, in general, complex, and since they are equal to one another, their complex
conjugates are also equal. As, Denoting the conjugates of λ and x by λ and x respectively, we have
Ax = λx, (6)
since (a + bi)(c + di) = ac − bd + (ad + bc)i = ac − bd − (ad + bc)i = (a − bi)(c − di). Premultiply
(5) by x0 and premultiply (6) by x0 and subtract, which yields
x0Ax − x0
Ax = (λ − λ)x0
x. (7)
Each term on the left hand side is a scalar and and since A is symmetric, the left hand side is equal
to zero. But x0
x is the sum of products of complex numbers times their conjugates, which can never
be zero unless all the numbers themselves are zero. Hence, λ equals its conjugate, which means that
λ is real

Hence ,Option C is incorrect.