Properties of the Symmetric Bilinear FormThe symmetric bilinear form is represented by the symmetric matrix AAA. Key properties:The matrix AAA has real eigenvalues, as it is symmetric.The bilinear form is positive definite if all eigenvalues are positive, negative definite if all are negative, and indefinite if it has both positive and negative eigenvalues.From the problem, since (v, v) > 0 and (w ,w) < 0 , the bilinear form is indefinite. This means AAA has one positive eigenvalue and one negative eigenvalue..

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CSIR NET MATHEMATICAL SCIENCES

Linear Algebra

Question

rank A = 1

rank A = 0

Solution

Correct option is D

Properties of the Symmetric Bilinear Form

The symmetric bilinear form is represented by the symmetric matrix $A$ . Key properties:

The matrix $A$ has real eigenvalues, as it is symmetric.
The bilinear form is positive definite if all eigenvalues are positive, negative definite if all are negative, and indefinite if it has both positive and negative eigenvalues.

From the problem, since (v, v) > 0 and (w ,w) < 0 , the bilinear form is indefinite. This means $A$ has one positive eigenvalue and one negative eigenvalue.