Proving Symmetry and Rank of B=A - aXX^T | Unit Eigenvector Help

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The discussion centers on proving the symmetry and rank of the matrix B defined as B = A - aXX^T, where A is a symmetric nxn matrix of rank r (r ≥ 1) and X is a unit eigenvector corresponding to a non-zero eigenvalue a. It is established that B is symmetric and that the null space N(A) is a proper subspace of N(B), leading to the conclusion that rank B = r - 1. The user successfully demonstrated that X is in N(B) but not in N(A), confirming the relationship between the null spaces.

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If A is a symmetric nxn mx of rank<=r>=1 and X is a unit eigenvector of A, with eigenvalue a not= 0, let B=A - aXX^T. Show that B is symmetric and that N(A) is a proper subspace of N(B). Conclude that rank B=<r-1.
i could show X is in N(B) but not in N(A). Does anyone know how I can prove it in general? Then,I could prove N(A) is a proper subspace of N(B). Thanks!
 
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