How about eigenvalues of a positive matrix and a positive rank one matrix - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T07:28:49Z http://mathoverflow.net/feeds/question/92552 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/92552/how-about-eigenvalues-of-a-positive-matrix-and-a-positive-rank-one-matrix How about eigenvalues of a positive matrix and a positive rank one matrix Kan He 2012-03-29T11:06:29Z 2012-03-29T14:59:51Z <p>Assume that A, B are positive n by n matrices and the rank of B is 1, B=xx*. If the eigenvalues of A are a_1≥a_2≥...≥a_n, and x is not the eigenvector of A, then there are d_i≥0 such that eigenvalue of A+B are a_1+d_1, a_2+d_2,...,a_n+d_n. Is it true? Are d_is non-negative?</p> <p>My e-mail is kanhemath@yahoo.com.cn If you any point, please let me know.</p> <p>Thanks! </p> <p>Yours He </p> http://mathoverflow.net/questions/92552/how-about-eigenvalues-of-a-positive-matrix-and-a-positive-rank-one-matrix/92572#92572 Answer by Felix Goldberg for How about eigenvalues of a positive matrix and a positive rank one matrix Felix Goldberg 2012-03-29T14:59:51Z 2012-03-29T14:59:51Z <p>Yes, provided you assume Hermitianity. Then, even more is true.</p> <p>Take A to be Hermitian (or real symmetric, if you like) matrix. As for B, it can be any positive semidefinite matrix (including your rank 1 case and without regard to the eigenvectors of A). Then your assertion follows from Weyl's Theorem about the eigenvalues of the sum of Hermitian matrices. This is actually stated as Problem 1 on page 198 of the Horn &amp; Johnson <em>Matrix Theory</em>.</p> <p>Here's a Google Books link to it:</p> <p><a href="http://books.google.ie/books?id=PlYQN0ypTwEC&amp;pg=PA198&amp;dq=interlacing+horn+johnson&amp;hl=en&amp;sa=X&amp;ei=2HZ0T_DrIZOAhQeeqKSmBQ&amp;redir_esc=y#v=onepage&amp;q&amp;f=false" rel="nofollow">http://books.google.ie/books?id=PlYQN0ypTwEC&amp;pg=PA198&amp;dq=interlacing+horn+johnson&amp;hl=en&amp;sa=X&amp;ei=2HZ0T_DrIZOAhQeeqKSmBQ&amp;redir_esc=y#v=onepage&amp;q&amp;f=false</a></p> <p>Since B is positive semidefinite, $\lambda_{1}(B) \geq 0$.</p> <p>As you can see there, you can even bound the $d_{i}$'s from above.</p>