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Let $A$ be a $7\times 7$ matrix such that it has rank $3$ and $a$ be $7\times 1$ column vector. Then least possible rank of $A+(a a^T)$ is? ($a^T$ is transpose of the vector)

Intuitively I think it's $2$ as $a a^T$ has rank $1$, and the most it could affect is one row of $A$.

hardmath
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Nitish
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  • For an introduction to posting with math notation, see that portion of the site FAQ and its links to other resources about MathJax and $\LaTeX$. – hardmath Feb 15 '17 at 15:03
  • While not literally the case that $aa^T$ can only affect one row of $A$, it is a good special case to motivate your intuition. Hint: Consider left multiplying $A+aa^T$ by an invertible matrix which reduces the general case to the single row affected case of your intuition. – hardmath Feb 15 '17 at 15:06

1 Answers1

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your intuition is right. instead of working with rows as suggested by hardmath, i will work with the columns of $A.$ without loss of generality, we can assume that the first three columns $a_1, a_2$ and $a_3$ are linearly independent.the column space of $Aa + aa^\top$ is $\{Ax + aa^\top x = Ax + (a^\top x)\, a\ \}$ the dimension space is two iff there is an $x$ such that $a_1x_j + (a^\top x) a = 0,\ j = 1, 2, 3.$ in other cases the dimension is either three or four.

abel
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  • can you elaborate a bit on the last part of the question, where you state that "the dimension space is two iff there is an $x$ such that $$a_1x_j+(a^⊤x)a=0, j=1,2,3$$ – grey Jun 17 '23 at 18:47