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Let A be a m x n complex matrix with rank A = r . If we are allowed to modify at most one row ( or at most one column , but not both ) of A , then what effect will it have on rank A ?

Thanks for any help .

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For $A\in\mathbb R ^{1\times 1}, A = (0): rank(A)$ is 0, but $rank((1))= 1$, and I only modified 1 row. So the rank can increase. Are there some other conditions about $m$ and $n$? –  Stefan Nov 21 '12 at 18:33
    
@Stefan Sorry , I haven't taken that into account –  Ester Nov 21 '12 at 18:36
    
Actually, if you are allowed to modifiy a row arbitrarily, you can always increase the rank by 1, unless $A$ already has full row rank. –  user1551 Nov 21 '12 at 18:58
    
@user1551 Yes,I actually want to prove that if s be the rank of the modified matrix , then |s-r|<=1 –  Ester Nov 21 '12 at 19:05

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Let $\tilde{A} \in \mathbb{C}^{m \times n}$ be the matrix obtained by changing a row (or) column of $A \in \mathbb{C}^{m \times n}$. Then the claim is that $$-1 \leq \text{rank}(A) - \text{rank}(\tilde{A}) \leq 1$$

Proof:

We will assume that the $k^{th}$ row of $A \in \mathbb{C}^{m \times n}$ has been modified to get $\tilde{A} \in \mathbb{C}^{m \times n}$. Then note that $$\tilde{A} = A + \begin{bmatrix} 0 & 0 & 0 & \cdots & 0 & 0 \\ 0 & 0 & 0 & \cdots & 0 & 0\\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & 0 & \cdots & 0 & 0\\ r_{k,1} & r_{k,2} & r_{k,3} &\cdots & r_{k,n-1} & r_{k,n} \\ 0 & 0 & 0 & \cdots & 0 & 0\\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & 0 & \cdots & 0 & 0\end{bmatrix}_{m \times n}$$ where $r_{k,\ell} \in \mathbb{C}$. Note that this can be written as

$$\tilde{A} = A + \begin{bmatrix} 0 \\ 0 \\ 0 \\ \vdots \\ 1 \\ \vdots \\ 0\end{bmatrix}_{m \times 1}\begin{bmatrix} r_{k,1} & r_{k,2} & r_{k,3} &\cdots & r_{k,n-1} & r_{k,n}\end{bmatrix}_{1 \times n}$$ Hence, it is equivalent to adding a rank $1$ perturbation to the matrix $A$. Hence, $$\text{rank}(\tilde{A}) \in \{\text{rank}(A)-1,\text{rank}(A), \text{rank}(A)+1 \}$$

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The rank of a matrix is equal to both the number of linearly independent rows and the number of linearly independent columns. Therefore, if you change at most one row or column, the rank can either increase by one, decrease by one, or stay the same. However, you can't say much more than this since you don't know anything about the structure of the matrix or how you're making the change.

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@Wong How are you proving the above statement ? –  Ester Nov 21 '12 at 18:52

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