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How do we prove that

$rank(A) = rank(AA^T) = rank(A^TA)$ ?

Is it always true?

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You may also be interested in this answer. –  EuYu Oct 16 '12 at 21:48
@Belgi, Thanks for curiosity. Is it a rule to accept one of the answers to all your questions? Does it not close the question? For some of the answer, the correctness of responses is not verifiable for me. But I always increase the vote of responses that increase something to my understanding, after I see them. –  user25004 Oct 16 '12 at 23:11

4 Answers 4

up vote 2 down vote accepted

It is always true. One of the important theorems one learns in linear algebra is that $$ \mathrm{Nul}(A^T)^{\perp} = \mathrm{Col}(A), \quad \mathrm{Nul}(A)^{\perp} = \mathrm{Col}(A^T).$$

Therefore $\mathrm{Nul}(A^T) \cap \mathrm{Col}(A) = \{0\}$, and so forth. Now consider the matrix $A^TA$. Then $\mathrm{Col}(A^TA) = \{A^TAx\} = \{A^Ty: y \in \mathrm{Col}(A)\}$. But since the null space of $A^T$ only intersects trivially with $\mathrm{Col}(A)$, then $\mathrm{Col}(A^TA)$ must have the same dimension as $\mathrm{Col}(A)$, which gives us the equality of ranks.

We can replace $A$ with $A^T$ to prove the other equality.

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The meaning of the equality is: the rank of a matrix is equal to the number of nonzero singular values of a matrix.

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This is only true for real matrices. For instance $\begin{bmatrix} 1 & i \\ 0 &0 \end{bmatrix}\begin{bmatrix} 1 & 0 \\ i &0 \end{bmatrix}$ has rank zero. For complex matrices, you'll need to take the conjugate transpose.

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Here is a common proof.

All matrices in this note are real. Think of a vector $X$ as an $m\!\times\!1$ matrix. Let $A$ be an $m\!\times\!n$ matrix.

We will prove that $A A^T X = 0$ if and only if $A^T X = 0$.

It is clear that $A^T X = 0$ implies $AA^T X = 0$.

Assume that $AA^T X = 0$ and set $Y = A^T\!X$. Then $X^T\!A\, Y = 0$, and thus $(A^T\!X)^T Y = 0$. That is $Y^T Y = 0$. This implies $Y = A^T X = 0$.

We just proved that the $m\!\times\!m$ matrix $AA^T$ and the $n\!\times\!m$ matrix $A^T$ have the same null space. Consequently they have the same nullity. The nullity-rank theorem states that $$ {\rm Nul} AA^T + {\rm Rank} AA^T = m = {\rm Nul} A^T + {\rm Rank} A^T. $$

Hence ${\rm Rank} AA^T = {\rm Rank} A^T$.

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