[LA] Rank: matrix의 속성

Definition: Rank ← matrix 속성

The rank of a matrix $A$, denoted by rank $A$,
is the dimension of the column space of $A$.

• Matrix를 이루는 column vectors에서 linearly independent 인 것들의 수를 의미

 

row space의 dimension 의 경우를 강조하여 row rank라고 부르고,

column space의 경우를 강조하여 column rank라고도 부르는 경우가 있으나,

동일한 matrix에 대해 이 둘은 같기 때문에 그냥 rank라고 지칭하는게 일반적임.

 

$m \times n$ matrix $A$에서 다음이 성립.

 

$$ \text{Column Rank}(A) \le n \\ \text{Row Rank}(A) \le n \\ \text{Rank}(A) \le \text{min}(m,n)$$

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Important Properties of Rank

• $\text{Rank}(A^\top) = \text{Rank}(A)$ : row rank와 column rank가 같음을 기억.
• $\text{Rank}(A_{m \times n}B_{n \times q})=\text{min}(\text{Rank}(A_{m \times n}),\text{Rank}(B_{n \times q})) \le \text{min}(m,n,q)$ : SfM에서 Observation Matrix에서 사용되는 성질.
• $\text{Rank}(A^\top A) = \text{Rank}(A A^\top) = \text{Rank} (A^\top) = \text{Rank}(A)$
• $A_{m\times m}$ is not singular iff $\text{Rank}(A_{m \times m})=m$: 이 경우 $A$ is a full rank matrix.

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Example 3 :

Determine the rank of the matrix

$$
A \sim \begin{bmatrix}2 & 5 & -3 & -4 & 8 \\ 0 & -3 & 2 & 5 & -7 \\
0 & -6 & 4 & 14 & -20 \\ 0 & -9 & 6 & 5 & -6\end{bmatrix}
$$

• Solution
  • Reduce $A$ to echelon form.
  • The matrix $A$ has 3 pivot columns, so rank $A$ = 3.

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Theorem 14 : Rank Theorem

If a matrix $A$ has $n$ columns, then rank $A$ + dim $\text{Nul }A = n$.

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2024.07.08 - [.../Linear Algebra] - [LA] Null Space

[LA] Null Space

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Reference

Linear Algebra and Its Applications, 5th ed., David C. Lay: Chapter 4.6 Rank

https://www.google.co.kr/books/edition/Linear_Algebra_and_Its_Applications_Fift/lZ4mzgEACAAJ?hl=ko

 

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