91影视

Consider a $n脳m$matrix A where $ker\left(A\right)=\left\{\stackrel{鈫�}{v}鈭�\overline{){\mathrm{鈩�}}^n}A\stackrel{鈫�}{v}=0\right\}andIm\left(A\right)=\left\{\stackrel{鈫�}{w}鈭�\overline{){\mathrm{鈩�}}^m}A\stackrel{鈫�}{v}=\stackrel{鈫�}{w}\right\}$.

By the Rank-Nullity theorem, the sum of the dimensions of the kernel and the image of a matrix is equal to the number of column of that matrix.

As the matrix A is$n脳m$ , then$A^{T}A$ is an$n脳n$ matrix, and therefore the number of their column is the same means.

$$\begin{gathered} dim\left\{ker\left(A\right)\right\}+dim\left\{lm\left(A\right)\right\}=m \\ dim\left\{ker\left(A^{T}A\right)\right\}+dim\left\{lm\left(A^{T}A\right)\right\}=m \end{gathered}$$

Theorem: Dimension of the image.

For a matrix A ,$dim\left(imA\right)=rank\left(A\right)$ .

By the theorem, the value$dim\left\{imA\right\}=rank\left\{A\right\}$ and$dim\left\{im\left(A^{T}A\right)\right\}=rank\left(A^{T}A\right)$ .

Substitute$rank\left\{A\right\}$ for$dim\left\{lmA\right\}$ in the$dim\left\{ker\left(A\right)\right\}+dim\left\{im\left(A\right)\right\}=m$ as follows.

$$\begin{gathered} dim\left\{ker\left(A\right)\right\}+dim\left\{im\left(A\right)\right\}=m \\ rank\left(A\right)+dim\left\{ker\left(A\right)\right\}=m \\ rank\left(A\right)=-dim\left\{ker\left(A\right)\right\} \end{gathered}$$

Substitute$rank\left(A^{T}A\right)$ for$dim\left\{im\left(A^{T}A\right)\right\}$ in the$dim\left\{ker\left(A^{T}A\right)\right\}+dim\left\{im\left(A^{T}A\right)\right\}=m$ as follows.

$$\begin{gathered} dim\left\{ker\left(A^{T}A\right)\right\}+dim\left\{im\left(A^{T}A\right)\right\}=m \\ rank\left(A^{T}A\right)+dim\left\{ker\left(A^{T}A\right)\right\}=m \\ rank\left(A^{T}A\right)=-dim\left\{ker\left(A^{T}A\right)\right\} \end{gathered}$$

Compare the equations$rank\left\{A\right\}=m-dim\left\{ker\left(A\right)\right\}$ and$rank\left(A^{T}A\right)=m-dim\left\{ker\left(A^{T}A\right)\right\}$ as follows.

$rank\left(A\right)=rank\left(A^{T}A\right)$

Hence, the equation$rank\left(A\right)=rank\left(A^{T}A\right)$ for all $n脳m$matrices.