20211005 Hermite matrix and some properties

Hermite matrix ：$a_{ij}$ And $a_{ji}$ conjugate , That is, the real part is equal , The imaginary part is the opposite .

Hermite Some properties of matrix

(1) set up $\mathbf{A}\in {\mathbf{C}}_r^{m\times n}(r>0)$, be ${\mathbf{A}}^{\mathrm{H}}\mathbf{A}$ yes Hermite matrix , And their eigenvalues are non negative real numbers ;
(2) $\mathrm{rank}\left({\mathbf{A}}^{\mathrm{H}}\mathbf{A}\right)=\mathrm{rank}\mathbf{A}$;
(3) set up $\mathbf{A}\in {\mathbf{C}}^{m\times n}$, be $\mathbf{A}=\mathbf{O}$ If and only if ${\mathbf{A}}^{\mathrm{H}}\mathbf{A}=\mathbf{O}$. These conclusions ask the reader to prove .

Proof. (1) Hermite Matrix means A yes A The conjugate transposition of , because ${\mathbf{A}}^{\mathrm{H}}\mathbf{A}={\left({\mathbf{A}}^{\mathrm{H}}\mathbf{A}\right)}^{\mathrm{H}}$, therefore ${\mathbf{A}}^{\mathrm{H}}\mathbf{A}$ yes
Hermite matrix . because ${\mathbf{x}}^{\mathrm{H}}{\mathbf{A}}^{\mathrm{H}}\mathbf{A}\mathbf{x}={\left(\mathbf{A}\mathbf{x}\right)}^{\mathrm{H}}\mathbf{A}\mathbf{x}⩾0$
For any nonzero $\mathbf{x}$, therefore ${\mathbf{A}}^{\mathrm{H}}\mathbf{A}$
The eigenvalues of are all nonnegative real numbers .

(2) For a certain $\mathbf{x}\in {\mathbf{C}}^n$, If $\mathbf{A}\mathbf{x}=0$, Can be launched ${\mathbf{A}}^{\mathrm{H}}\mathbf{A}\mathbf{x}=0$; For a certain $\mathbf{x}\in {\mathbf{C}}^n$, If ${\mathbf{A}}^{\mathrm{H}}\mathbf{A}\mathbf{x}=0$, be ${\mathbf{x}}^{\mathrm{H}}{\mathbf{A}}^{\mathrm{H}}\mathbf{A}\mathbf{x}={\left(\mathbf{A}\mathbf{x}\right)}^{\mathrm{H}}\mathbf{A}\mathbf{x}=0$, You can get $\mathbf{A}\mathbf{x}=0$. The zero space is the same , Zero space has the same dimension .

Consider the number of columns of the matrix = The maximum number of linearly independent groups （ Rank ）+ Zero space dimension ,${\mathbf{A}}^{\mathrm{H}}\mathbf{A}$ and $\mathbf{A}$ Same number of columns for , Zero space is the same , So the rank is the same .

(3) If $\mathbf{A}=\mathbf{O}$, be ${\mathbf{A}}^{\mathrm{H}}\mathbf{A}=\mathbf{O}$ establish ; If ${\mathbf{A}}^{\mathrm{H}}\mathbf{A}=\mathbf{O}$, For any $\mathbf{x}\in {\mathbf{C}}^n$, Yes ${\mathbf{x}}^{\mathrm{H}}{\mathbf{A}}^{\mathrm{H}}\mathbf{A}\mathbf{x}={\left(\mathbf{A}\mathbf{x}\right)}^{\mathrm{H}}\mathbf{A}\mathbf{x}=0$, That is, for any $\mathbf{x}\in {\mathbf{C}}^n$, Yes $\mathbf{A}\mathbf{x}=0$, On the other hand $\mathbf{A}$ Any row vector of , With all $\mathbf{x}\in {\mathbf{C}}^n$ vertical , that $\mathbf{A}$ All lines of are $\mathbf{O}$, in other words $\mathbf{A}=\mathbf{O}$.

Definition 4. 11 set up $\mathbf{A}\in {\mathbf{C}}_r^{m\times n}(r>0),{\mathbf{A}}^{\mathrm{H}}\mathbf{A}$ The eigenvalue of is
$${\lambda }_1⩾{\lambda }_2⩾\cdots ⩾{\lambda }_r>{\lambda }_{r+1}=\cdots ={\lambda }_n=0$$ said ${\sigma }_i=\sqrt{{\lambda }_i}(i=1,2,\cdots ,n)$ by $\mathbf{A}$ The singular value of ; When $\mathbf{A}$ Zero matrix , Its singular values are $0.$

Easy to see ,
(1) matrix $\mathbf{A}$ The number of singular values of is equal to $\mathbf{A}$ Columns of .

(2) $\mathbf{A}$ The number of nonzero singular values of is equal to $\mathrm{rank}A$.

Proof: $\mathbf{A}$ Zero singular value of , That is to say ${\mathbf{A}}^{\mathrm{H}}\mathbf{A}$ Zero eigenvalue of , That is to say ${\mathbf{A}}^{\mathrm{H}}\mathbf{A}\mathbf{x}=\mathbf{O}$ Solution , that $\mathbf{A}$ The number of zero singular values of is equal to ${\mathbf{A}}^{\mathrm{H}}\mathbf{A}$ The dimension of the zero space of .

in addition ,$\mathbf{A}$ The number of all singular values of is equal to ${\mathbf{A}}^{\mathrm{H}}\mathbf{A}$ Dimension of .

from (2) It can be seen that , The number of columns in a matrix = The maximum number of linearly independent groups （ Rank ）+ Zero space dimension , be $\mathbf{A}$ The number of nonzero singular values of is equal to $\mathrm{rank}A$.