91影视

The single value decomposition produces orthonormal bases of v鈥檚 and u鈥檚 for the four fundamental sub-spaces.

Using those bases, A becomes a diagonal matrix $\underset{}{鈭�}$and${\mathbf{Av}}_{\mathbf{i}}\mathbf{=}{\mathbf{蟽}}_{\mathbf{i}}{\mathbf{u}}_{\mathbf{i}}$ role="math" localid="1664177451234" ${\mathbf{蟽}}_{\mathbf{i}}$= singular value.

Consider$\mathrm{A}\stackrel{藱}{{\mathrm{v}}_1}=3\stackrel{藱}{{\mathrm{u}}_1}$and$\mathrm{A}\stackrel{藱}{{\mathrm{v}}_2}=2\stackrel{藱}{{\mathrm{u}}_2}$where$\left\{\stackrel{藱}{{\mathrm{v}}_1},{\stackrel{藱}{\mathrm{v}}}_2\right\}$and$\left\{{\stackrel{藱}{\mathrm{u}}}_1,\stackrel{藱}{{\mathrm{u}}_2}\right\}$are pairs of orthogonal unit vectors and ${蟽}_1,{蟽}_2<5$.

Any unit vector v in${\mathrm{鈩�}}^2$can be written as$\stackrel{藱}{\mathrm{v}}=\mathrm{肠辞蝉胃}\stackrel{藱}{{\mathrm{v}}_1}+\mathrm{蝉颈苍胃}\stackrel{藱}{{\mathrm{v}}_1}$. Hence,

$\mathrm{A}\stackrel{藱}{\mathrm{v}}=\mathrm{肠辞蝉胃A}\stackrel{藱}{{\mathrm{v}}_1}+\mathrm{蝉颈苍胃A}\stackrel{藱}{{\mathrm{v}}_1}$

Since,

$\mathrm{A}\stackrel{藱}{{\mathrm{v}}_1}$$={\mathrm{蟽}}_1\stackrel{藱}{{\mathrm{u}}_1}$

$\mathrm{A}\stackrel{藱}{{\mathrm{v}}_2}$$={\mathrm{蟽}}_2\stackrel{藱}{{\mathrm{u}}_2}$

$\mathrm{A}\stackrel{藱}{\mathrm{v}}$$=\mathrm{肠辞蝉胃}{蟽}_1\stackrel{藱}{{\mathrm{u}}_1}$$+{\mathrm{蝉颈苍胃蟽}}_2\stackrel{藱}{{\mathrm{u}}_2}$

$\mathrm{A}\stackrel{藱}{\mathrm{v}}$$={蟽}_1\mathrm{肠辞蝉胃}\stackrel{藱}{{\mathrm{u}}_1}$$+{\mathrm{蟽}}_2\mathrm{蝉颈苍胃}\stackrel{藱}{{\mathrm{u}}_2}$

Find $||\mathrm{A}\stackrel{藱}{\mathrm{v}}||$as follows:

$=5$

This implies that $||{\mathrm{Ae}}_1||<5,||{\mathrm{Ae}}_2||<5$, where ${\mathrm{e}}_1,{\mathrm{e}}_2$are standard basis vectors.

Let $\mathrm{A}=\left[\begin{array}{cc}\mathrm{a}& \mathrm{b}\\ \mathrm{c}& \mathrm{d}\end{array}\right],$then ${\mathrm{Ae}}_1={\left(\mathrm{a},\mathrm{c}\right)}^{\mathrm{T}}.$

Then, $||{\mathrm{Ae}}_1||<5$implies that $\sqrt{{\mathrm{a}}^2+{\mathrm{c}}^2}<5$, which means $\mathrm{a},\mathrm{c}<5.$

Similarly, $\mathrm{b},\mathrm{d}<5.$

Hence, all the entries of $\mathrm{A}$are less than 5.

The given statement is TRUE.