91Ó°ÊÓ

• q(x) is the quadratic form, defined by symmetric matrix $\mathrm{A}∈{\mathrm{R}}^{\mathrm{n}×\mathrm{n}}$, that is $\mathrm{q}\left(\mathrm{x}\right)={\mathrm{x}}^{\mathrm{T}}\mathrm{Ax}$with rank (A) = r and it has p positive eigen values.
• As rank (A) = r , and (n - r ) eigenvalues are zero and since p of the r nonzero eigen values are positive and (r - p) eigenvalue are negative.
• Let${\mathrm{λ}}_1,{\mathrm{λ}}_2,.....,{\mathrm{λ}}_p$be the positive eigenvalues and the negative eigenvalues are role="math" localid="1659700981027" ${\mathrm{λ}}_{p+1},{\mathrm{λ}}_{p+2},.....,{\mathrm{λ}}_r$.
• ${\mathrm{λ}}_{r+1},{\mathrm{λ}}_{p+2},.....,{\mathrm{λ}}_r$are all zero eigenvalues.
• As per theorem 8.2.2, we know that for orthonormal eigenbasis $B=\left\{{\stackrel{→}{v}}_1,{\stackrel{→}{v}}_2,...{\stackrel{→}{v}}_n\right\}$and corresponding eigenvalue ${\mathrm{λ}}_1,{\mathrm{λ}}_2,...,{\mathrm{λ}}_{\mathrm{p}},{\mathrm{λ}}_{\mathrm{p}+1},....,\mathrm{λ°ù},{\mathrm{λ}}_{\mathrm{r}+1},....,{\mathrm{λ}}_{\mathrm{n}}$ of A, the above quadratic form is diagonalizable as $\mathrm{q}\left(\mathrm{x}\right)={\mathrm{λ}}_1{\mathrm{c}}_1^2+{\mathrm{λ}}_2{\mathrm{c}}_2^2+...+{\mathrm{λ}}_{\mathrm{p}}{\mathrm{c}}_{\mathrm{p}}^2+{\mathrm{λ}}_{\mathrm{p}+1}{\mathrm{c}}_{\mathrm{p}+1}^2+...+{\mathrm{λ}}_{\mathrm{r}}{\mathrm{c}}_{\mathrm{r}}^2+{\mathrm{λ}}_{\mathrm{r}+1}{\mathrm{c}}_{\mathrm{r}+1}^2+...+{\mathrm{λ}}_{\mathrm{n}}{\mathrm{c}}_{\mathrm{n}}^2$, $∴\mathrm{q}\left(\mathrm{x}\right)={\mathrm{λ}}_1{\mathrm{c}}_1^2+{\mathrm{λ}}_2{\mathrm{c}}_2^2+...+{\mathrm{λ}}_{\mathrm{p}}{\mathrm{c}}_{\mathrm{p}}^2+{\mathrm{λ}}_{\mathrm{p}+1}{\mathrm{c}}_{\mathrm{p}+1}^2+...+{\mathrm{λ}}_{\mathrm{r}}{\mathrm{c}}_{\mathrm{r}}^2$where ${\mathrm{λ}}_{\mathrm{i}}=0\mathrm{for}\mathrm{i}=\mathrm{r}+1,...\mathrm{n}$
• Here ${\mathrm{c}}_{\mathrm{i}}\mathrm{s}$are the coordinates of x with respect to the eigenbasis B
• Let us define $C=\left\{{\stackrel{→}{w}}_1,{\stackrel{→}{w}}_2,....,{\stackrel{→}{w}}_n\right\}$where ${\stackrel{→}{\mathrm{w}}}_{\mathrm{i}}=\frac{{\stackrel{→}{\mathrm{v}}}_{\mathrm{i}}}{\sqrt{\left|{\mathrm{λ}}_{\mathrm{i}}\right|}}\mathrm{for}\mathrm{i}=1,2...\mathrm{r}\mathrm{and}{\stackrel{→}{\mathrm{w}}}_{\mathrm{j}}={\stackrel{→}{\mathrm{v}}}_{\mathrm{j}}$for$\mathrm{j}=\mathrm{r}+1,...,\mathrm{n}$then, is orthogonal basis of as defined.

, where ${\mathrm{λ}}_1=0\mathrm{for}\mathrm{i}=\mathrm{r}+1,...,\mathrm{n}$