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If and only if all of the eigenvalues of a symmetric matrix are positive, it is said to be positive-definite. The product of the eigenvalues is the determinant. If and only if the determinant of a square matrix is not zero, it is invertible. As a result, a positive definite symmetric matrix can be said to be invertible.

Let an invertible matrix $n脳n$A and an orthogonal basis ${\stackrel{鈫�}{v}}_1,{\stackrel{鈫�}{v}}_2,....,{\stackrel{鈫�}{v}}_n$for A with associated eigen values are ${位}_1<{位}_2<{位}_n$.Any vector in$R^n$ can be represented as

$\stackrel{鈫�}{v}=c_1{\stackrel{鈫�}{v}}_1+c_2{\stackrel{鈫�}{v}}_2+....+c_n{\stackrel{鈫�}{v}}_n$

To evaluate the dot product as$A\stackrel{鈫�}{v}.\stackrel{鈫�}{v}$ follows

$$\begin{gathered} A\stackrel{鈫�}{v}.\stackrel{鈫�}{v}=\left({位}_1{c}_1{\stackrel{鈫�}{v}}_1+{位}_1{c}_1{\stackrel{鈫�}{v}}_1+......+{位}_n{c}_n{\stackrel{鈫�}{v}}_n\right).\left(c_1{\stackrel{鈫�}{v}}_1+c_1{\stackrel{鈫�}{v}}_1+......+c_n{\stackrel{鈫�}{v}}_n\right) \\ A\stackrel{鈫�}{v}.\stackrel{鈫�}{v}={位}_1{c}_1^2+{位}_2{c}_2^2+....+{位}_n{c}_n^2 \end{gathered}$$

If the eigen values are positive then the dot product$A\stackrel{鈫�}{v}.\stackrel{鈫�}{v}$ will be positive. If the eigen values are negative then the dot product$A\stackrel{鈫�}{v}.\stackrel{鈫�}{v}$ will be negative. If the matrix A has positive as well as negative eigen values is${位}_1<0<{位}_n$ ,then there exists non-zero vectors$\stackrel{鈫�}{v}$ such that$A\stackrel{鈫�}{v}$ an orthogonal to$\stackrel{鈫�}{v}$ Matrix A have both positive and negative eigen values then there exists non-zero vectors$\stackrel{鈫�}{v}$ in$R^n$ such that$A\stackrel{鈫�}{v}$ is orthogonal to$\stackrel{鈫�}{v}$.