Question

Show that any positive definite matrix A can be written as, where B is a positive definite matrix.

Short answer

$A=B^2$, where B is a positive definite matrix.

Step-by-step solution

Given Information:

$A=B^2$

Determining is the B is positive definite Matrix:

Consider the quadratic form$\mathrm{q}\left(\overrightarrow{\mathrm{x}}\right)=\overrightarrow{\mathrm{x}}·\mathrm{A}\overrightarrow{\mathrm{x}}$, where A is a$\mathrm{n}\times \mathrm{n}$symmetric matrix. If$\mathrm{q}\left(\mathrm{x}\right)$is positive for all nonzero$\overrightarrow{x}$in${\mathrm{ℝ}}^n$, we say A is positive definite. S is an orthogonal matrix that has the following properties:

$S^{-1}AS=D$

When D is a diagonal matrix, all of the entries are positive. As a result, A can be written as:

$A=SD{S}^{-1}$

We can write D as a diagonal matrix with positive diagonal elements.

$D=D_1^2$

Whereis a diagonal matrix with positive diagonal entries, Equationcan now be written as follows:

$A={\mathrm{SDS}}^{-1}$

$$\begin{gathered} \Rightarrow A=S\left(D_1^2\right)S^{-1} \\ \Rightarrow A=S\left(D_1·D_1\right)S^{-1} \\ \Rightarrow A=\left(S{D}_1{S}^{-1}\right)·\left(S{D}_1{S}^{-1}\right) \\ \Rightarrow A=B^2 \end{gathered}$$

${\mathrm{SBS}}^{-1}={\mathrm{D}}_1$is a diagonal matrix with positive diagonal elements, hence B is positive definite.

Determining the Result:

$A=B^2$, where B is a positive definite matrix.