# Can we recover the covariance matrix from its eigen-decomposition?

by hxd1011   Last Updated January 12, 2018 22:19 PM

I feel it is a very basic question, but I am still confused about it. Can we recover the covariance matrix from its eigen-decomposition?

For example, if we have a covariance matrix (of 2D Gaussian distribution) as

$$\Sigma=\begin{bmatrix} 2 & 1 \\ 1 & 2 \end{bmatrix}$$

The eigen values are $[3,1]$, and eigen-vectors are

$$\begin{bmatrix} \sqrt 2/2 & -\sqrt 2/2 \\ \sqrt 2/2 & \sqrt 2/2 \end{bmatrix}$$

On the other hand, can we recover $Sigma$ from its decomposition?

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