Krdytkyu

I have two symmetric real matrices I am interested in, $A_{ntimes n}$ and $B_{mtimes m}$. If I do the following operations:

$$EAE^{T}=FBF^{T}$$

where $E$ and $F$ are of dimensions $ptimes n$ and $ptimes m$ respectively.

Supposed that the $A$ and $B$ can be eigen-decomposed as $A=Q_ALambda_A Q_A^{T}$, $B=Q_BLambda_B Q_B^{T}$, and thus the previous equation can be written as

$$EQ_ALambda_A Q_A^{T}E^{T}=FQ_BLambda_B Q_B^{T}F^{T}$$

where $Q$ and $Lambda$ are matrices containing eigenvectors and eigenvalues, respectively

or in the form as

$$(EQ_ALambda_A^{1/2}) (EQ_ALambda_A^{1/2} )^T=(FQ_BLambda_B^{1/2}) (FQ_BLambda_B^{1/2} )^T.$$

If I further let $K=EQ_ALambda_A^{1/2}$ and $L=FQ_BLambda_B^{1/2}$, then I obtain

$$KK^T=LL^T.$$

Here come my questions:

1. What are the relations between $Q_A$ and $Q_B$?


2. To simplify the first question, my strategy is to obtain $KK^{T}=LL^{T}$ as mentioned, and find the relations between $K$ and $L$ first. That is, what are the relations between matrices with the same Gramian matrix?
   I found the answer in wiki, which says, "...in finite-dimensions it determines the vectors up to isomorphism, i.e. any two sets of vectors with the same Gramian matrix must be related by a single unitary matrix." But I still can not understand what it exactly means.

Thus, can anyone help me with these two questions or give me any references to read?