If $AA^T = BB^T$, and $A, B$ are real matrices, what can we say about real matrices $A$ and $B$? Is it true that $A = \pm B$
We know number of rows of $A$ and $B$ should be equal.
If $AA^T = BB^T$, and $A, B$ are real matrices, what can we say about real matrices $A$ and $B$? Is it true that $A = \pm B$
We know number of rows of $A$ and $B$ should be equal.
Take $A$ to be any orthogonal matrix and $B$ to be the identity matrix, in the same dimension of $A$.
No, this is not true. For example $A=\left(\begin{array}\, 1 & \,0\\0 & -1\end{array}\right)$ and $B$ the 2x2 identity matrix.
Assume $A,B$ are $n\times n$ matrices. The determinant product formula implies $|\det A|=|\det B|$.
Also note that by comparing the entries of $AA^T$ and $BB^T$ you get equations for the entries of $A$ and $B$, i.e. $\sum^n_{k=1}a_{ik}a_{jk}=\sum^n_{k=1} b_{ik}b_{jk}$ for all $i,j=1,\dots,n$ (similar for $n\times m$ matrices).
Consider that any unitary matrix $U$ (unitary is $UU^T = \mathbf{I}$) when applied to any given $A$ could give the matrix $B$: $AA^T = A\underbrace{UU^T}_{\mathbf{I}}A^T = (AU)(AU)^T = BB^T$
There are many unitary matrices not just $\pm\mathbf{I}$.
The general form for $U$ in two dimensions is called a Givens rotation: $\pmatrix{c & s \\ -s & c}$ where c and s are cos and sin, any numbers that satisfy $c^2 + s^2 = 1$