For questions about congruent matrices.
Two matrices $A$ and $B$ over a ring are called congruent if there exists an invertible matrix $P$ over the same ring such that $$P^TAP=B.$$ Matrix congruence is an equivalence relation.
Questions tagged [matrix-congruences]
58 questions
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Is Same signs of determinants of $A$ and $B$ $\implies P^tAP =B$
Let $$A =
\begin{pmatrix}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & b \\
\end{pmatrix}
$$
and
$$ B=
\begin{pmatrix}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0…
Largest Prime
- 513
0
votes
1 answer
How to conclude that matrices $A$ and $B$ are congruent?
Let $A$ and $B$ be real $n×n$ matrices. Which of the following statements is false ?
If $A^{-1}$ and $B^{-1}$ are congruent then so are $A$ and $B$.
If $A^t$ and $B^t$ are congruent then so are $A$ and $B$. Where $A^t$ denotes transpose of matrix…
Largest Prime
- 513