If $A$ and $B$ are two $n\times n$ matrices, and given that $B$ is symmetric, then is the matrix $C=\text{trn}(A)BA$ necessarily symmetric?

Question

If $A$ and $B$ are two $n\times n$ matrices, and given that $B$ is symmetric, then is the matrix $C=\text{trn}(A)BA$ necessarily symmetric?

I know that given the symmetric matrices $A$ and $B$, then $AB$ is symmetric if and only if $A$ and $B$ follow commutative property of multiplication, i.e., if $AB = BA$... but is that the case here?

It is not clear what $C=\text{trn}(A)BA$ is supposed to mean. Is trn your notation for the trace? — Ben Grossmann, Mar 24 '22 at 12:37
I assume $trn(A)$ means the transpose of $A$, usually denoted $A^T$. Do you know that $(AB)^T = B^T A^T$? — Jaap Scherphuis, Mar 24 '22 at 12:39
Just use the definition of a symmetric matrix. What is $C^T$? — KBS, Mar 24 '22 at 12:44

score 0 · Accepted Answer · answered Mar 24 '22 at 13:06

0

Recall that a matrix is called symmetric if it's equal to its own transpose,ie:$X^{T}=X$

As noted by the other users in the comments, also recall that $(PQR)^T=R^TQ^TP^T$

$$(A^TBC)^T=\cdots=\cdots$$

Can you finish?

answered Mar 24 '22 at 13:06

DatBoi

4,055

trn(C)trn(B)A ? – Mr Reddit Mar 24 '22 at 16:29
@MrReddit thats correct. Now what does the question say about the transpose of $B$? – DatBoi Mar 24 '22 at 16:48
it says that B is symmetric which means it's equal to its own transpose – Mr Reddit Mar 25 '22 at 18:48
Ah okay i get it now thanks alot :) – Mr Reddit Mar 25 '22 at 18:50

If $A$ and $B$ are two $n\times n$ matrices, and given that $B$ is symmetric, then is the matrix $C=\text{trn}(A)BA$ necessarily symmetric?

1 Answers1