Vector times symmetric matrix

Question

For most of you here, this is probably quite basic.

As for a symmetric matrix $A$ the first row equals the first column, multiplying the matrix with a column vector $b$ equals multiplying the transposed vector $b'$ with the symmetric matrix, i.e. if $A=A'$ then $$Ab=b'A$$ Could you please confirm? Is there a better derivation than the verbal one above?

Also, while I have found many sources online about matrix algebra, I have not found this property, yet. is there a reliable source on the web where one could find this? I was hoping that such a site would also contain additional information which might help to answer some other questions I have.

score 3 · Accepted Answer · answered Apr 19 '15 at 21:06

3

Except in dimension $1$, your claim is not correct. However, because transposition swaps factors we have $$ (Ab)^T=b^TA^T=b^TA$$ that is multiplying with a column vector from the right equals the transpose of multiplying the transposed vector from the left.

answered Apr 19 '15 at 21:06

Hagen von Eitzen

374,180

Thank you! Yes, sorry, I overlooked that Ab gives a column vector while b'A gives a row vector - thanks for clarifying! – user70160 Apr 19 '15 at 21:24
except in dimnesion 1? column vectors are in dimension 1 only so his statement is correct? – james black Apr 18 '18 at 06:35

Vector times symmetric matrix

1 Answers1