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Explain this step : -

To prove uniqueness,let $A = R + S$ where $R$ is symmetric and $S$ is skew−symmetric ∴ $A′ = (R+S)′ = R′+S′ = R−S$ { ∵ $R′= R$ and $S′= −S$ by Definition of symmetric and skew−symmetric matrices } ∴ $1/2 (A+A′) = 1/2 ( R+S + R−S) = R = P$

$1/2 (A−A′) = 1/2 (R+S − R+S) = S = Q.$

Hence, the representation $A = P + Q$ is unique. Hence, it is proved that every square matrix can be uniquely expressed as a sum of symmetric and skew-symmetric matrix.

I don't know anything about proving uniqueness. The solution i am referring to is from this website: https://www.ques10.com/p/5984/show-that-every-square-matrix-can-be-uniquely-ex-1/

Kenta S
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Sahil Bhatti
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    PLease see https://math.meta.stackexchange.com/questions/5020/ if you wish to make your question more readable. – Angina Seng Dec 02 '19 at 04:28

1 Answers1

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Proving uniqueness means that, if $A=R+S$ and $A=P+Q$ with $R,P$ symmetric and $S,Q$ skew-symmetric, then $R=P$ and $S=Q$. To see this note that $$ P+Q=A=R+S \implies Q-S=R-P $$ The LHS is skew-symmetric while the RHS is symmetric. But the only matrix that is both skew symmetric and symmetric is the null matrix, so that $Q-S=0=R-P$, which gives what you want.

Reveillark
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  • what does uniqueness means in general ? thanks in advance – Sahil Bhatti Dec 02 '19 at 04:32
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    Uniqueness of an object satisfying some condition means there exists only one such object. Said another way, if you had two things which satisfy the condition, then they're actually equal. For example, the condition "$x$ is a positive number and $x^2=x$" is satisfied by a unique object, namely the number $1$. – Reveillark Dec 02 '19 at 04:33