Prove that every square matrix is expressible as the sum of a symmetric
and a skew symmetric matrix
Answers
Symmetric matrix is a matrix whose transpose is that matrix itself and skew-symmetric matrix is a matrix whose transpose is negative of that matrix.
Let A be any square matrix. Then,
A=
2
1
(A+A
T
)+
2
1
(A−A
T
)=P+Q, (say)
where, P=
2
1
(A+A
T
),Q=
2
1
(A−A
T
).
Now, P
T
=(
2
1
(A+A
T
))
T
=
2
1
(A+A
T
)
T
[∵(KT)
T
=K⋅A
T
]
⇒ P
T
=
2
1
[A
T
+(A
T
)
T
] [∵(A+B)
T
=A
T
+B
T
]
⇒ P
T
=
2
1
(A
T
+A) [∵(A
T
)
T
=A]
⇒ P
T
=
2
1
(A+A
T
)=P
∴ P is symmetric matrix
Also, Q
T
=
2
1
(A−A
T
)
T
=
2
1
[A
T
−(A
T
)
T
]=
2
1
[A
T
−A]
⇒ Q
T
=−
2
1
[A−A
T
]=−Q
∴ Q is skew-symmetric matric
Thus, A=P+Q where P is a symmetric matrix and Q is a skew-symmetric matrix.
Hence, A is expressible as the sum of symmetric and a skew-symmetric matrix
Uniqueness :
If possible, let A=R+S, where R is symmetric and S is skew-symmetric, then,
A
T
=(R+S)
T
=R
T
+S
T
⇒ A
T
=R−S [∵R
T
=RandS
T
=−S]
Now, A=R+S and A
T
=R−S
⇒ R=
2
1
[A+A
T
]=P and S=
2
1
(A−A
T
)=Q
Hence , A is uniquely expressible as the sum of a symmetric and a skew-symmetric matrix.