Lesson 10 of 13 · 7 min
Symmetric and skew symmetric
NCERT §3.6
Two square tables sit on the counter's noticeboard: road distances between three schools, and the net number of pens each class has lent another.
The lesson in notes
In short
A square matrix is symmetric if A′ = A, that is aᵢⱼ = aⱼᵢ: the entries mirror across the leading diagonal.
A square matrix is skew symmetric if A′ = −A, that is aⱼᵢ = −aᵢⱼ. Putting j = i gives aᵢᵢ = −aᵢᵢ, so every diagonal entry of a skew symmetric matrix is 0.
For any square matrix A, A + A′ is symmetric and A − A′ is skew symmetric. Proof: (A + A′)′ = A′ + A and (A − A′)′ = A′ − A = −(A − A′).
Every square matrix splits as A = ½(A + A′) + ½(A − A′), a symmetric part plus a skew symmetric part, and the split is unique.
Worked example: B with rows (2, −2, −4), (−1, 3, 4), (1, −2, −3) has symmetric part P with rows (2, −3/2, −3/2), (−3/2, 3, 1), (−3/2, 1, −3) and skew part Q with rows (0, −1/2, −5/2), (1/2, 0, 3), (5/2, −3, 0); P + Q = B.
The only matrix that is both symmetric and skew symmetric is the zero matrix, since A = −A forces A = O.
If A and B are symmetric of the same order, AB is symmetric exactly when AB = BA, and AB − BA is always skew symmetric.