Transpose, symmetric and skew symmetric

The transpose of A, written A′ (or Aᵀ), turns rows into columns: row 1 of A becomes column 1 of A′, and so on. An m × n matrix becomes n × m.

Example: the transpose of [1, 2, 3; 4, 5, 6] is [1, 4; 2, 5; 3, 6].

Its rules:

• (A′)′ = A: transposing twice gets you back.
• (kA)′ = kA′ and (A + B)′ = A′ + B′.
• (AB)′ = B′A′: the order reverses. (Like taking off shoes before socks.)

Two special kinds of square matrix:

• Symmetric: A′ = A. The matrix is a mirror image across its diagonal: aᵢⱼ = aⱼᵢ. Example: [2, 7; 7, 5].
• Skew symmetric: A′ = –A, so aᵢⱼ = –aⱼᵢ. Every diagonal entry must equal its own negative, so it is 0. Example: [0, 3; –3, 0].

For any square matrix A:

• A + A′ is symmetric, since its transpose is A′ + A, the same thing.
• A – A′ is skew symmetric, since its transpose is A′ – A = –(A – A′).

So every square matrix splits into a symmetric part and a skew symmetric part:

A = ½(A + A′) + ½(A – A′)

Example: A = [1, 5; 3, 2]. A′ = [1, 3; 5, 2]. Symmetric part ½(A + A′) = [1, 4; 4, 2]; skew part ½(A – A′) = [0, 1; –1, 0]. Adding them gives back A.