Symmetric And Skew-symmetric

Question

Every square matrix can be written uniquely as the sum of a symmetric matrix and a skew-symmetric matrix; the symmetric part of a square matrix A is correctly given by which expression?

RankGuru · Accepted Answer

Any square matrix decomposes uniquely into symmetric and skew-symmetric parts, a structural theorem frequently tested in JEE Advanced. A symmetric matrix S satisfies S^T = S, while a skew-symmetric matrix K satisfies K^T = -K. Writing A = S + K and taking the transpose gives A^T = S - K. Adding the two relations, A + A^T = 2S, so S = (1/2)(A + A^T), which indeed satisfies S^T = S. Subtracting gives K = (1/2)(A - A^T). Option (1/2)(A - A^T) is the skew-symmetric part, not the symmetric part. Option A + A^T is symmetric but lacks the normalizing factor 1/2, so it is twice the symmetric part. Option A^T - A is skew-symmetric and negatively scaled. Hence the symmetric part is (1/2)(A + A^T). Plausibility check: transposing (1/2)(A + A^T) gives (1/2)(A^T + A), identical to the original, confirming it is genuinely symmetric and correctly normalized.

Symmetric And Skew-symmetric

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