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Matrices

Symmetric and Skew-Symmetric Matrices

Understand

A square matrix \(A\) is symmetric if \(A^T = A\) (i.e. \(a_{ij} = a_{ji}\)), and skew-symmetric if \(A^T = -A\) (i.e. \(a_{ij} = -a_{ji}\)).

In a skew-symmetric matrix, putting \(i = j\) gives \(a_{ii} = -a_{ii}\), so every diagonal entry is 0.

Key theorem

For any square matrix \(A\):

  • \(A + A^T\) is symmetric, and \(A - A^T\) is skew-symmetric.
  • \(A = \tfrac12(A + A^T) + \tfrac12(A - A^T)\): every square matrix is the sum of a symmetric and a skew-symmetric matrix (and this split is unique).

Key Concepts

  • Symmetric: mirror image across the main diagonal.
  • Skew-symmetric: mirrored entries are negatives; diagonal is zero.

Formula Bank

Symmetric + skew-symmetric split

\[A = \tfrac12(A + A^{T}) + \tfrac12(A - A^{T})\]

Key Points

Skew-symmetric ⇒ zero diagonal

From \(a_{ii} = -a_{ii}\), every diagonal entry of a skew-symmetric matrix is 0.

Counting free entries

An \(n\times n\) symmetric matrix is fixed by \(\frac{n(n+1)}{2}\) entries; a skew-symmetric one by \(\frac{n(n-1)}{2}\) entries (its diagonal is zero).

Solved Examples

Splitting a 3 × 3 matrix

Express \(A = \begin{bmatrix} 1 & 2 & 4 \\ 6 & 8 & 1 \\ 3 & 5 & 7 \end{bmatrix}\) as the sum of a symmetric and a skew-symmetric matrix.

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Topic Summary

\(A^T = A\): symmetric. \(A^T = -A\): skew-symmetric (zero diagonal). \(A = P + Q\) with \(P = \tfrac12(A + A^T)\), \(Q = \tfrac12(A - A^T)\).

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