Symmetric and Skew-Symmetric Matrices

A matrix is called symmetric if it equals its own transpose. That is, a square matrix A is symmetric when A^T = A, meaning the entry in row i and column j equals the entry in row j and column i for every pair (i, j). Symmetric matrices appear everywhere in mathematics, physics, and statistics — from quadratic forms and covariance matrices in statistics to stiffness matrices in engineering and the adjacency matrices of undirected graphs. A matrix is called skew-symmetric (or antisymmetric) if its transpose equals its negative: A^T = -A. This forces every entry on the main diagonal to be zero (since a_{ii} = -a_{ii} implies a_{ii} = 0), while entries above the diagonal are the negatives of their mirror images below it. Skew-symmetric matrices naturally encode rotations and cross-product operations in physics and mechanics. Together, these two families form a beautiful decomposition theorem: every square matrix A can be written uniquely as the sum of a symmetric matrix and a skew-symmetric matrix. The symmetric part is (A + A^T)/2 and the skew-symmetric part is (A - A^T)/2. This splitting is fundamental in many branches of linear algebra and applied mathematics.

Key takeaways

Symmetric Matrices: Definition
Skew-Symmetric Matrices: Definition
Key Properties of Symmetric Matrices
Key Properties of Skew-Symmetric Matrices
The Fundamental Decomposition

Symmetric Matrices: Definition

An n×n matrix A is symmetric if A^T = A, i.e., a_{ij} = a_{ji} for all 1 ≤ i, j ≤ n. For example, the matrix [[3, 7, -1], [7, 0, 4], [-1, 4, 2]] is symmetric because the (1,2) entry equals the (2,1) entry (both 7), the (1,3) entry equals the (3,1) entry (both -1), and so on. Only square matrices can be symmetric.

Skew-Symmetric Matrices: Definition

An n×n matrix A is skew-symmetric if A^T = -A, i.e., a_{ij} = -a_{ji} for all i, j. Setting i = j gives a_{ii} = -a_{ii}, so 2a_{ii} = 0, forcing every diagonal entry to be 0. A typical 3×3 skew-symmetric matrix looks like [[0, b, c], [-b, 0, d], [-c, -d, 0]] for real numbers b, c, d.

Key Properties of Symmetric Matrices

1) The sum of two symmetric matrices is symmetric: if A^T = A and B^T = B, then (A+B)^T = A^T + B^T = A + B. 2) A scalar multiple of a symmetric matrix is symmetric: (cA)^T = cA^T = cA. 3) For any matrix B (not necessarily square), the product B^T B is always symmetric, since (B^T B)^T = B^T (B^T)^T = B^T B. This fact is central in least-squares problems. 4) All eigenvalues of a real symmetric matrix are real numbers — this is part of the Spectral Theorem.

Key Properties of Skew-Symmetric Matrices

1) The sum of two skew-symmetric matrices is skew-symmetric. 2) All eigenvalues of a real skew-symmetric matrix are either zero or purely imaginary complex numbers. 3) The determinant of any odd-order skew-symmetric matrix is 0 (a non-trivial result). 4) The product of two skew-symmetric matrices is generally NOT skew-symmetric; instead, their commutator [A, B] = AB - BA is skew-symmetric.

The Fundamental Decomposition

Every n×n matrix A can be written uniquely as A = S + K, where S = (A + A^T)/2 is symmetric and K = (A - A^T)/2 is skew-symmetric. Verification: S^T = (A + A^T)^T / 2 = (A^T + A)/2 = S ✓. K^T = (A - A^T)^T / 2 = (A^T - A)/2 = -K ✓. And S + K = (A + A^T)/2 + (A - A^T)/2 = 2A/2 = A ✓. Uniqueness follows because if A = S₁ + K₁ = S₂ + K₂, then S₁ - S₂ = K₂ - K₁; the left side is symmetric and the right is skew-symmetric, and the only matrix that is both symmetric and skew-symmetric is the zero matrix.

The Key Equation

A = \underbrace{\frac{A + A^T}{2}}_{S \text{ (symmetric)}} + \underbrace{\frac{A - A^T}{2}}_{K \text{ (skew-symmetric)}}

A — Original matrix (dimensionless): Any n×n real (or complex) square matrix.
A^T — Transpose of A (dimensionless): The matrix obtained by reflecting A across its main diagonal; (A^T)_{ij} = A_{ji}.
S = \tfrac{A+A^T}{2} — Symmetric part (dimensionless): The unique symmetric matrix in the decomposition; satisfies S^T = S.
K = \tfrac{A-A^T}{2} — Skew-symmetric part (dimensionless): The unique skew-symmetric matrix in the decomposition; satisfies K^T = -K and has all-zero diagonal.

Worked Example

Problem. Let A = [[4, 1, -3], [7, 0, 2], [-1, 5, 6]]. (a) Find the symmetric part S and the skew-symmetric part K of A. (b) Verify that S + K = A.

S = \begin{pmatrix}4 & 4 & -2\\ 4 & 0 & \tfrac{7}{2}\\ -2 & \tfrac{7}{2} & 6\end{pmatrix},\quad K = \begin{pmatrix}0 & -3 & -1\\ 3 & 0 & -\tfrac{3}{2}\\ 1 & \tfrac{3}{2} & 0\end{pmatrix}

Answer. S = [[4, 4, -2], [4, 0, 7/2], [-2, 7/2, 6]] (symmetric); K = [[0, -3, -1], [3, 0, -3/2], [1, 3/2, 0]] (skew-symmetric); and S + K = A.

Figure

Diagram

Origins · 1 minArthur Cayley and how this was discovered

Arthur Cayley

1821–1895 · British

Cayley was one of the founders of modern matrix theory. His 1858 paper 'A Memoir on the Theory of Matrices' introduced the formal algebraic study of matrices, including matrix multiplication, the matrix inverse, and the characteristic polynomial. The classification of matrices by symmetry properties — including symmetric and skew-symmetric forms — developed directly from this foundational framework. Cayley worked as a lawyer for 14 years while simultaneously writing over 200 research papers in mathematics, before eventually becoming a professor at Cambridge at age 42.

Matrix Transpose — The transpose operation A → A^T is the foundation of both definitions: symmetric means A^T = A, and skew-symmetric means A^T = -A.
Matrix Multiplication — The product B^T B is always symmetric — a key result connecting matrix multiplication to symmetric matrices, used in statistics and optimization.
Eigenvalues and Eigenvectors — The Spectral Theorem guarantees that real symmetric matrices have only real eigenvalues and orthogonal eigenvectors — one of the most powerful results in linear algebra.
Quadratic Forms — Every quadratic form Q(x) = x^T A x can be associated with a symmetric matrix; the symmetric decomposition ensures we can always replace A with its symmetric part without changing the form.