Definition · applies to matrices

High school

A symmetric matrix is a square matrix that is unchanged when flipped across its main diagonal: the entry in row $i$ and column $j$ equals the entry in row $j$ and column $i$.

Undergraduate

A symmetric matrix is a square matrix that is equal to its transpose:

\[A^{\top} = A.\]

Properties

  • A real symmetric matrix has real eigenvalues and an orthonormal basis of eigenvectors (the spectral theorem), so $A = Q \Lambda Q^{\top}$ with $Q$ orthogonal and $\Lambda$ diagonal.
  • Every square matrix is the sum of a symmetric matrix and a skew-symmetric matrix: \[A = \frac{1}{2}(A + A^{\top}) + \frac{1}{2}(A - A^{\top}).\]

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