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Symmetric Matrices

Symmetric-matrix checker. Enter A to test whether A=Aᵀ, with residual—for symmetric matrices.

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A = Aᵀ ⇔ aᵢⱼ = aⱼᵢ
space-separated entries, semicolon-separated rows, e.g. 1 2;3 4

📖 Tutorial｜Symmetric Matrices

1. Definition
A square matrix with A = Aᵀ, i.e. aᵢⱼ = aⱼᵢ, is symmetric.
2. Symbols
SymbolMeaning
Aᵀtranspose
aᵢⱼ=aⱼᵢsymmetry condition
3. How this tool works
  • A symmetric matrix mirrors about the main diagonal;
  • It must be square;
  • Symmetric matrices have real eigenvalues and are orthogonally diagonalizable.
4. Steps
  1. Enter A;
  2. Click Calculate to compare A and Aᵀ;
  3. See whether it is symmetric.
5. Example
Example: A=1 2 3;2 4 5;3 5 6. a₁₂=2=a₂₁, a₁₃=3=a₃₁, a₂₃=5=a₃₂, so it is symmetric.
6. Pitfalls
Only square matrices can be symmetric;
Symmetry is about the main diagonal, not the anti-diagonal;
Near-equality within 1e-6 is treated as equal.

❓ FAQ｜Symmetric Matrices

What is a symmetric matrix?
A square matrix satisfying A=Aᵀ, i.e. aᵢⱼ=aⱼᵢ.
Can a non-square matrix be symmetric?
No; symmetric matrices must be square.
Properties?
All eigenvalues are real; eigenvectors for distinct eigenvalues are orthogonal; orthogonally diagonalizable.
Is a diagonal matrix symmetric?
Yes, since aᵢⱼ=0=aⱼᵢ for i≠j.
What tolerance is used?
Entries within 1e-6 are treated as equal.