Symmetric matrix

A matrix $A=(a_{i,j})_{i,j}$ is symmetric if it equals its transpose $A^{T}$ , which means that $a_{i,j}=a_{j,i}$ for all positive integers $i,j$ such that $a_{i,j}$ is defined.

Remarks

Note that any symmetric matrix needs to be a square matrix.

Practice this concept with exercises

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Soient $A = \begin{pmatrix} 2 & 3 & 1 \\ 3 & 5 & 5 \\ 1 & 5 & 2 \end{pmatrix}$ , $B = \begin{pmatrix} 0 & 1 \\ 0 & 0\end{pmatrix}$ , $C = \begin{pmatrix} 1 & 2 \\ 2 & 3\end{pmatrix}$ , $D$ une matrice diagonale et $E = \begin{pmatrix} 1 & 2 & 3 \\ 2 & 8 & 7 \\ 3 & 7 & 12\end{pmatrix}$ .
Pour chaque matrice, dire si elle symétrique ou non, en justifiant.

Open exercise

Problems using this concept (2)

The canonical inner product on real matrices An inequality on symmetric matrices

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