Knowen - Definition of symmetric matrix

Created almost 9 years ago, updated 5 months ago

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Definition. A matrix $A$ is symmetric if it equals its transpose: $A = A^T$.

Equivalently, $a_{ij} = a_{ji}$ for all $i, j$. A symmetric matrix is always square.

Example. $A = \begin{pmatrix} 1 & 3 & 0 \\ 3 & -2 & 4 \\ 0 & 4 & 5 \end{pmatrix}$ is symmetric.

Properties:

The sum of symmetric matrices is symmetric
If $A$ and $B$ commute, then $AB$ is symmetric
For any matrix $M$, both $MM^T$ and $M^T M$ are symmetric
The diagonal entries of a symmetric matrix can be arbitrary, but the matrix is determined by $n(n+1)/2$ independent entries

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Symmetric matrices

**Definition.** A matrix $A$ is *symmetric* if it equals its transpose: $A = A^T$.

Equivalently, $a_{ij} = a_{ji}$ for all $i, j$. A symmetric matrix is always square.

**Example.** $A = \begin{pmatrix} 1 & 3 & 0 \\ 3 & -2 & 4 \\ 0 & 4 & 5 \end{pmatrix}$ is symmetric.

**Properties:**
- The sum of symmetric matrices is symmetric
- If $A$ and $B$ commute, then $AB$ is symmetric
- For any matrix $M$, both $MM^T$ and $M^T M$ are symmetric
- The diagonal entries of a symmetric matrix can be arbitrary, but the matrix is determined by $n(n+1)/2$ independent entries

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