Linear and multilinear algebra; matrix theory Definition

Symmetric matrix

Definition

Let $R$ be a ring and let $A = (a_{ij})$ be an $n \times n$ square matrix over $R$ . The matrix $A$ is symmetric when $A^T = A$ , where $A^T$ is the transpose of $A$ . Equivalently, $a_{ij} = a_{ji}$ for all $i$ and $j$ .

Remarks

The matrix $\begin{pmatrix} 1 & 7 \\ 7 & 2 \end{pmatrix}$ is symmetric. For each $m \times n$ matrix $B$ over a commutative ring, $B^T B$ and $B B^T$ are symmetric. A real symmetric matrix has real eigenvalues and an orthonormal basis of eigenvectors, by the spectral theorem. Symmetric matrices represent symmetric bilinear forms in a basis.

Reading path

To understand this page from the ground up, read these 4 pages in order. The foundations come first.

Show the path as a graph

Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

no evidence level

Matrix definition
Square matrix definition
Transpose definition
Symmetric matrix definition

Where this leads

18 pages build on this one.

Tag	`0N28QN4CYA`
URI	`m2://0N28QN4CYA`
Kind	definition
Writing	W1 stated
MSC	15A99, 15B57
Defines	Symmetric matrix
Uses	Transpose, Square matrix

Symmetric matrix

Definition

Remarks

Reading path

Where this leads

Further reading

What this page needs for W2

What links here

Used by