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symmetric matrix (Definition)

Definition:
Let $ A=(a_{ij})$ be a square matrix of order $ n$. The matrix $ A$ is symmetric if $ a_{ij} = a_{ji}$ for all $ 1 \leq i \leq n, 1 \leq j \leq n$.

$ A = \begin{pmatrix} a_{11} & \cdots & a_{1n} \ \vdots & \ddots & \vdots \ a_{n1} & \cdots & a_{nn} \end{pmatrix}$

Properties:

  1. $ A^t = A$ where $ A^t$ is the matrix transpose

Examples:

  • $ \begin{pmatrix} a & b \ b & c \end{pmatrix}$
  • $ \begin{pmatrix} a & b & c \ b & d & e \ c & e & f \end{pmatrix}$



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See Also: self-dual, Hessian matrix, skew-Hermitian matrix

Other names:  symmetric
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Cross-references: transpose, matrix, order, square matrix
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This is version 9 of symmetric matrix, born on 2001-11-20, modified 2006-09-20.
Object id is 974, canonical name is SymmetricMatrix.
Accessed 18618 times total.

Classification:
AMS MSC15-00 (Linear and multilinear algebra; matrix theory :: General reference works )

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