Symmetric matrix . It is said of the square matrix that is equal to its transposition .
Summary
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- 1 Definition
- 2 Properties
- 3 See also
- 4 Sources
Definition
Let A be a square matrix of dimension m . If the element in row i and column j of A is denoted by A (i, j) , then matrix A is symmetric if A (i, j) = A (j, i) .
Example: the identity matrix is a symmetric matrix.
Properties
- The inverseof a regular symmetric matrix is symmetric.
- The attached matrixof a symmetric matrix is symmetric.
- The symmetric sum is symmetric. The product is if, and only if, it is also commutative.
- The eigenvalues (eigenvalues) of a square, real, and symmetric matrix are real.
- A real square matrix, A, is symmetric if, and only if, it is diagonalizable by an orthogonal step matrix, Q.
