This article defines a property that can be evaluated for a square matrix. In other words, given a square matrix (a matrix with an equal number of rows and columns) the matrix either satisfies or does not satisfy the property.
View other properties of square matrices
Suppose is a positive integer and is a matrix. We say that is symmetric if the following holds:
Note that the condition is not required; the above definition is equivalent to:
Finally, because the equality condition is symmetric in , it suffices to check it for , so the above definition if equivalent to:
We can encode a symmetric matrix by simply storing the entries on or above the diagonal. This is a total of entries instead of the total of entries.
For the equal entries and , , we will use as the label. This is purely a matter of convention.
|General description of symmetric matrix||Number of free parameters (equals )|
For both the trace and the determinant, there does not seem to be a simpler way of computing them than the standard approach for square matrices.
|Matrix operation||Description in terms of encoding||Number of additions||Number of multiplications||Number of inverse computations||Set of symmetric matrices closed under operation?|
|Addition of two symmetric matrices||Add the corresponding entries of the two matrices||0||0||Yes|
|Multiplication of two symmetric matrices||Usual matrix multiplication algorithms; symmetry does not seem to offer speedup||for naive matrix multiplication||for naive matrix multiplication||0||No|
Relation with other properties
|Property||Meaning||Proof of implication||Proof of strictness (reverse implication failure)||Intermediate notions|
|scalar matrix||scalar multiple of identity matrix||Bisymmetric matrix, Diagonal matrix|FULL LIST, MORE INFO|
|diagonal matrix||all off-diagonal entries are zero||||
|bisymmetric matrix||both symmetric and centrosymmetric||||
|symmetric Toeplitz matrix||Bisymmetric matrix|FULL LIST, MORE INFO|