Symmetric matrix

In linear algebra, a symmetric matrix is a square matrix that is equal to its transpose. Formally,

This article is about a matrix symmetric about its diagonal. For a matrix symmetric about its center, see Centrosymmetric matrix.

The sum and difference of two symmetric matrices is symmetric.

This is not always true for the : given symmetric matrices $A$ and $B$ , then $AB$ is symmetric if and only if $A$ and $B$ commute, i.e., if $AB=BA$ .

product

For any integer $n$ , $A^{n}$ is symmetric if $A$ is symmetric.

If $A^{-1}$ exists, it is symmetric if and only if $A$ is symmetric.

Rank of a symmetric matrix $A$ is equal to the number of non-zero eigenvalues of $A$ .

Horn, Roger A.; Johnson, Charles R. (2013), Matrix analysis (2nd ed.), Cambridge University Press, 978-0-521-54823-6

ISBN

, Encyclopedia of Mathematics, EMS Press, 2001 [1994]

"Symmetric matrix"

A brief introduction and proof of eigenvalue properties of the real symmetric matrix

How to implement a Symmetric Matrix in C++

$_$_$DEEZ_NUTS#0__titleDEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#0__subtitleDEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#8__titleDEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#8__descriptionDEEZ_NUTS$_$_$

Because equal matrices have equal dimensions, only square matrices can be symmetric.

The entries of a symmetric matrix are symmetric with respect to the main diagonal. So if $a_{ij}$ denotes the entry in the $i$ th row and $j$ th column then

for all indices $i$ and $j.$

Every square diagonal matrix is symmetric, since all off-diagonal elements are zero. Similarly in characteristic different from 2, each diagonal element of a skew-symmetric matrix must be zero, since each is its own negative.

In linear algebra, a real symmetric matrix represents a self-adjoint operator^[1] represented in an orthonormal basis over a real inner product space. The corresponding object for a complex inner product space is a Hermitian matrix with complex-valued entries, which is equal to its conjugate transpose. Therefore, in linear algebra over the complex numbers, it is often assumed that a symmetric matrix refers to one which has real-valued entries. Symmetric matrices appear naturally in a variety of applications, and typical numerical linear algebra software makes special accommodations for them.

$_$_$DEEZ_NUTS#3__descriptionDEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#5__titleDEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#5__descriptionDEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#5__heading--0DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#5__description--0DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#5__heading--1DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#5__description--1DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#4__titleDEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#4__descriptionDEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#4__heading--0DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#4__description--0DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#4__heading--1DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#4__description--1DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#6__titleDEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#6__subtextDEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#6__quote--0DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#6__name--0DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#6__company_or_position--0DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#6__quote--1DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#6__name--1DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#6__company_or_position--1DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#2__titleDEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#2__subtextDEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#2__quote--0DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#2__name--0DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#2__company_or_position--0DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#2__quote--1DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#2__name--1DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#2__company_or_position--1DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#2__quote--2DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#2__name--2DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#2__company_or_position--2DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#7__titleDEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#7__subtextDEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#7__quote--0DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#7__name--0DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#7__company_or_position--0DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#7__quote--1DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#7__name--1DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#7__company_or_position--1DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#7__quote--2DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#7__name--2DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#7__company_or_position--2DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#7__quote--3DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#7__name--3DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#7__company_or_position--3DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#7__quote--4DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#7__name--4DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#7__company_or_position--4DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#7__quote--5DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#7__name--5DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#7__company_or_position--5DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#7__quote--6DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#7__name--6DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#7__company_or_position--6DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#7__quote--7DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#7__name--7DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#7__company_or_position--7DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#7__quote--8DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#7__name--8DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#7__company_or_position--8DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#7__quote--9DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#7__name--9DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#7__company_or_position--9DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#7__quote--10DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#7__name--10DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#7__company_or_position--10DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#9__titleDEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#9__subtextDEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#9__quote--0DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#9__name--0DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#9__company_or_position--0DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#9__quote--1DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#9__name--1DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#9__company_or_position--1DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#9__quote--2DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#9__name--2DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#9__company_or_position--2DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#9__quote--3DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#9__name--3DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#9__company_or_position--3DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#9__quote--4DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#9__name--4DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#9__company_or_position--4DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#9__quote--5DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#9__name--5DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#9__company_or_position--5DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#9__quote--6DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#9__name--6DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#9__company_or_position--6DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#10__titleDEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#10__subtextDEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#10__quote--0DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#10__name--0DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#10__company_or_position--0DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#10__quote--1DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#10__name--1DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#10__company_or_position--1DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#10__quote--2DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#10__name--2DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#10__company_or_position--2DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#10__quote--3DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#10__name--3DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#10__company_or_position--3DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#10__quote--4DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#10__name--4DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#10__company_or_position--4DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#10__quote--5DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#10__name--5DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#10__company_or_position--5DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#10__quote--6DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#10__name--6DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#10__company_or_position--6DEEZ_NUTS$_$_$