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Symmetric Matrices as a Vector Space

Symmetric Matrices as a Vector Space

February 07, 2025 Education

Understanding Symmetric Matrices as a Vector Space

When discussing symmetric matrices in the context of linear algebra, it is often useful to examine whether the set of these matrices forms a vector space. In this article, we will explore the key properties that must be satisfied to confirm that the set of symmetric matrices indeed constitutes a vector space. This analysis is crucial for both theoretical understanding and practical applications in various fields, such as physics, computer science, and engineering.

Definition of Symmetric Matrices

A symmetric matrix is a square matrix A that satisfies the following property:

AT A

Here, AT denotes the transpose of the matrix A. The transpose of a matrix is obtained by flipping the matrix over its diagonal, essentially switching the rows and columns.

Introduction to the Set of Symmetric Matrices

Let Sn denote the set of all n × n symmetric matrices. A generic symmetric matrix can be represented as:

A  begin{pmatrix} a_{11}  a_{12}  cdots  a_{1n} a_{12}  a_{22}  cdots  a_{2n} vdots  vdots  ddots  vdots a_{1n}  a_{2n}  cdots  a_{nn} end{pmatrix}

Note that the elements aij satisfy aij aji, ensuring the symmetry of the matrix.

Properties of a Vector Space

To show that Sn is a vector space, we must verify the following properties:

Closure under Addition

If A and B are symmetric matrices, then:

A   B  begin{pmatrix} a_{11}   b_{11}  a_{12}   b_{12}  cdots  a_{1n}   b_{1n} a_{12}   b_{12}  a_{22}   b_{22}  cdots  a_{2n}   b_{2n} vdots  vdots  ddots  vdots a_{1n}   b_{1n}  a_{2n}   b_{2n}  cdots  a_{nn}   b_{nn} end{pmatrix}


Since (A   B)T  AT   BT  A   B, the result of the addition is also a symmetric matrix. This proves closure under addition.


Closure under Scalar Multiplication


For a symmetric matrix A and a scalar c, we have:


cA  begin{pmatrix} ca_{11}  ca_{12}  cdots  ca_{1n} ca_{12}  ca_{22}  cdots  ca_{2n} vdots  vdots  ddots  vdots ca_{1n}  ca_{2n}  cdots  ca_{nn} end{pmatrix}


Again, (cA)T  cAT  cA, confirming that the scalar multiple is also symmetric. This proves closure under scalar multiplication.


Existence of the Zero Vector


The zero matrix 0 is symmetric, and it acts as the zero vector in this space. For any symmetric matrix A, adding the zero vector results in A itself, maintaining the symmetry.


Existence of Additive Inverses


For any symmetric matrix A, its additive inverse -A is also symmetric. Specifically, (-A)T  -AT  -A. Thus, every symmetric matrix has an additive inverse within the set Sn.


Associativity and Commutativity of Addition


Matrix addition is associative and commutative. These properties hold true for all symmetric matrices, making them valid for any matrix in Sn.


Distributive Properties


Scalar multiplication distributes over matrix addition, and scalar addition distributes over scalar multiplication. These distributive properties, when applied to symmetric matrices, ensure that these operations behave in a consistent and predictable manner.


Conclusion


Since the set of symmetric matrices satisfies all the properties of a vector space, it is concluded that the set of all n × n symmetric matrices forms a vector space. This conclusion is supported by the verification of closure under addition and scalar multiplication, the existence of the zero vector and additive inverses, and the satisfaction of the additional vector space properties.

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