Symmetric Matrix as a subspace

In summary, S = {A € Mn,n | A = AT} is a subspace of the vector space Mn,n due to its satisfaction of conditions for a subset to be a subspace, such as closure under vector addition and scaler multiplication. This is because symmetric matrices, which are defined by A = AT, fulfill these conditions.
  • #1
seyma
8
0
My question is;
Let S = {A € Mn,n | A = AT } the set of all symmetric n × n matrices
Show that S is a subspace of the vector space Mn,n

I do not know how to start to this if you can give me a clue for starting, I appreciate.
 
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  • #2
What are the conditions that a subset be a subspace? It's not hard to show that S satisfies them.
 
  • #3
edit: A=A^T

It must satisfies the operations in S and it requires that it must be closed under vector addition and scaler multiplication. I got it: symmetic matrix satisfies those condition :) thanks:)
 

FAQ: Symmetric Matrix as a subspace

What is a symmetric matrix?

A symmetric matrix is a square matrix that is equal to its transpose, meaning that the elements above the main diagonal are equal to the elements below the main diagonal.

How can a symmetric matrix be represented as a subspace?

A symmetric matrix can be represented as a subspace by considering it as a subset of the vector space of all n x n matrices. This is because a symmetric matrix satisfies the properties of a subspace, such as closure under addition and scalar multiplication.

What are the advantages of representing a symmetric matrix as a subspace?

Representing a symmetric matrix as a subspace allows for easier manipulation and analysis of the matrix. It also simplifies calculations and allows for the use of linear algebra techniques to solve problems involving symmetric matrices.

How can we determine if a given matrix is a subspace of a symmetric matrix?

In order for a matrix to be a subspace of a symmetric matrix, it must satisfy the properties of a subspace, such as closure under addition and scalar multiplication. Additionally, it must also be square and equal to its transpose, just like a symmetric matrix.

Can a symmetric matrix have more than one subspace?

Yes, a symmetric matrix can have multiple subspaces. This is because a subspace can be any subset of the vector space that satisfies the properties of a subspace, and a symmetric matrix can have many different subsets that satisfy these properties.

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