Symmetric Matrices as a Subspace of all Square Matrices

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Theorem: subspace W of a vector space V over R is a subset of V which also has the properties that W is closed under addition and scalar multiplication. That is, For all x, y in W, x and y are in V and for any c in R, cx + y is in W.
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Let W_n be the subset of all M_(n x n)(R) such that A_ij = A_ji (ie. the set of all real symmetric square matrices).

To show that W_n is a subspace of M_(n x n)(R) it is sufficient to demonstrate that the three conditions of the theorem are satisfied.

S1) In M_(n x n)(R) there exists a matrix Z:{z_ij = 0, for all 1 < i,j < n} (ie. a matrix such that for every row i or column j consists of the zero vector).

Since z_ij = z_ji it is a symmetric matrix and must also be in W_n.

To satisfy S2,3) it is necessary to show the property of transpose as it relates to addition and scalar multiplication of matrices.

Let A, B be matrices in M_(n x n)(R), let a,b be scalars in R. Then let O = aA + bB and let P = a(A^t) + b(B^t). Then:

O_ij = a(A_ij) + b(B_ij) = P_ji => P_ij = O_ji

S2) If A,B are in W_n, with A = A^t and B = B^t then:

(A + B)^t = A^t + B^t = A + B => A + B is in W_n.

Hence W_n is closed under addition.

S3) If A is in W_n then for all a in R:

(aA) = a(A^t) = aA => aA is in W_n.

Hence W_n is closed under scalar multiplication.

Thus by theorem we see that W_n ≤ M_(n x n)(R). []

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