basis for Snn

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Oct 14th 2012, 05:16 PM
tenrear3

basis for Snn

Another one. The M n,n is a vector space, and that Snn( the set of n x n symmetric matrices) is a subspace
Find a basis for the space of all 2 x 2 symmetric matrices.
B)Find a basis for the space of all 3 x 3 symmetric matrices
C) Determine the dimensions of the space of n x n symmetric ( Hint: this essentially a counting argument.From examining the bases in the first two parts determine what goes into the basis required.
Oct 14th 2012, 06:42 PM
tenrear3

Re: basis for Snn

Mnn is a space of all m x n matrices Symmetric matrix Snn are in M and is a subset of all symmetric matrices, exchanges rows and columns such as A=AT (A,B) are in Symmetric matrices such as (aij=aji or bij=bji) an example of this is A+B==aij+bij, therefore (A+B)T = (aji+bji)=(BT)+(AT) hope this helps.
Oct 15th 2012, 01:26 AM
Deveno

Re: basis for Snn

a 2x2 symmetric matrix is of the form:

$M = \begin{bmatrix}a&b\\b&c \end{bmatrix}$

this suggests the basis set:

$B = \left\{\begin{bmatrix}1&0\\0&0 \end{bmatrix}, \begin{bmatrix}0&1\\1&0 \end{bmatrix}, \begin{bmatrix}0&0\\0&1 \end{bmatrix} \right\}$

linear independence and spanning should be clear (that is: you should be able to prove these on demand).

for 3x3 symmetric matrices, prove the following is a basis set:

$B' = \left\{\begin{bmatrix}1&0&0\\0&0&0\\0&0&0 \end{bmatrix}, \begin{bmatrix}0&1&0\\1&0&0\\0&0&0 \end{bmatrix}, \begin{bmatrix}0&0&1\\0&0&0\\1&0&0 \end{bmatrix}, \begin{bmatrix}0&0&0\\0&1&0\\0&0&0 \end{bmatrix}, \begin{bmatrix}0&0&0\\0&0&1\\0&1&0 \end{bmatrix}, \begin{bmatrix}0&0&0\\0&0&0\\0&0&1 \end{bmatrix}\right\}$

does this suggest what to do for the nxn case? (hint: consider the diagonal and off-diagonal entries separately).
Oct 15th 2012, 02:02 PM
tenrear3

Re: basis for Snn

Thank can anyone help with part C given Deveno's answer.
Oct 15th 2012, 02:33 PM
Deveno

Re: basis for Snn

another hint: you should have n(n+1)/2 matrices. where did i get this number?