Finding the members of the Lie algebra of SO (n)

[c3po]

Homework Statement

Show that the members of the Lie algebra of SO(n) are anti-symmetric nxn matrices. To start, assume that the nxn orthogonal matrix R which is an element of SO(n) depends on a single parameter t. Then differentiate the expression:

R.R^T= I

with respect to the parameter t, keeping in mind that I is a constant matrix. Then you must consider that the element M of the Lie algebra is defined as:

M = (dR/dt) t=0

And that R(0) is the identity matrix.

Homework Equations

(A.B)^T = B ^T.A ^T

The Attempt at a Solution

d/dt[R(t).R^T(t)] = 0

I was introduced to linear algebra and group theory very recently and am having trouble doing any of the proofs that I am assigned for homework. I feel that this problem is probably easy, but it is surely not coming to me easily at all. . .

Please help!

 

[c3po]

I figured this out for the case of a 2x2 rotational matrix, but how would I generalize this for nxn matrices?

 

[Dick]

Homework Statement

Show that the members of the Lie algebra of SO(n) are anti-symmetric nxn matrices. To start, assume that the nxn orthogonal matrix R which is an element of SO(n) depends on a single parameter t. Then differentiate the expression:

R.R^T= I

with respect to the parameter t, keeping in mind that I is a constant matrix. Then you must consider that the element M of the Lie algebra is defined as:

M = (dR/dt) t=0

And that R(0) is the identity matrix.

Homework Equations

(A.B)^T = B ^T.A ^T

The Attempt at a Solution

d/dt[R(t).R^T(t)] = 0

I was introduced to linear algebra and group theory very recently and am having trouble doing any of the proofs that I am assigned for homework. I feel that this problem is probably easy, but it is surely not coming to me easily at all. . .

Please help!

Use the product rule. Evaluate at $t=0$. Remember a matrix $A$ being antisymmetric means $A=(-A^T)$. You want to show $R'(0)$ is antisymmetric.

 

[Hamza Rasheed]

Can you please provide complete solution? I have same question

 

[Dick]

Can you please provide complete solution? I have same question

No, that's not something we do. Try it and post your work if you want guidance. This isn't even hard if you give it some thought.

 

[Hamza Rasheed]

Thanks. I solved it :)