If A is an n × n matrix, then
A = S + K,
where S is symmetric and K is skew symmetric. Let
A= [1 3 -2;4 2 2;5 1 2]
Find the matrices S and K described above
can some0ne explain how to get these two matricies?
thanks
To find the matrices S and K as described, you need to decompose the matrix A into its symmetric and skew-symmetric parts.
Here's how you can do it step by step:
1. Start with matrix A:
A = [1 3 -2; 4 2 2; 5 1 2]
2. Find the symmetric part, S:
For a matrix to be symmetric, its transpose should be equal to itself. So, to obtain the symmetric part, add A to its transpose and divide by 2:
S = (A + A') / 2
where A' is the transpose of A.
For the given example:
A' = [1 4 5; 3 2 1; -2 2 2]
A + A' = [1+1 3+4 -2+5; 4+3 2+2 2+1; 5-2 1+2 2+2] = [2 7 3; 7 4 3; 3 3 4]
S = (A + A') / 2 = [1 7/2 3/2; 7/2 2 3/2; 3/2 3/2 2]
3. Find the skew-symmetric part, K:
For a matrix to be skew-symmetric, its transpose should be the negation of itself. To obtain the skew-symmetric part, subtract A from its transpose and divide by 2:
K = (A' - A) / 2
For the given example:
A' = [1 4 5; 3 2 1; -2 2 2]
A' - A = [1-1 4-3 5+2; 3-4 2-2 1-2; -2-5 2-2 2-2] = [0 1 7; -1 0 -1; -7 0 0]
K = (A' - A) / 2 = [0 1/2 7/2; -1/2 0 -1/2; -7/2 0 0]
So, for the given matrix A = [1 3 -2; 4 2 2; 5 1 2], the symmetric matrix S = [1 7/2 3/2; 7/2 2 3/2; 3/2 3/2 2] and the skew-symmetric matrix K = [0 1/2 7/2; -1/2 0 -1/2; -7/2 0 0].