How to find the projection of a matrix?

Question

Mathematics

Good evening. Please tell me how to solve this kind of problems Find the projection of the matrix

(1, 3, 3)
--------------
(1, 1, -1)
---------- ----
(-1, 5, 1)

to the space of symmetric matrices parallel to the space of skew-symmetric matrices from M_3(R)
IMPORTANT! I do not ask you to decide for me, but at least explain what you need to use

Answer 1

An interesting problem. Do you study at the Faculty of Applied Mathematics?
General considerations can be found here:
https://ocw.mit.edu/courses/mathematics/18-06sc-li...
(Projection in higher dimensions)
https://en.wikipedia.org/wiki/Projection_matrix
Try using elementary decomposition of a square matrix into a sum of symmetric and skew-symmetric:

( 1, 3,  3)   (1, 2, 1)   ( 0, 1,  2)
( 1, 1, -1) = (2, 1, 2) + (-1, 0, -3)
(-1, 5,  1)   (1, 2, 1)   (-2, 3,  0)