Untitled

Here is a simple trick for turning any square matrix into a symmetric one.

A = magic(4) + eye(4);
B = A+A'
B = 
    34     7    12    17
     7    24    17    22
    12    17    14    27
    17    22    27     4

Picking a symmetric matrix at random, there is little chance that it will be positive definite. Fortunately, the built-in Cholesky factorization chol always detects this property. The following would cause an error if run:

R = chol(B)

There is a different trick for making an SPD matrix from (almost) any other matrix.

B = A'*A
B = 
   411   213   224   377
   213   393   385   234
   224   385   383   233
   377   234   233   381

R = chol(B)
R = 
  20.273134932713294  10.506515184106888  11.049105170140576  18.596038612245525
                   0  16.811101649985090  15.996120561162783   2.297317496515087
                   0                   0   2.245306645409047  -4.105343043039244
                   0                   0                   0   3.613286419735975

norm( R'*R - B )
ans = 
     7.051978355127676e-14

A word of caution: chol does not check symmetry; in fact, it doesn't even look at the lower triangle of the input matrix.

chol( triu(B) )
ans = 
  20.273134932713294  10.506515184106888  11.049105170140576  18.596038612245525
                   0  16.811101649985090  15.996120561162783   2.297317496515087
                   0                   0   2.245306645409047  -4.105343043039244
                   0                   0                   0   3.613286419735975