Spectral decomposition in Mata

[Anna Polsi]

Hi Mata users,

I would like to know if there is a built-in function in Mata that calculates the square root of a matrix using the spectral decomposition or do I have to write the routine to do so?


Thanks,

Anna

[William Lisowski]

It is not clear from your question exactly what your starting point is, either in terms of the matrix or in terms of your current knowledge of Mata. Perhaps your answer lies in the Mata routines documented following the links in the Eigensystems, powers, & transcendental section of the output of

help m4_matrix

[Anna Polsi]

Thank you William for your reply. That's true, I haven't expressed myself well.

Precisely, I wanted to do in Mata what sqrtm does in Matlab, where "X = sqrtm(A) returns the principal square root of the matrix A, that is, X*X = A".

Starting from a mxm matrix A, I want to find A^-1/2. I procededed manually with the decomposition as follows A^-1/2 = X*D*X^-1, where X is the matrix of eigenvectors and D the diagonal matrix of eigenvalues of inverse square-roots.
In Mata,

mata:

A = (5, -4, 1, 0, 0\ -4, 6, -4, 1, 0\ 1, -4, 6, -4, 1\ 0, 1, -4, 6, -4\ 0, 0, 1, -4, 6)
X = J(1,5,0)
L = J(5,5,0)

symeigensystem(A, X, L)

D=diag((sqrt(L)):^(-1))

invsqrtA = X*D*invsym(X)

end

I was just wondering whether there was a built-in function able to compute the square-root of a function immidiately without doing it manually.

Thanks again.

[William Lisowski]

Perhaps this example will start you in a useful direction.

. mata:
------------------------------------------------- mata (type end to exit) ----------------------
: A = (5, -4, 1, 0, 0\ -4, 6, -4, 1, 0\ 1, -4, 6, -4, 1\ 0, 1, -4, 6, -4\ 0, 0, 1, -4, 6)

: sA = matpowersym(A,-.5)

: A
[symmetric]
        1    2    3    4    5
    +--------------------------+
  1 |   5                      |
  2 |  -4    6                 |
  3 |   1   -4    6            |
  4 |   0    1   -4    6       |
  5 |   0    0    1   -4    6  |
    +--------------------------+

: invsym(sA*sA)
[symmetric]
                  1              2              3              4              5
    +----------------------------------------------------------------------------+
  1 |             5                                                              |
  2 |            -4              6                                               |
  3 |             1             -4              6                                |
  4 |  -1.66533e-15              1             -4              6                 |
  5 |  -1.66533e-16   -2.99760e-15              1             -4              6  |
    +----------------------------------------------------------------------------+

: end
------------------------------------------------------------------------------------------------

[Anna Polsi]

Thanks, William, for your useful suggestion.