flip half of matrix over the diagonal to make a symmetric matrix

Dear all, If I have a half of a matrix, e.g

1
2 3
4 5 6
7 8 9 10
...

I want to flip it over the diagonal to make a symmetric matrix:

1 2 4 7
2 3 5 8
4 5 6 9
7 8 9 10

Please help. Thanks

[Jan]

Tghe solution depends on how the triangular "array" is stored. Are there zeros in the upper right elements?

[John D'Errico]

Is the matrix stored as a matrix, so only the lower triangle, with zeros as the upper triangle. Or is there junk in the upper triangle? Or do you have the elements of the lower triangle, stored in a vector?

All of these things are pertinent to any efficient solution.

Answer 1

[Azzi Abdelmalek]

A=[1 0 0 0
  2 3 0 0
  4 5 6 0
  7 8 9 10]
[n,m]=size(A);
B=A'+A
B(1:n+1:end)=diag(A)

[Tom Davis]

Or, with fewer lines:

A = (A+A') - eye(size(A,1)).*diag(A)

[Junho Kweon]

Oh my.. I like fewer version. :)

[Bill Tubbs]

I think you can do it in one line like this:

B = triu(A.',1) + tril(A)  % Takes bottom half of A to make B symmetric

Also, this does not do a conjugate transpose.

Answer 2

[Simon Liljestrand]

A=[1 0 0 0
2 3 0 0
4 5 6 0
7 8 9 10];
B=A'+triu(A',1)';

[Stephen23]

+1 Neat. Also possible with fewer transposes:

>> B = A+tril(A,-1).'
B =
    1    2    4    7
    2    3    5    8
    4    5    6    9
    7    8    9   10

[Simon Liljestrand]

Ah, that makes it a little more elegant still!

Answer 3

[Ben McSeveney]

If I have a column vector e.g.

1
2
3

How do I quickly create a symmetric matrix i.e.

[1 2 3;
 2 1 2;
 3 2 1]

?

[James Tursa]

Please delete this answer and instead open up a new question.

[Jos (10584)]

for which the answer will be toeplitz

v = 1:5
toeplitz(v)

[Tom Davis]

[a,circshift(a,1),circshift(a,2)]
triu(a' - a + ones(size(a,1))) + tril(a - a' + ones(size(a,1))) - eye(size(a,1))

Answer 4

[Walter Bova]

A = (A+A') - eye(size(A)).*A

[sun]

This formula A = (A+A') - eye(size(A)).*A is correct.

Answer 5

[Rohit Sachdeva]

As most people have pointed out, I just wanted to add another way of doing this:

B = (A+A') - diag(diag(A));

The (A+A') part is clear to most of us. This is how the 2nd term works:

• First diag(.) extracts the diagonal elements of A.
• The next diag(.) creats a matrix with just those diagonal elements.
• Finally we subtract that matrix of diagonal elements from the (A+A') as required.

This eliminates the need of the eye(.) function. Hope it helps!

[Steven Lord]

In general, this doesn't work.

A = magic(4)

A = 4x4

16 2 3 13 5 11 10 8 9 7 6 12 4 14 15 1

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B = (A+A') - diag(diag(A))

B = 4x4

16 7 12 17 7 11 17 22 12 17 6 27 17 22 27 1

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It does work if the matrix is real and one of the triangular parts already contains all 0 values.

C = triu(A)

C = 4x4

16 2 3 13 0 11 10 8 0 0 6 12 0 0 0 1

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D = (C+C') - diag(diag(C))

D = 4x4

16 2 3 13 2 11 10 8 3 10 6 12 13 8 12 1

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It doesn't work if the matrix is complex even if the matrix is triangular.

format shortg
C(1, 2) = 2+3i

C = 
           16 +          0i            2 +          3i            3 +          0i           13 +          0i
            0 +          0i           11 +          0i           10 +          0i            8 +          0i
            0 +          0i            0 +          0i            6 +          0i           12 +          0i
            0 +          0i            0 +          0i            0 +          0i            1 +          0i

D = (C+C') - diag(diag(C))

D = 
           16 +          0i            2 +          3i            3 +          0i           13 +          0i
            2 -          3i           11 +          0i           10 +          0i            8 +          0i
            3 +          0i           10 +          0i            6 +          0i           12 +          0i
           13 +          0i            8 +          0i           12 +          0i            1 +          0i

D is not symmetric, it is however Hermitian.

issymmetric(D)

ans = logical
   0

ishermitian(D)

ans = logical
   1

But if you used the non-conjugate transpose then the result is symmetric but not Hermitian:

E = (C+C.')-diag(diag(C))

E = 
           16 +          0i            2 +          3i            3 +          0i           13 +          0i
            2 +          3i           11 +          0i           10 +          0i            8 +          0i
            3 +          0i           10 +          0i            6 +          0i           12 +          0i
           13 +          0i            8 +          0i           12 +          0i            1 +          0i

issymmetric(E)

ans = logical
   1

ishermitian(E)

ans = logical
   0