I was recently asked how to create a tridiagonal matrix in SAS/IML software. For example, how can you easily specify the following symmetric tridiagonal matrix without typing all of the zeros?

```proc iml; m = {1 6 0 0 0, 6 2 7 0 0, 0 7 3 8 0, 0 0 8 4 9, 0 0 0 9 5 };```

This matrix can be created in two steps. The first step is to use the DIAG function to create a matrix that contains specific values on the diagonal and zeros elsewhere:

```/** create diagonal matrix **/ d = 1:5; m = diag(d);```

The second step relies on the fact that SAS/IML matrices are stored in row-major order. Therefore, the indices of the upper diagonal of m are 2, 8, 14, and 20. In general, for an n x n matrix, the following statements compute the indices of the upper diagonal and assign values to those locations:

```n = ncol(m); /** size of matrix **/ d1 = 6:9; /** values for upper diag **/ uppIdx = do(2, n*n, n+1); /** indices of upper diag **/ m[uppIdx]= d1; /** set upper diag **/```

Similarly, the following statements assign the lower diagonal:

```lowIdx = do(n+1, n*n, n+1); /** indices of lower diag **/ m[lowIdx] = d1; /** set lower diag **/ print m;```

Of course, you can also use the indices of the upper or lower diagonal to extract the values, rather than to assign the values.

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Distinguished Researcher in Computational Statistics

Rick Wicklin, PhD, is a distinguished researcher in computational statistics at SAS and is a principal developer of SAS/IML software. His areas of expertise include computational statistics, simulation, statistical graphics, and modern methods in statistical data analysis. Rick is author of the books Statistical Programming with SAS/IML Software and Simulating Data with SAS.