# Symbolic Math Solves a Linear Algebra Conundrum

I'd like to introduce this week's guest blogger Alan Weiss. Alan writes documentation for mathematical toolboxes here at MathWorks.

An economist I know (well, he's my son) asked me a question the other day. I was able to answer him quickly by using Symbolic Math Toolbox™ calculations. You might find the steps to be an interesting case of a somewhat nonstandard use of symbolic mathematics

### Contents

#### A Linear Algebra Question

My son was working through a paper that described a computational algorithm. He got stuck on one step of the algorithm:

If $A$ is a positive definite (symmetric) matrix, find a **lower triangular** matrix $G$ such that

$$GAG^{-1} = D, \mbox{ where }D\mbox{ is diagonal.}$$

He called me asking if there was some linear algebra that he just never learned. I knew how to find a matrix $G$ that would diagonalize $A$ using the `eig` function. But a lower triangular version? I didn't know how to do that. I told him I would call him back after some investigation.

#### A Symbolic Approach

After spending a couple of minutes fruitlessly looking through a linear algebra book, I decided to see what Symbolic Math Toolbox had to say about the equations that any such matrix $G$ would have to satisfy. Here were my steps.

syms a b c g1 g2 g3 real % Create symbolic variables G = [g1,0;g2,g3] % Lower triangular matrix H = inv(G) % Its inverse A = [a,b;b,c] % Symmetric matrix A D = G*A*H % D is supposed to be diagonal

G = [ g1, 0] [ g2, g3] H = [ 1/g1, 0] [ -g2/(g1*g3), 1/g3] A = [ a, b] [ b, c] D = [ a - (b*g2)/g3, (b*g1)/g3] [ (a*g2 + b*g3)/g1 - (g2*(b*g2 + c*g3))/(g1*g3), (b*g2 + c*g3)/g3]

Look at the upper right entry in D. If D is diagonal, then b*g1 = 0. This means b = 0 or g1 = 0. But if b = 0 then A is already diagonal. And if g1 = 0 then the first row of G is zero, so G is not invertible.

In other words, I had just proved that there is no such thing as a lower triangular matrix that diagonalizes a symmetric matrix in this way.

I called my son back and explained the calculation. We decided that the paper he was reading had a misstatement, and he went ahead and used `eig` to diagonalize the matrix in his application. He will get in touch with the author of the paper to explain the misstatement in the algorithm.

I was quite happy with this result. Symbolic math calculations had saved us both a lot of time searching for an algorithm that doesn't exist.

#### Feedback

Do you use symbolic math calculations in cute or innovative ways? Tell us about it here.

**Category:**- Symbolic

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