Determinant of a positive semi definite matrix

Question

Is it possible that the determinant of a positive semi definite matrix is equal to 0. It is coming to be zero in my case. I have a diagonal matrix with diagonal elements non zero. When I try to calculate the determinant of this matrix it is coming out to be 0. Why is it so?

Accepted Answer

This is the reason why computing the determinant is never a good idea. Yeah, I know. Your book, your teacher, or your boss told you to do so. They were probably wrong. Why? Determinants are poorly scaled beasts. Even if you compute the determinant efficiently (many algorithms fail to do even that) you don't really want a determinant most of the time.

Consider this simple positive definite matrix.

```
A = eye(1000);
```

What is the determinant? I need not even bother. It is 1. But, if you insist...

```
det(A)
ans =
     1
```

OK, so that works. How about if we simply multiply that entire matrix by a small constant, 0.1 for example. What is the determinant? You might say there is no reason to bother, as we already know the determinant. It must be just det(A)*0.1^1000, so 1e-1000.

```
det(A*0.1)
ans =
     0
```

What did we do wrong here? Where this failed is we forgot to remember we were working in floating point arithmetic. Since the dynamic range of a double in MATLAB goes down only to essentially

```
realmin
ans =
      2.2250738585072e-308
```

then smaller numbers turn into zero - they underflow. Anyway, most of the time when we compute a determinant, we are doing so for the wrong reasons anyway. If they want you to test to see if a matrix is singular, then use rank or cond, not det.

Determinant of a positive semi definite matrix

Solution

Problem