# Is every Toeplitz matrix invertible?

1. Nov 15, 2008

### maverick280857

Hi

Given a square matrix $R_{X}$ that is Toeplitz, is it necessarily invertible? I am not convinced about this.

In communication theory, a finite duration impulse response (FIR) filter in discrete-time is constructed for purposes of linear prediction of a random process X(t). The autocorrelation matrix of X is found to be a Toeplitz matrix and textbooks go one step further in trying to find the optimal predictor coefficients, by taking the inverse of this matrix, in a certain step.

I was just curious whether every Toeplitz matrix is invertible (apart from the trivial cases like the zero matrix, of course) or whether the invertibility is solely a function of the nature of the random process X(t).

TIA
Cheers

2. Nov 15, 2008

### morphism

A Toeplitz matrix is one which has constant diagonals, correct? If so, then the answer is no, there are singular Toeplitz matrices. Just take one that is strictly upper triangular.

3. Nov 15, 2008

### maverick280857

Thanks, and yes, I just discovered that the matrix that arises in the communication theoretic application I mentioned is not just Toeplitz but also symmetric with a nonzero determinant. So in that particular case, it is invertible.