Is every Toeplitz matrix invertible?

[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

 

[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.

 

[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.