# Derivative of Log Determinant of a Matrix w.r.t a scalar parameter

• fbelotti
In summary, the conversation discusses solving a derivative involving a scalar parameter and a symmetric tridiagonal matrix. The question is whether there is a rule for this type of problem. The conversation also mentions the differentiation of the logarithm of a matrix, and one participant suggests using Taylor's theorem. Another participant realizes that the derivative simplifies to the logarithm of the determinant of the matrix.
fbelotti
Hi All,

I'm trying to solve the following derivative with respect to the scalar parameter $\sigma$

$$\frac{\partial}{\partial \sigma} \ln|\Sigma|,$$

where $\Sigma = (\sigma^2 \Lambda_K)$ and $\Lambda_K$ is the following symmetric tridiagonal $K \times K$ matrix
$$\Lambda_{K} = \left( \begin{array}{ccccc} 2 & -1 & 0 & \cdots & 0 \\ -1 & 2 & -1 & \cdots & 0 \\ 0 & -1 & \ddots & \ddots & \vdots \\ \vdots & \ddots & \ddots & \ddots & -1 \\ 0 & 0 & \ldots & -1 & 2 \\ \end{array}\right).$$

Is there a rule for these case?

Have you thought about what the logarithm of a matrix means?

Typically to define a function for matrices that is consistent with the usual elementary functions, you use Taylor's theorem in the indeterminate x and replace x with the matrix. The differentiation is straightforward, I think.

kevinferreira said:
Have you thought about what the logarithm of a matrix means?

joeblow said:
Typically to define a function for matrices that is consistent with the usual elementary functions, you use Taylor's theorem in the indeterminate x and replace x with the matrix. The differentiation is straightforward, I think.

Am I missing something here? fbelotti is taking the derivative of the determinant of a matrix. The matrix logarithm shouldn't need to come into this at all, no?

fbelotti, if your matrix is just ##\Sigma = \sigma^2 \Lambda_K##, then by the property of determinants, ##|cB| = c^n |B|## for an nxn matrix B, are you not just taking the derivative of ##\log(|\sigma^2 \Lambda_K|) = \log(\sigma^{2K} |\Lambda_K|)##, where ##|\Lambda_K|## is just a constant?

Oops. Only now noticed the determinant.

Mute, you are perfectly right. Many thanks for pointing that out. It was too simple... maybe it was too late and I was too tired...

## 1. What is the derivative of log determinant of a matrix with respect to a scalar parameter?

The derivative of log determinant of a matrix with respect to a scalar parameter is the logarithm of the determinant of the matrix multiplied by the derivative of the matrix with respect to the scalar parameter.

## 2. Why is the derivative of log determinant of a matrix with respect to a scalar parameter important?

The derivative of log determinant of a matrix with respect to a scalar parameter is useful in many applications such as optimization, machine learning, and signal processing. It helps in finding the optimal values of the scalar parameter that maximize or minimize the determinant of the matrix.

## 3. How is the derivative of log determinant of a matrix with respect to a scalar parameter calculated?

The derivative of log determinant of a matrix with respect to a scalar parameter is calculated using the chain rule and the derivative of the matrix with respect to the scalar parameter. This can be done analytically or numerically using numerical differentiation methods.

## 4. Can the derivative of log determinant of a matrix with respect to a scalar parameter be negative?

Yes, the derivative of log determinant of a matrix with respect to a scalar parameter can be negative. This indicates that the determinant of the matrix is decreasing with respect to the scalar parameter. Similarly, a positive derivative indicates an increasing determinant.

## 5. Is the derivative of log determinant of a matrix with respect to a scalar parameter always defined?

No, the derivative of log determinant of a matrix with respect to a scalar parameter is not always defined. It depends on the differentiability of the matrix with respect to the scalar parameter. If the matrix is not differentiable, then the derivative of log determinant does not exist.

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