TITLE: How to calculate natural log of determinant involving identity matrix?
QUESTION [0 upvotes]: I have an expression given by $$\ln \left(\text{det}\left(\frac{y^{H}y}{N}\text{I}_{N}\right)\right)$$ where $N$ is a scalar and product of $y^{H}y$ is a matrix. How can I simplify this expression?

REPLY [2 votes]: The determinant of a scalar multiple of the $n\times n$ identity matrix is
$$\det(\alpha I_n) = \alpha^n\det(I) = \alpha^n$$
and the logarithm is
$$\log\det(\alpha I_n) = n\log(\alpha)$$
Setting $\,\alpha = \frac 1n y^Hy\;$ yields
$$\log\det\left(\frac {y^Hy}{n}\,I_n\right) = n\log\left(\frac {y^Hy}{n}\right)$$
{"set_name": "stack_exchange", "score": 0, "question_id": 3840644}