What are some Hermitian matrices with known formula for determinant?

Asked Jul 28 '21 at 17:49

Active Jul 28 '21 at 18:05

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I'm wondering if there are some Hermitian matrices with known formula for determinant. For example, according to wiki the determinant of the tri-diagonal matrix

could be calculated by the recursive formula

$$f_n=a_nf_{n-1}-c_{n-1}b_{n-1}f_{n-2}. $$

What are some other matrices with known formula for calculating the determinant? I'm especially interested in known formula for Hermitian matrices.

edited Jul 28 '21 at 18:05

asked Jul 28 '21 at 17:49

duang

1

You can research the Vandermonde Determinant, it has a known formula with a very simple derivation. – benmcgloin Jul 28 '21 at 17:52
The determinant of a diagonal matrix is the product of the diagonal entries – J. W. Tanner Jul 28 '21 at 17:53
Triangular matrices have an easy formula- they’re also the product of diagonal entries. Look up the block matrix determinant as well – FShrike Jul 28 '21 at 18:03
@FShrike Thanks! I already know the determinant formula for the diagonal/triangular matrices, but rn I'm especially looking for determinant of Hermitian matrices. I'll edit the title to make it clear. – duang Jul 28 '21 at 18:05
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@duang You might find symmetric and real circulant matrices to be interesting. – Ben Grossmann Jul 28 '21 at 20:25
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@duang You might also be interested in matrices of the form $\lambda I_n \pm BB^T$, where $B$ is an $n \times k$ matrix with $k$ small (equal to $1,2$ or $3$). The determinant is given by $\lambda^{n-k} \det(\lambda I_k \pm B^TB)$, which is often a much nicer expression. See the W-A determinant identity. The case of $k=1$ in particular has come up on this site fairly frequently. – Ben Grossmann Jul 28 '21 at 20:29

What are some Hermitian matrices with known formula for determinant?

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