0
votes
2answers
46 views

Proving a Simple equation

I have a not so smart question; but I just cannot figure it out ! Suppose that I have a real $2 \times 2 $ matrix $(a_{ij})$ of non-zero determinant, and let $z \in \mathbb{C} $ be such that $ ...
4
votes
2answers
87 views

$T=-T^{*}$, show that $T+\alpha I$ is invertible.

Please don't answer the question. Just tell me if I am in the right direction. I should be able to solve this. We are given $T=-T^{*}$, show that $T+\alpha I$ is invertibe for all real alphas that ...
0
votes
1answer
50 views

Determinant formula and invertibility.

I am working on a problem where I need to find the determinant of $$ \begin{bmatrix} b & a & & \\ & b & a \\ & & & \ddots \\ & & & & ...
0
votes
0answers
273 views

Determinant of a general circulant matrix

I'm dealing with a problem that is comparable to "How do I calculate the circulant determinant $C(1, a, a^2, a^3,\dots , a^{n-1})$?", yet slightly more difficult: I was asked to determine the ...
7
votes
2answers
1k views

Determinant of an $n\times n$ complex matrix as an $2n\times 2n$ real determinant

If $A$ is an $n\times n$ complex matrix. Is it possible to write $\vert \det A\vert^2$ as a $2n\times 2n$ matrix with blocks containing the real and imaginary parts of $A$? I remember seeing such a ...
4
votes
1answer
2k views

Are complex determinants for matrices possible and if so, how can they be interpreted?

I've been asked to compute the determinant of a 3x3 matrix with complex entries. I have done so using the normal expansion along a row or column method that I would use were the entries real. My ...