Hi!
It just dawned on me that any such matrices (I suppose there are only 4 places A could go ^^, ) are related by simple permutations. Since any permutation matrix has determinant + or - 1 then what you say must be true.
Thank you for the enlightenment! =)
The Cauchy expansion says that
\text{det} \begin{bmatrix}
A & x \\[0.3em]
y^T & a
\end{bmatrix}
= a \text{det}(A) - y^T \text{adj}(A) x ,
where A is an n-1 by n-1 matrix, y and x are vectors with n-1 elements, and a is a scalar.
There is a proof in Matrix Analysis by Horn and...
Hi all!
I have no application in mind for the following question but it find it curious to think about:
Say that we have a square matrix where the sum of the elements in each row and each column is zero. Clearly such a matrix is singular. Suppose that no row or column of the matrix is the...
That's a fair point.
I was playing around with a different matrix - Hermitian and also symmetric about the anti-diagonal. Turns out that the eigenvectors are closely related to the eigenvectors of the matrix R above so I was curious about their structure. It seems reasonable that the upper and...
Hi folks!
I've encountered the matrix below and I'm curious about its properties;
R=
\begin{pmatrix}
0 & N-S\\
N+S & 0
\end{pmatrix}
where R, N and S are real matrices, R is 2n by 2n, N is n by n symmetric and S is n by n skew-symmetric.
Clearly R is symmetric so the...
Quick question on eigenvectors;
Are there any general properties of a matrix that guarantee that a zero will or will not appear as an element in an eigenvector?
Thank you!