[Math] determinant of a very large matrix in MATLAB

determinantMATLABmatrices

I have a very large random matrix which its elements are either $0$ or $1$ randomly. The size of the matrix is $5000$, however when I want to calculate the determinant of the matrix, it is either $Inf$ or $-Inf$. Why it is the case (as I know thw determinant is a real number and for a finite size matrix with finite elements, it cannot be $Inf$) and how can I remedy any possible mistake?

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I did a search for "determinant of random matrix" and found this article:

https://people.math.osu.edu/nguyen.1261/cikk/loglaw.pdf

It shows that the log of the determinant of a n by n random matrix is usually about $n\log(n)/2 $.

Therefore, for large $n$, any computation of the determinant will almost certainly overflow.

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[Math] Typing a large size block matrix on matlab

Not really sure whether this is what you are asking, but inputting a matrix in Matlab is quite straightforward:

A = [1+3i 2+4i 12-1i;
2 2i 2+2i]

Combining matrices is also quite simple:

B = [A A+1 2*A;
A-1 A-2 A+1i]

Matrix determinant lemma for non-invertible matrices

Would this help?

For the sake of simplicity, say that $$ D=\begin{bmatrix}0&0\\0&\tilde{D}\end{bmatrix}, $$ where $\tilde{D}$ is an $(m-k)\times(m-k)$ nonsingular matrix (so that the leading zero block is $k\times k$).

You can "fix" the singular block by a suitable diagonal matrix and then compensate it back to the low-rank update. Say, $\Delta=\mathrm{diag}(\delta_i)_{i=1}^k$ be a nonsingular diagonal matrix which we want to add to the leading block of $D$ to make it nonsingular. Note that if $E_k$ are the first $k$ columns of the identity matrix then $$ D_+=\begin{bmatrix}\Delta & 0 \\ 0 & \tilde{D}\end{bmatrix}=D+E_k\Delta E_k^T $$ so $$ D+XX^T=D_+-E_k\Delta E_k^T+XX^T =D_++[E_k,X][-E_k \Delta,X]^T=:D_++YZ^T. $$ You can apply the lemma to this matrix. If $k$ (the number of zero diagonal entries of $D$) is sufficiently small, this should not much harm the efficiency.

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Related Solutions

[Math] Typing a large size block matrix on matlab

Matrix determinant lemma for non-invertible matrices

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