# Tag Info

Accepted

### When do the nonzero eigenvalues of a directed graph Laplacian have the same absolute value?

This is false. Run this code in SageMath; you can do this at sagecell.sagemath.org if you do not have SageMath already installed on your own computer. ...
• 12.5k

### How to prove that each element of $A(A^TA)^{-1}A^Ty$ is greater than 0, if $A(i,j)>0$ and $y=[1, 1, 1, ..., 1]^T$

This does not hold in general. Note that $\beta=A(A^TA)^{-1}A^Ty\in \mathbb{R}^m$ belongs to the image ${\rm Im} A$ (that is, to the column space of $A$). Also, $A^T\beta=A^TA(A^TA)^{-1}A^Ty=A^Ty$, so ...
• 106k
1 vote
Accepted

• 123k
1 vote

### Reference request: "Higher order eigentuples" as generalized eigenvectors?

This seems to be equivalent to finding invariant two dimensional subspaces. One way to algebraically describe two dimensional subspaces is the vector space $V'=\Lambda^2V$ (the second wedge product). ...
Accepted

### “Smallest” non-zero linear combination of vectors to obtain a non-negative vector

I will show that $\delta_j\le j-1$. I suspect that in fact $\delta_j=j-1$, but I don't have a complete argument for it and it might be wrong. Let $L$ be the lattice spanned by the columns of $A$ and ...
• 551
Accepted

### Reference request: "Higher order eigentuples" as generalized eigenvectors?

The observation that $MV = V\Lambda$ is essentially an invariant subspace relation is commonly used in methods to solve algebraic Riccati equations via Hamiltonian matrices; for instance, the Schur ...
• 19.8k

• 3,731