## Thursday, August 8, 2019

### screencast software, OBS, windows, mac, linux

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## Wednesday, August 7, 2019

### Yuan method on adjacency matrix controllability

Yuan clearly used weighted matrix, and j->i as direction. So, column to row indicate direction?
Wikipedia, “In directed graphs, the in-degree of a vertex can be computed by summing the entries of the corresponding column, and the out-degree can be computed by summing the entries of the corresponding row.” The question now is does i->j and j->i matters? It can be tested using a star shaped network with outward and inwarding arrows. A quick exam on these show both star networks should have the same number of minimal control nodes. Well, eigen values of a matrix and its transpose are the same, see .
So, i->j and j->i does not matter! This is somewhat shocking to me.

### Eigenvalues of a Matrix and its Transpose are the Same

https://yutsumura.com/eigenvalues-of-a-matrix-and-its-transpose-are-the-same/

Recall that the eigenvalues of a matrix are roots of its characteristic polynomial.
Hence if the matrices $A$ and ${A}^{\mathrm{T}}$ have the same characteristic polynomial, then they have the same eigenvalues.
So we show that the characteristic polynomial ${p}_{A}\left(t\right)=det\left(A-tI\right)$ of $A$ is the same as the characteristic polynomial ${p}_{{A}^{\mathrm{T}}}\left(t\right)=det\left({A}^{\mathrm{T}}-tI\right)$ of the transpose ${A}^{\mathrm{T}}$.
We have

Therefore we obtain ${p}_{{A}^{\mathrm{T}}}\left(t\right)={p}_{A}\left(t\right)$, and we conclude that the eigenvalues of $A$ and ${A}^{\mathrm{T}}$ are the same.

### Remark: Algebraic Multiplicities of Eigenvalues

Remark that since the characteristic polynomials of $A$ and the transpose ${A}^{\mathrm{T}}$ are the same, it furthermore yields that the algebraic multiplicities of eigenvalues of $A$ and ${A}^{\mathrm{T}}$ are the same.