0
votes
0answers
19 views

Natural logarithm of a square matrix without eigen-analysis

I'm trying to find a method to determine the natural logarithm of a square nonsingular matrix without using eigenvalues or eigenvectors. So far, I've only found this method: ...
1
vote
0answers
24 views

Basis for a general function of tr$X^k$

I have a function of a $4\times 4$-matrix variable $X$ which is a general function of $\left<X^k\right>$ where $\left<\cdot\right>$ denotes the trace. My question is this: is there a ...
2
votes
6answers
278 views

Definition of matrix exponential

Is there an alternative definition of a matrix exponential so I can use it to prove that $$e^{A}=\sum_{m=0}^{\infty} \frac{1}{m!}(A)^m \;?$$ Thanks a lot in advance!
1
vote
4answers
74 views

Differentiating second order term of Taylor polynomial (multivariable)

I am trying to derive Newton step in an iterative optimization. I know the step is: $$\Delta x=-H^{-1}g$$ where H is Hessian and $g$ is gradient of a vector function $f(x)$ at $x$. I also know the ...
1
vote
0answers
72 views

Why the ith coefficient of $|\lambda I-A|$ is the sum of all $i$-th order principle minors of $A$?

I come across a theorem that $f(\lambda )=|\lambda I-A|$, which equals to $\lambda ^{n}-a_{1}\lambda ^{n-1}+\alpha _{2}\lambda ^{n-2}-...(-1)^{n}a_{n}$ where $a_{i}$ is the sums of all ith order ...
2
votes
1answer
1k views

Taylor expansion for matrices

Is it possible to define a Taylor expansion for matrices ? Can I use functional derivative ? More precisely I have to calculate something like : $\ln(A+B)$ using a Taylor expansion, where $A$ and $B$ ...
0
votes
1answer
357 views

Need help understanding Hessian matrix for direction estimation

Additional context: $H = |δ^2f / δx_iδx_j|$ is the Hessian matrix. $(3)$ From my previous question: What are the functionality of δ symbol and $δr^T$?, I got a few questions: I have read more ...
1
vote
1answer
62 views

What are the functionality of δ symbol and $δr^T$?

I got two questions here: Does anybody know what is the functionality of the small delta letter δ here? Is it simply the same as the rate of change just like the big delta letter Δ? And for the ...