# Tag Info

1 vote

### Why does the matrix exponential $e^A$ always exist?

It is worth to treat the matrix $A$ as a linear bounded operator $\mathcal{L}(\mathbb{R}^n,\mathbb{R}^n)$. Therefore defining the operator $\exp A:\mathbb{R}^n\to \mathbb{R}^n$ as a limit of the ...

1 vote

### How to take the derivative of a matrix with respect to itself?

The answer is a lot easier than the previous posters are indicating $\frac{d}{dX}(A*X)=A$ define $A=I_m$ where $m$ is the number of rows of $X$ because $I_m*X=X$ this is an identity and you get your ...
1 vote
Accepted

### Derivative of the inverse of a symmetric matrix w.r.t itself

In the single-variable case, we have that $$\dfrac{d}{dt}C(t)^{-1}=-C(t)^{-1}\dfrac{dC(t)}{dt}C(t)^{-1}.$$ This can obtained by differentiating the expression $C(t)C(t)^{-1}=I$ on both sides with some ...
1 vote

Accepted

### Differentiating with respect to a vector transpose

The plain simple answer is that the Differentiation of $β^T$($X^T$X)B becomes ($X^T$X)$\hat \beta$ ($\hat \beta$ is the Beta estimate) $B^T$ disappears as usual. `

1 vote
Accepted

### What the rank of a matrix with the elements of one column to be infinity?

The following two statements are in direct contradiction: Suppose the $m\times m$ real matrix $A$ if all the elements in the first column and first row are assigned to be infinity Infinity is not a ...
1 vote
Accepted

### Derivative of transpose of matrix

$X↦X^⊤$ is a linear map and so equal to its own derivative. A better question to ask is can we express this function in the standard form of a linear map, which is as a matrix-vector product in the ...
The set of real $n \times n$ matrices forms a real vector space with respect to the usual addition of matrices. Your identification of this space with the canonical vector space $\mathbb{R}^{n^2}$ is ...