Intereting Posts

Does a continuous point-wise limit imply uniform convergence?
How can a Markov chain be written as a measure-preserving dynamic system
Divisor on curve of genus $2$
Sum of all natural numbers is 0?
Problems and Resources to self-study medium level math
Does this inequality have any solutions in $\mathbb{N}$?
Intuitive proofs that $\lim\limits_{n\to\infty}\left(1+\frac xn\right)^n=e^x$
Pseudo-inverse of a matrix that is neither fat nor tall?
The distance function on a metric space
Reflection across a line?
Closed form of $\int_0^1(\ln(1-x)\ln(1+x)\ln(x))^2\,dx$
How is $\lim_{x \to a}\left(\frac{x^n – a^n}{x – a}\right) = n\times a^{n-1}$?
Evaluate $\sum\limits_{k=1}^n k^2$ and $\sum\limits_{k=1}^n k(k+1)$ combinatorially
About irrational logarithms
What exactly is a number?

$x, a$ in $\mathbb R^n$, $A$ in $\mathbb R^{n\times n}$. Compute $d(x^T a)/dx$ and $d(x^T A x)/dx$.

I’m not sure about how to think about these and how to do these. Can someone explain how to derive the expressions for the two?

Finally, what happens when we have $A$ and $X$, BOTH in $\mathbb R^{n\times n}$, and we want to find $dTrace(XA)/dX$?

- Calculate unknown variable when surface area is given. (Calculus)
- Indefinite integral question: $\int \frac{1}{x\sqrt{x^2+x}}dx$
- Can this standard calculus result be explained “intuitively”
- Is $\int_{-\infty}^{\infty} \sin x \, \mathrm{dx}$ divergent or convergent?
- Limit of $\sqrt{x^2-6x+7}-x$ as x approaches negative infinity
- Find a minimum of $x^2+y^2$ under the condition $x^3+3xy+y^3=1$

- How to show that $\lim_{n \to \infty} a_n^{1/n} = l$?
- Evaluating $\int_0^\pi \frac{x}{(\sin x)^{\sin (\cos x)}}dx$
- Proving the integral of an inverse function
- Real-analytic periodic $f(z)$ that has more than 50 % of the derivatives positive?
- Is there a name for function with the exponential property $f(x+y)=f(x) \cdot f(y)$?
- A improper integral with complex parameter
- $\int {e^{3x} - e^x \over e^{4x} + e^{2x} + 1} dx$
- When does a Square Matrix have an LU Decomposition?
- Lagrange diagonalization theorem - what if we omit assumption about the form being symmetric
- If $A^2 = I$ (Identity Matrix) then $A = \pm I$

Let’s use the convention that members of $\mathbb{R}^{n}$ are column

vectors. Recall that

$$

x^{T}a=\sum_{i=1}^{n}x_{i}a_{i}

$$

and for a scalar $c$,

$$

\frac{dc}{dx}\equiv\left(\begin{array}{c}

\frac{dc}{dx_{1}}\\

\frac{dc}{dx_{2}}\\

\vdots\\

\frac{dc}{dx_{n}}

\end{array}\right).

$$

Therefore,

$$

\frac{d\left(x^{T}a\right)}{dx}=\left(\begin{array}{c}

a_{1}\\

a_{2}\\

\vdots\\

a_{n}

\end{array}\right)=a.

$$

You can follow the same arguments to get

$$

\frac{d\left(x^{T}Ax\right)}{dx}=2Ax.

$$

See a list of identities at https://en.wikipedia.org/wiki/Matrix_calculus#Scalar-by-matrix_identities.

Note that $x^ta=a^tx$ and so

$$

\frac{d(x^tA)}{dx}=\frac{d}{dh}a^t(x+h)|_{h=0}=a^t.

$$

Similarly, since $h^tAx=x^tA^th$,

$$

\frac{d(x^tAx)}{dx}=\frac{d}{dh}(x+h)^tA(x+h)|_{h=0}=\frac{d}{dh}(x^tAh+h^tAx)|_{h=0}=x^t(A+A^t).

$$

Finally, since

$$

tr(XA)=\sum_{i=1}^n\sum_{j=1}^n X_{ij}A_{ji}

$$

we have

$$

\frac{d tr(XA)}{d X_{ij}}=A_{ji},

$$

which gives

$$

\frac{d tr(XA)}{d X}=A^t.

$$

All this must be interpreted properly if used to make other computations!

- Proving $\binom{2n}{n}\ge\frac{2^{2n-1}}{\sqrt{n}}$
- Proving identity $ \binom{n}{k} = (-1)^k \binom{k-n-1}{k} $. How to interpret factorials and binomial coefficients with negative integers.
- I need help to advance in the resolution of that limit: $ \lim_{n \to \infty}{\sqrt{\frac{n!}{n^n}}} $
- Finding a closed form for $\cos{x}+\cos{3x}+\cos{5x}+\cdots+\cos{(2n-1)x}$
- Non-aleph infinite cardinals
- How do we describe standard matrix multiplication using tensor products?
- Showing an ideal is prime in polynomial ring
- Osgood condition
- Do mathematicians, in the end, always agree?
- How many ways to arrange people on a bench so that no woman sits next to another woman?
- Proving The Average Value of a Function with Infinite Length
- When is $X^n-a$ is irreducible over F?
- Probability that the last ball is white?
- Limit of $f(x+\sqrt x)-f(x)$ as $x \to\infty$ if $|f'(x)|\le 1/x$ for $x>1$
- Proof of Convergence: Babylonian Method $x_{n+1}=\frac{1}{2}(x_n + \frac{a}{x_n})$