Intereting Posts

Does the fibres being equal dimensional imply flatness?
How to solve $ \int_0^{\pi} \sin{2x}\sin{x} dx $?
Law of exponent for complex numbers
Finding an equation of circle which passes through three points
number of subsets of even and odd
Reference for Algebraic Geometry
Prove the statement $\forall x\in\mathbb N (x > 1\to\exists k\in\mathbb N\exists m \in\mathbb N (m \equiv 1 (\text{mod }2) \wedge x = 2^km))$.
proof that union of a sequence of countable sets is countable.
Proof for power functions
Find limit $a_{n+1} = \int\limits^{a_n}_0 \bigl( 1+\frac 1 4 \cos^{2n+1}t \bigr) \, dt$
Calculate integrals involving gamma function
Why this proof $0=1$ is wrong?(breakfast joke)
Cardinality of sets regarding
Why define norm in $L_p$ in that way?
Why does the diophantine equation $x^2+x+1=7^y$ have no integer solutions?

What is the derivative of $\log |AA^T|$ with respect to $A$, where $|A|$ denotes the determinant of $A$?

- Determinant of matrix with trigonometric functions
- $A,B\in M_{n}(\mathbb{R})$ so that $A>0, B>0$, prove that $\det (A+B)>\max (\det(A), \det(B))$
- Proving that $\lim\limits_{h\to 0}\frac{f(x+h)-2f(x)+f(x-h)}{h^2}=f''(x)$
- Construct a function which is continuous in $$ but not differentiable at $2, 3, 4$
- Computing determinant of a specific matrix.
- Partial Derivative v/s Total Derivative
- Simple related rates derivative question
- How should I calculate the $n$th derivative of $f(x)=x^x$?
- Finding Maximum Area of a Rectangle in an Ellipse
- Derivative of a vector

Let $f : \mathbb{R}^{m \times n} \to \mathbb{R}$ be defined by

$$f (X) = \log |X X^T|$$

The directional derivative of $f$ in the direction of $V \in \mathbb{R}^{m \times n}$ is

$$\begin{array}{rl} D_V f (X) &= \displaystyle\lim_{h \to 0} \frac{1}{h} \left( \log |(X + h V) (X + h V)^T| – \log |X X^T| \right)\\\\ &= \displaystyle\lim_{h \to 0} \frac{1}{h} \left( \log |X X^T + h V X^T + h X V^T + h^2 V V^T| – \log |X X^T| \right)\end{array}$$

If $X$ has full row rank, i.e., if $\operatorname{rank} (X) = m$, then $X X^T$ is invertible. Hence,

$$\begin{array}{rl} D_V f (X) &= \displaystyle\lim_{h \to 0} \frac{1}{h} \left( \log |X X^T + h V X^T + h X V^T + h^2 V V^T| – \log |X X^T| \right)\\\\ &= \displaystyle\lim_{h \to 0} \frac{1}{h} \left( \log |X X^T| + \log | I_m + h (X X^T)^{-1} \left( V X^T + X V^T + h V V^T \right)| – \log |X X^T| \right)\\\\ &= \displaystyle\lim_{h \to 0} \frac{1}{h} \, \log | I_m + h (X X^T)^{-1} \left( V X^T + X V^T + h V V^T \right)|\\\\ &= \displaystyle\lim_{h \to 0} \frac{1}{h} \, \log ( 1 + h \operatorname{tr} ((X X^T)^{-1} \left( V X^T + X V^T + h V V^T \right)))\\\\ &= \displaystyle\lim_{h \to 0} \operatorname{tr} ((X X^T)^{-1} \left( V X^T + X V^T + h V V^T \right))\\\\ &= \operatorname{tr} ((X X^T)^{-1} \left( V X^T + X V^T \right))\\\\ &= 2 \, \operatorname{tr} (X^T (X X^T)^{-1} V)\end{array}$$

Assume that $A\in M_{n,m}$ where $n\leq m$ and $rank(A)=n$ and let $f(A)=\log(\det(AA^T))$. Then $Df_A:H\rightarrow tr((AA^T)^{-1}(HA^T+AH^T))$ or, if $A$ is a derivable matrix function, then $f'(A)=tr((AA^T)^{-1}(A’A^T+AA’^T))$.

Hint 1:

$\partial \log\det(M)=trace(M^{-1} \partial M)$

Hint 2:

Use the chain rule.

I hope this helps. Please let us know if you couldn’t figure it out.

- Cardinal number subtraction
- Dense subset of $C(X)$
- If a function is Riemann integrable, then it is Lebesgue integrable and 2 integrals are the same?
- Why is the continuum hypothesis believed to be false by the majority of modern set theorists?
- show $\lim_{x\to 0}\frac{e^x-1}{x}=1$ without L'Hopital
- $C \otimes A \cong C \otimes B$ does not imply $A \cong B$
- How is the hyperplane bundle cut out of $(\mathbb{C}^{n+1})^\ast \times \mathbb{P}^n$?
- Intuitive explanation of a positive-semidefinite matrix
- Is there a set theory that handles collections of proper classes?
- Factorise $y^2 -3yz -10z^2$
- How $a+bi$ becomes $\left(\matrix{a & -b\\b & a}\right)$?
- Show $(A^o)^c=\overline{A^c}$
- How to compute a lot of digits of $\sqrt{2}$ manually and quickly?
- Unique Limits in T1 Spaces
- Can $(X_1,X_2) \cap (X_3,X_4)$ be generated with two elements from $k$?