0
votes
0answers
40 views

Function with constant derivative

We have a column matrix $P_i$ defined as follows $P_i= {\begin{pmatrix} a_i \\ b_i \\ c_i \end{pmatrix}}_{3\times 1}\tag 1 $. Given Data All $a_i,b_i,c_i$ are constants It is given that $i$ can ...
2
votes
0answers
62 views

Definite Integral involving matrices

We have a definite integral of the form given below $ f(t) = \int_0^1 e^{\alpha X(t)} \frac{dX(t)}{dt} e^{(1-\alpha) X(t)}\,d\alpha \tag 1$ Given Data in the question $X(t)$ is a ...
1
vote
1answer
45 views

Matrix - Commutative property

I have a rotation matrix represented as $R(t)=e^{B(t)},\tag 1$ where $B(t)$ is a skew symmetric matrix (since any rotation matrix can be expressed as a matrix exponent of a skew symmetric matrix), ...
0
votes
1answer
28 views

Matrix Solution

I have matrix integral equation of the following form ${f^{'}(x)}_{1 \times 1}A_{3\times 3}=P_{3\times3} (1-x)+Q_{3 \times 3}x \tag 1$ . All dimensions are indicated in equation itself. " ' " ...
0
votes
2answers
42 views

Matrix Exponent - equivalent of a rotation matrix

For every Rotation Matrix,there is a Matrix Exponent representation where the power is a skew symmetric matrix. More clearly if I have a rotation matrix ${R}_{3 \times 3}$ then there will be a skew ...
1
vote
0answers
62 views

Matrix exponent form

We have an equation of matrix exponent $ Ae^{Ax}R-e^{Ax}R (P_1 +P_2 x) = Y \tag1$ Given condition $A,R,P_1,P_2,Y$ are constant $3 \times 3 $ matrices. R is invertible,orthonormal,determinent ...
0
votes
2answers
60 views

Multiplication and derivation of 3D matrix

I have $A(q)=\begin{bmatrix}q_1 &q_2 & q_3\\ 2q_1 &3q_2 & 4q_3\\ 2q_1 &3q_1 & 10\\ \end{bmatrix}\tag 1$ $ q= {\left(\begin{array}{c}q_1\\q_2\\q_3\\q_4\\q_5\\q_6 ...
0
votes
2answers
80 views

A Matrix Integral Equation

We have an integral equation on matrix. ${\Im(t)}=\Im(0)+\int_{0}^{t} \Im(s)[K(s)]_{ \times }ds \tag 1$ $[\hspace{.2cm} ]_{\times}$ is skew symmetric matrix with diagonals zero and is non ...
2
votes
0answers
38 views

Finding a solution basis of differential equation

Find a solution basis of $$y'=\left[ \begin{matrix}3&-4&-2\\2&-3&-2\\0&0&1\\ \end{matrix} \right]y \,\text{ and find the solution } \Phi \text{ with } \Phi(0) = (1,1,1).$$ I'm ...
1
vote
1answer
92 views

find a matrix transform

Given a vector $v={(v_1,v_2,...,v_n)}^T$, I would like to find some matrix operations on $v$ to create an $n \times n$ matrix $X$ such that its entry $X_{i,j} $ satisfy (1), (2), (3), (4), ...
0
votes
0answers
14 views

Numerical solution of first order ODE

I have an in-homogeneous ODE. $R'(x)-(C_1 +C_2 x) R(x) = R_1-C_1 R_0\, x \tag 1$. What I know is the constant matrix $ R(0)$ as initial condition. Question:- how to find out R(1) by numerical ...
0
votes
2answers
74 views

Matrix exponential Differentiation

We have the equation $e^X = \sum_{k=0}^\infty{1 \over k!}X^k.$, where X is a matrix of dimension $3 \times 3$ . Now I have a function $f(x)=C_1x+C_2*\frac{x^2}{2} $ where $C_1,C_2,f(x)$ has ...
2
votes
1answer
39 views

Matrix time derivative

Given a complex, square matrix $A$ that is diagonalizable, is it possible to write a simple formula for $\frac{d}{dt} A^t$ for a real, positive power $t$ and for $A$ a smooth function of $t$?
0
votes
1answer
46 views

Please help me work out matrix derivative

$\mathbf{X}$ is a m by n matrix; $\mathbf{\theta}$ is $n \times 1$ vector, and $\mathbf{y}$ is $ m \times 1 $ vector; Let $$ J(\theta) = \frac 1 {2m} (\mathbf{X} \mathbf{\theta} - ...
6
votes
2answers
472 views

Why integration operator has no eigen values?

Let $V$ be the vector space of all functions from $\mathbb R$ into $\mathbb R$ which are continuous. Let $T$ be the linear operator on $V$ defined by $$(Tf)(x) = \int_0^x f(t) dt$$ Prove that ...
8
votes
3answers
87 views

Find the expansion for $\det(I+\epsilon A)$ where $\epsilon$ is small without using eigenvalue.

I'm taking a linear algebra course and the professor included the problem that prove $$ \rm{det}(I+\epsilon A) = 1 + \epsilon\,\rm{tr}\,A + o(\epsilon) $$ Since the professor hasn't covered the ...
0
votes
1answer
28 views

Finding determinant of 4*4 Matrix via LU Decomposition?

What is the shortcut way of finding the determinant of a 4 by 4 matrix (and I assume this applies to any n by n square matrix greater than 2) once you have found an LU or PLU decomposition? Given ...
3
votes
0answers
17 views

$\sqrt{X}$ where $X$ is a positive definite matrix is smooth $C^{\infty}$ [duplicate]

I'm trying to prove the following statement. Let $P_n \subset Mat_{nxn}(\mathbb R)$ be the set of all symmetric positive definite matrices with real entries of size $n$x$n$. Let $\sqrt{}:P_n \to ...
0
votes
1answer
38 views

Why is this proof valid - inverse function theorem

Question from worksheet, I don't fully understand the solution the teacher gave. Question: let $S$ be the set of symmetric positive definite matrices of dimension $n$x$n$. Let $T: S \to S$, ...
0
votes
1answer
35 views

Derivative of a Matrix with respect to a vector

I know that for two k-vectors, say $A$ and $B$, $\partial A/\partial B$ would be a square $k \times k$ matrix whose $(i,j)$-th element would be $\partial A_i/\partial B_j$. But could someone please ...
1
vote
2answers
63 views

Double integral requiring orthogonal transformations and quadric forms

I have $\int_{-\infty}^\infty \int_{-\infty}^\infty exp(-x^T Ax) \;\mathrm{d}x_1 \; \mathrm{d}x_2$ $A = \left[ \begin{align} 3 && 2 \\ 2 && 3 \end{align} \right]$ Where $x^T = ...
2
votes
2answers
202 views

Rotate Existing Vector

Hello and apologies if the title of the question is not very precise. Question: I am reading the document talking about the simulation of photons in tissues using a Monte Carlo simulation. The exact ...
0
votes
0answers
40 views

Differentiating Exponential matrix Expression

To give the scalar version first: For the well known Ornstein-Uhlenbeck process: $dr(t)=\alpha(b-r(t))dt+\sigma dW(t)$ It is well known that the variance is: $\sigma_r^2=\sigma^2 \int_u^t\exp^{-2 ...
1
vote
1answer
35 views

How to evaluate a limit that involves matrices

I've stumbled upon this problem while I was browsing through the contents of an admission exam . I've struggled tremendously with this exercise and I've got no idea what do to next , it's eating me ...
0
votes
1answer
36 views

Derivative of $f(x)=\|Ax\|_2^2$

I'm trying to find the derivative of $f(x)=\|Ax\|_2^2$ where $A$ is some matrix and $\|u\|_2$ is the euclidean norm of $u$, $\|u\|_2 = \sqrt{u_1^2+u_2^2+\cdots+u_n^2}$ I know how to do this by ...
0
votes
1answer
37 views

Using Gram-Scmidt to obtain an orthonormal basis for the column space

How would I use the method of Gram-Schmidt to obtain an orthonormal basis for the column space of the matrix? $$A = \begin{bmatrix} 2 & 3 & 1 \\ -1 & -1 & 1 \\ 1 & 0 & 1 \\ 2 ...
2
votes
1answer
68 views

What is $\frac{\partial }{{\partial A}}tr({A^T}BA)$

What is $\frac{\partial }{{\partial A}}tr({A^T}BA)$? My thoughts: $\frac{\partial }{{\partial A}}tr(AB{A^T}) = A(B + {B^T})$, hence $\frac{\partial }{{\partial {A^T}}}tr({A^T}B{A^{}}) = ...
0
votes
2answers
47 views

Need help finding Jacobian matrix of diffeomorphism of spheres

Let $S_a \subset \mathbb{R}^{n+1}$ and $S_b \subset \mathbb{R}^{n+1}$ be two spheres of radius $a$ and $b$ respectively. So $S_a$ are $n$-dimensional. Let $F:S_a \to S_b$ be the diffeomorphism $F(s) ...
0
votes
0answers
16 views

Criterion of removal of equations from overdetermined system

Consider the problem of solving overdetermined system Ax = b; In the problem I am trying to solve (from the field of spectral unmixing) number of unknowns usually varies between N = 2 and 5 and the ...
0
votes
1answer
54 views

Matrices word problem?

Julie and Bill are waiters at the Ogling Ogre Convention Center, which is well-known for serving the most deliciously disgusting meals to its guests. One of their tasks was to count the 2-eyed and ...
0
votes
2answers
74 views

Eigen values and Eigen vectors

Let A be a 4x4 matrix with real entries such that $ \ -1,1,2,-2 \ $ are its eigen values.If $B=A^4-5A^2+5I$ ,where $I$ denotes the 4x4 identity matrix ,then which of the following statements are ...
-1
votes
1answer
38 views

Matrices in the plane,polygon assignment, help. please?

a.) Given polygon P with vertices (1,5), (4,8), (8,5), (6,2) and (2,1), find the following: Find the area of P. Tip: Make sure to move COUNTERCLOCKWISE from point to point to ensure you get a ...
0
votes
1answer
46 views

Determinant of a square matrix with main diagonal of zeros?

How can I show that the determinant of a square matrix A of dimension NxN with all elements equal to $-\delta$ except the main diagonal composed by zeros, is equal to $-(N-1)\times \delta^N$?
2
votes
2answers
139 views

Derivative of a matrix: Outer product chain rule

I ran into a seemingly simple matrix calculus question that I can't seem to find the solution to. Suppose I have the following matrices: $X_{(t \times n)}, V_{(n \times m)}$, and $\Phi_{(t\times m)} ...
1
vote
2answers
279 views

Real world situation with System of Equation with 3 variables?

Where do you run into a real world situation involving 3 variables and 3 equations? Can someone think of a specific example from business, etc? I recall taking an operations research course that ...
0
votes
1answer
65 views

Check my answer - Differential of $P(A)=\det(A^{-1}-A)$

We are asked to find the differential of $P: GL_n(\mathbb R) \to \mathbb R$, $P(A)=\det(A^{-1}-A)$ and show it is differentiable. If we define $f(A)=\det(A)$ and $g(A)=A^{-1}-A$ then it is clear ...
2
votes
1answer
99 views

Differentiate vector norm by matrix

I've been trying to perform the following differentiation of a neural network: $$\frac{\delta||h(XW)\alpha-y||^2}{\delta W} = \frac{\delta}{\delta W}\sum_i(h(XW)_i\alpha-y_i)^2$$ Where $X$ and $W$ ...
1
vote
1answer
36 views

differential (Jacobi Matrix) of $f(A)=A^2$ where $A$ is a matrix - check my answer

I just want a quick verification that what I did here is correct: let $f(A)=A^2$ where $A$ is a n by n matrix with real entries. then $$D_f(A)=\lim_{t \to 0} \frac{f(A+tA)-f(A)}{t} = \lim_{t \to 0} ...
0
votes
1answer
36 views

How do I find the differential of these functions?

We are given 2 functions: 1) $f: \operatorname{Mat}_{n\times n}(\mathbb R) \to\operatorname{ Mat}_{n\times n} (\mathbb R)$, $f(A)=A^m$, $m>0$. and 2) $g: \operatorname{GL}_n (\mathbb R) \to ...
1
vote
2answers
67 views

What is the derivative of a vector with respect to its transpose?

I've already looked at Vector derivative w.r.t its transpose $\frac{d(Ax)}{d(x^T)}$, but I wasn't able to find the direct answer to my question in that question. What is the value of $$\frac{d}{dx} ...
2
votes
1answer
62 views

For a matrix $A$, is $\|A\| \leq {\lambda}^{1/2}$ true?

In class I saw a proof that went something along these lines: Define $\|A\| = \sup \dfrac{\|Av\|}{\|v\|}$ for v in V, where the norm used is the standard (Does this even exist?) Euclidean norm in V. ...
0
votes
1answer
31 views

Product rule type formula for $\nabla \cdot (M(x)v(x))$ where $M(x)$ is a matrix and $v(x)$ is a vector?

Let $M(x)$ be a $n\times n$ matrix with each element depending on $x$ a variable on $\mathbb{R}^n$. Let $v(x)$ be a vector. Is there a nice product rule formula for $\nabla \cdot (M(x)v(x))$?
2
votes
1answer
246 views

Product rule when differentiate matrix products

I want to differentiate the following expression with respect to $b$ $(Y-Xb)'(Y-Xb)$ Where $Y$ is $n\times1$ and $X$ is $n\times k$ and $b$ is $k\times1$, ' denotes transpose. If i do it term by ...
0
votes
1answer
92 views

Find a symmetric matrix for f(x)

Consider the function $f(x,y,z)=2y^{2}+2xy+2xz+2yz$, Find a symmetric matrix $A$ such that $f(x,y,z)$ can be written in the form $(Ax)x=(Ax)^{T}x$, where $x^T = [x y z]$.
0
votes
1answer
35 views

convexity of a Hessian matrix.

Suppose I have $f(x_{1},x_{2}) = x_{1}^2 + x_{2}^2, S = \mathbb{R}^2$. How do I determine whether the function is concave or convex based off of the Hessian of what is above? I know the Hessian is ...
0
votes
1answer
28 views

simple partial derivative of constant times matrix?

Is the partial derivative of $cX$ w.r.t the real matrix $X$, given by $c$ or by $cI$, where $I$ is the identity, and $c$ is a constant scalar? please give a simple reasoning.
2
votes
2answers
61 views

transforming a vector from cartesian to spherical and cylindrical co-ordinate system

I know the formula(which i don't know how to copy here but it was in matrix form) for transforming a vector from cartesian system to spherical or cylindrical coordinate system. But, I want to know its ...
1
vote
2answers
68 views

Matrix differential problem

How to derive the Jacobian and Hessian matrices of $f(X)=\textrm{tr}\left[(R+XX^H)^{-1}\right]$? where $X$ is a matrix. Thanks in advance.
0
votes
0answers
67 views

Differentiating an expression including matrix values for a scalar value

I have to differentiate the folowing expression which includes matrix values wrt to a scalr value. $\sigma^2(T_r \Delta_r(k)^{-1}T_r'+m^2(T_{p-r}\Delta(k)^{-1}T_{p-r}') + ...
1
vote
0answers
131 views

Area optimization: Packing rectangles inside rectangle

Background: I am scrambling to figure out the optimization algorithm to build sprite image, which is essentially a big container rectangular image, with multiple rectangular images. I have found an ...