0
votes
1answer
30 views

Finding the area of a triangle from vertices? Linear Algebra

I pretty much did this problem, but I failed to get the few last blanks where they ask the area. Its confusing, they say its half the volume of matrix (u v w) in the start of the question. which means ...
1
vote
1answer
35 views

How to factor and reduce a huge determinant to simpler form? Linear Algebra

So, I have learned about cofactor expansion. But the cofactor expansion I know doesn't reduce the number of rows and colums to one matrix. I usually pick a colum, multiply each element in the column ...
0
votes
3answers
25 views

How do you know that rows are independent and what are the 120 terms?

I am having trouble with the question below, help me out;
0
votes
2answers
55 views

Suppose $(x,y,z)$, $(1,1,0)$, and $(1,2,1)$ lie on a plane through the origin.

What determinant is zero? What equation does this give for the plane? I need some help here, am pretty stuck
2
votes
1answer
24 views

Acquiring $Df(\mathbf{x})$

Sorry for the probably easy and silly question, but I try to teach myself linear algebra and I am stucked at "the derivative as a matrix" part. I know how to differentiate partially and I know how ...
1
vote
1answer
50 views

When is a given matrix-valued function the Jacobian of something?

Let $F$ be an $n\times m$ matrix of real-valued functions which are defined and smooth on a neighborhood of a point $p\in \mathbb{R}^m$. Under what conditions is it possible to find a smooth function ...
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
37 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 $T: S \to S$, ...
2
votes
1answer
42 views

Find the differential of $f(A)=det(A^{-1}-A)$ where $A$ is invertible.

The question is if $A$ is an invertible matrix with real entries of size $n$. Is $f(A)=det(A^{-1}-A)$ differentiable? and what is the differential. I think I managed to show it's differentiable. the ...
2
votes
1answer
82 views

Gradient of matrix exponential function

Grateful if somebody could help me with the following. I am trying to find the gradient of the next expression: $$f(a_1, a_2, a_3, a_4)=\Vert R*y-x \Vert $$ where $y$ and $x$ are known 4x1 column ...
1
vote
0answers
62 views

How to compute time ordered Exponential?

So say you have a matrix dependent on a variable t: $$A(t)$$ how do you compute $$e^{A(t)}$$ It seems Sylvester's formula, my standard method of computing matrix exponentials can't be applied ...
2
votes
1answer
51 views

Proving a set of $2\times 3$ matrices is a manifold?

The way I have always been told to check if something is a manifold (I haven't had a whole lot of experience with them), is to check if the derivative of the function representing the loci of the ...
0
votes
1answer
32 views

Derivative of a function of a matrix $f(B) = x^T(AB)^ky$

I have a function of the form $f(B) = x^T(AB)^ky$ where $x$ and $y$ are column matrices, $A$ and $B$ are square matrices, and $B$ is a diagonal matrix, and $k$ is an integer constant. I want to find ...
2
votes
2answers
92 views

Question about Implicit function theorem

I was asked a simple question, show that $y+\sin y=x$ sets in the neighborhood of $(0,0)$ $y$ as a function of $x$, and find $\dfrac{dy}{dx}(0,0)$ Firstly, my naive solution would be: Since $lim_{y ...
1
vote
1answer
187 views

Determining the values of k for which the Matrix A has an inverse

I've been given this question in class, with the 3x3 matrix: 2 1 0 1 2 1 0 -3 k My job here is to find the values of k for which this matrix has an ...
0
votes
0answers
34 views

How to compute the Jacobian Matrix of the next system?

I have a little problem with notation and I do not know how to work it out. I have the next system $$ \dot x = A(x)x, $$ where $x \in \mathbb{R}^n$ and the square matrix $A(x)_{n\times n}$ has ...
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 ...
0
votes
2answers
74 views

Minimizing the following objective function with matrices

Suppose $A$ and $B$ are known matrices, and we are to find matrix $X$ that minimizes the following function, $$\frac{1}{2}||X||^2+\frac{1}{2}||X^TAX-B||^2$$ Taking the relevant derivative w.r.t $X$ ...
2
votes
1answer
94 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} ...
1
vote
2answers
64 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
37 views

Local maximality implies global maximality?

Let $S$ be the unit sphere in $\mathbb R^n.$ For a given $A\in\operatorname{M}_n(\mathbb R),$ define $f:\mathbb R^n\mapsto\mathbb R$ as $f(x)=\langle x,Ax\rangle.$ Suppose $a\in S$ is an element ...
5
votes
1answer
132 views

Extended matrix function

I have a continuous matrix-valued function $f:\mathbb{R}^d\mapsto {\cal M}_{k\times d}$, with $d<k$, such that $f(x)$ is full rank for all $x\in\mathbb{R}^k$. Can I extend this function to be a ...
0
votes
1answer
30 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 $\nabla \cdot (M(x)v(x))$?
0
votes
3answers
96 views

Why is this 'obviously' positive semi-definite?

Here is a snapshot from a book I am studying. I learned all about positive semi-definiteness, and in fact I know that this matrix they are showing is in fact PSD. What I do not know is how they ...
2
votes
1answer
1k views

Expressing the determinant of a sum of two matrices?

Can $$\det(A + B)$$ be expressed in terms of $$\det(A), \det(B), n$$ where $A,B$ are $n\times n$ matrices? # I made the edit to allow $n$ to be factored in.
2
votes
1answer
48 views

Second Order Derivative of a function $f:R^2\to R^2$

The Exercise: My Work: Part 1: $$ Df=\left( \begin{array}{ccc} D_1f_1 & D_2f_1\\ D_1f_2 & D_2f_2 \\\end{array} \right) $$ $$f_1(x,y)=\sin x+\sin y$$ $$f_2(x,y)=\cos x+\cos y$$ $$ ...
2
votes
0answers
44 views

Continuity of the orthogonal matrix-valued function

Given $d<k$. Suppose that $H:\mathbb{R}^k\rightarrow {\cal M}(\mathbb{R})_{d,k}$ is a continuous matrices-valued function such that $H(x)$ is full rank for every $x \in \mathbb{R}^k$. Now define a ...
3
votes
1answer
52 views

Vector by Matrix derivitive

According to wikipedia, there is no widely accepted definition of a Vector by Matrix derivative. I have a need of such a notion. For matrix w, and vector h. $$\mathbf{y=w \;h} $$ $$ ...
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
58 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 ...
0
votes
1answer
82 views

Finding the determinant of an $n\times n$ matrix… and the inverse

Finding the determinant of a $2\times 2$ matrix is easy and the inverse is even easier. Finding the determinant of a $3\times 3$ matrix and its inverse is a little more difficult but still doable. ...
1
vote
0answers
67 views

The inverse matrix |${\delta}$| does it have an application

The jacobian is the determinant of |${\delta}$| this means that |${\delta}$| is invertible. Does this inverse have any use in the real world? Maybe I am not clear in my question: Does the inverse of ...
1
vote
1answer
62 views

How to derive curl in spherical coordinates

This is one of those questions where I know I am making a dumb mistake someplace and I am trying to check where it is. $$ \begin{vmatrix} \frac{\bf\hat r}{r^2\sin\theta} & ...
5
votes
2answers
137 views

Derivative of the trace of matrix product $(X^TX)^p$

Let $X$ be a squared matrix, We know that $\frac {\partial tr(X^TX)}{\partial X}$ is $2X$ But how about the case of $\frac {\partial tr((X^TX)^2)}{\partial X}$ or even $\frac {\partial ...
0
votes
1answer
103 views

Divergence calculation for jacobians

Suppose that u is suitably regular (e.g. $C^2(\mathbb{R}^N,\mathbb{R}^N)$ or $W^{1,2}(\mathbb{R}^N)^N$) and we write $$\det (\nabla u)=\nabla u^1 \cdot\sigma$$ for some $\sigma$ (obtained via the ...
0
votes
1answer
237 views

gradient of vector 2-norm

I have a function $f(\Theta) = \frac{1}{2N}\| y-\mathcal{X}(\Theta)\|_2^2$. Matrix $\Theta\in\mathbb{R}^{m_1\times m_2}$, $y=[y_1,\cdots,y_N]^T\in\mathbb{R}^N$ is the observation vector, and we use ...
0
votes
1answer
52 views

Conditions for linear independence of extended vector systems

Assume $$g: R^n \times R^m \rightarrow R^n$$ $$h: R^n \times R^m \rightarrow R$$ $$(x,y) \in R^n \times R^m$$ I would like to show that the following vectors are linearly independent: ...
1
vote
4answers
64 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 ...
0
votes
0answers
52 views

Calculating the Jacobian for sampled data

I am reading about the Jacobian matrix which I interpret as a generalized gradient. I would like to take my investigations a bit further, so I have constructed some sample data which I would like to ...
1
vote
1answer
140 views

Is there a general form for the derivative of a matrix to a power?

Let $S:Mat(2,2) \rightarrow Mat(2,2)$ be the squaring map $S(A)=A^2$ then $[DS(A)]B=AB+BA$. I was wondering if there was a general form for this solution ($S(A)=A^n$, then $[DS(A)]B =$...). I have ...
1
vote
1answer
665 views

Deducing that a matrix is indefinite using only its leading principal minors

$A$ is indefinite iff $A$ fits none of the above criteria. Equivalently, $A$ has both positive and negative eigenvalues. Also equivalently, $x^TAx$ is positive for at least one $x$ and ...
0
votes
2answers
51 views

Non-elementwise Matrix Derivatives

Let A,B,C,D,X be matrices. I'd like to perform a Gradient Descent minimization to the loss functin $$ tr[(AXBX^TC-D)^T(AXBX^TC-D)] $$ My question is, how to take the gradient efficiently w.r.t. $B$? ...
0
votes
1answer
131 views

Which is the correct Hessian matrix (the standard matrix of a bilinear form)?

Please note the typo in the first entry: $\frac {\partial^2f} {\partial x_1\partial x_2}$ should instead be $\frac{\partial^2f} {\partial x_1\partial x_1}$. Also, this Hessian matrix need not be ...
0
votes
1answer
52 views

Change of variable formula, hermitian matrices

Let \begin{align} (d\mathbf{H})= \bigwedge_{1\leq j\leq k\leq N} d h_{jj}^{(1)} \bigwedge_{1\leq j< k\leq N}d h_{jk}^{(2)} \ ... \bigwedge_{1\leq j< k\leq N}\ d h_{jk}^{(\beta)} \end{align} ...
0
votes
1answer
112 views

Differentiation of a unitary matrix

Let $\mathbf{U}$ be a unitary matrix ($\mathbf{UU}^\dagger=\mathbf{1}$). What does this implies for $d( \mathbf{ U U }^\dagger)$? Is it mathematically sound to say: \begin{equation} d\mathbf{U} ...
1
vote
1answer
56 views

Derivative of a function of vector parameter. Problem with notation.

I have an error function $$err=\frac{1}{N}\left[\textbf{y}^T\ln{\textbf{x}}+(\textbf{1}-\textbf{y})^T\ln{(\textbf{1}-\textbf{x})}\right]$$ I need to find the gradient $\bigtriangledown_x{}err$, such ...
3
votes
1answer
53 views

Projection and direct sum

I want to show that for every projection $A^2=A$ we have that there exists a subspace $U_1 \subset ker(A)$ and $U_2$ such that $A|_{U_2} = id$ such that $V = U_1 \oplus U_2$. Does anybody here have a ...
7
votes
4answers
180 views

General question about matrix calculus with specific example (with attempted answer)

I'm struggling to find the right way to approach matrix calculus problems generally. As an example of a problem that is bothering me, I would like to calculate the derivative of $||Ax||$ (Euclidean ...
0
votes
1answer
48 views

How to Differentiate this Matrix product

I am trying to solve the matrix equations for linear regression and it leads me to the following differentiation. I cannot find an explanation on how to do it on the Internet, only the result being ...