Tagged Questions
2
votes
1answer
53 views
To show the inequality $\|A\|\geq\max\{\|u_1\|,\ldots,\|u_q\|,\|\vec{v_1}\|,\ldots,\|\vec{v_q}\|\}$
Let $A\in$ $\mathbb{C}^{p\times q}$ with column $u_1,\ldots,u_q$ and rows $\vec{v_1},\ldots,\vec{v_p}$. show that
$$\|A\|\geq\max\{\|u_1\|,\ldots,\|u_q\|,\|\vec{v_1}\|,\ldots,\|\vec{v_q}\|\}$$ and ...
0
votes
1answer
108 views
How would one use matrices to find a normal unit vector?
A recent class assignment involved finding a unit vector perpendicular to a plane, given two unit vectors to start with. The solution given involved using the cross product; I was wondering if such a ...
2
votes
1answer
91 views
Why does this equation converge to 1?
The following simple equation takes in an N-length (real) vector, and spits out a (real) number between 0 and 1. (I believe this means that it is a transformation mapping $\mathfrak{R}^N \rightarrow ...
2
votes
1answer
74 views
Is this vector derivative correct?
I want to comprehend the derivative of the cost function in linear regression involving Ridge regularization, the equation is:
$$L^{\text{Ridge}}(\beta) = \sum_{i=1}^n (y_i - \phi(x_i)^T\beta)^2 + ...
1
vote
1answer
106 views
Gradient vector function using sum and scalar
Could someone take a look on my attempt to compute the gradient for:
$$f(x) = \lambda \sum_{x = 1}^n g(x_i)$$
Where $x \in \mathbb{R^d}$, $\lambda \in \mathbb{R}$ and
$$g(x_i) = \begin{cases}
x_i - ...
1
vote
1answer
95 views
Vector derivative with inner function
I want to compute the gradient for the following function:
$$L(\beta) = \sum_{i=1}^n (y_i - \phi(x_i)^T \cdot \beta)^2 + \sum_{j = 1}^k l(\beta_j)$$
where $l(\beta_j) = \begin{cases}
\beta_j - ...
1
vote
0answers
135 views
Minimizing L1 Regularization
I have given a high dimensional input $x \in \mathbb{R}^m$ where $m$ is a big number. Linear regression can be applied, but in generel it is expected, that a lot of these dimensions are actually ...
1
vote
1answer
40 views
Vector derivative with power of two in it
I want to compute the gradient of the following function with respect to $\beta$
$$L(\beta) = \sum_{i=1}^n (y_i - \phi(x_i)^T \cdot \beta)^2$$
Where $\beta$, $y_i$ and $x_i$ are vectors. The ...
1
vote
2answers
184 views
Log-likelihood gradient and Hessian
Considering a binary classification problem with data $D = \{(x_i,y_i)\}_{i=1}^n$, $x_i \in \mathbb{R}^d$ and $y_i \in \{0,1\}$. Given the following definitions:
$f(x) = x^T \beta$
$p(x) = ...
4
votes
2answers
2k views
What is the proof that covariance matrices are always semi-definite?
Suppose that we have two different discreet signal vectors of $N^{th}$ dimension, namely $\textbf{x}[i]$ and $\textbf{y}[i]$, each one having a total of $M$ set of samples/vectors.
$\textbf{x}[m] = ...
1
vote
1answer
52 views
What does $Az=y$ and $A^Tx=0$ imply about the relationship between $x$ and $y$?
Suppose A is a $m \times n$ matrix and the vectors $x$ and $y$ are such that $Az=y$ for some vector $z$ and $A^T x=0$. Which one is correct?
$x^Ty=0$
$||x||_2=||y||_2$
$||x||_2 < ||y||_2$
$x=ay$ ...
0
votes
2answers
57 views
All the matrices that are orthogonal and have $q_1,q_2$
"Determine all the orthogonal matrices $Q=[q_1,q_2,q_3]$ that have as the first two columns the vectors $q_1=\frac{1}{\sqrt{6}}(-1,2,-1)^T, \ q_2=\frac{1}{\sqrt{3}}(1,1,1)^T$".
I used the ...
2
votes
2answers
168 views
Differentiating a function with respect to a vector
I need to differentiate the function $u$ shown below with respect to a vector $\psi$: ($a, c$ and $f$ are constants)
$u(\psi) =\left[\begin{array}{cccc}
a & f & 0 & 0\\
c & a & ...
0
votes
0answers
62 views
clever solution to decomposition of linear products?
There may be a better name for this class of problem, and if so feel free to edit!
Imagine a matrix consisting of the following columns: daily return, $\alpha_t$, $factor^1_t$, $factor^2_t$, ... and ...
1
vote
1answer
90 views
Why does this algorithm work?
Given a matrix, $P$, why does finding its eigenvalues, say they are $\{\lambda_1, \lambda_2\}$ then the general form of $p_{ij}^{(n)}=A_{ij}\lambda_1^n+B_{ij}\lambda_2^n$? Thanks.
Added: Context: $P$ ...
2
votes
2answers
151 views
equation in matrix form
i have an equation
$$\sum_{i=0}^m [w^{(i)}[(c-a^{(i)})-\frac{d^{(i)}(c-a^{(i)})\cdot d^{(i)}}{dis^2}]]=0$$
Where:
$a,b$ are two end points in a 3D Line. $d$ is a vector and vector $d=b-a$. the ...
1
vote
1answer
176 views
Wikipedia Article — Legendre Transform
I was reading the wiki article on Legendre Transform. I would be grateful if someone could explain the section at http://en.wikipedia.org/wiki/Legendre_transformation#Examples
ie how they arrived at ...
