Linear algebra is concerned with vector spaces of all dimensions and linear transformations between them, including systems of linear equations, bases, dimensions, subspaces, matrices, determinants, traces, eigenvalues and eigenvectors, diagonalization, Jordan forms, etc. For questions specifically ...

learn more… | top users | synonyms (1)

2
votes
2answers
73 views

Gradient and Hessian of function on matrix domain

Let $A \in R^{k \times p}$. Define $f(X) : R^{p \times k} \rightarrow R$ to be $f(X) = \log \det(XA + I_{p})$, where $I_{p}$ is a $p \times p$ identity matrix. I want to know what is the gradient and ...
0
votes
0answers
10 views

Completing a semi-vector space to a vector space

If we have a semi-vector space $U$ (as defined here), what do we have to additionally demand from $U$ such that we can complete it to a vector space $\tilde{U}$ via $U\xrightarrow{\iota}\tilde{U}$ and ...
2
votes
1answer
52 views

How do I prove this statement in linear algebra?

I had a test about a week ago and I want to know the answer to this question: Given an orthonormal basis $B$ spanned by $\{ v_1, v_2 , v_3 \}$ of ${\mathbb R}^3$ Prove that for every $v \in {\mathbb ...
0
votes
1answer
36 views

Trying to visualize and understand double dual space

Currently I am reading "Finite-dimensional vector spaces" by Paul Halmos. I would have a question regarding the theorem on page 25. It says: If $V$ is a finite-dimensional vector space, then ...
1
vote
2answers
46 views

Find non-diagonal matrices $A$ and $B$ such that $B^TAB$ is diagonal

Here $B^T$ denotes the transpose of $B$. $A$ and $B$ are invertible $3\times 3$ matrices with integer entries. $A$ is symmetric positive definite with at most two zero entries. We want the ...
1
vote
1answer
31 views

Solving a recurrence relation of conditional probability functions

Suppose you have the recurrence relation for a probability function Q: $$Q(n_1,n_2|n) = Q(n_1-1,n_2|n-1)\frac{n_1-1}{n-1} + Q(n_1,n_2-1|n-1)\frac{n_2-1}{n-1}$$ where $n = n_1 + n_2$ and the ...
1
vote
0answers
14 views

Application of Structure Theorem to Prove Simultaneous Diagonalizability and Group of Units of Cyclic Groups

I am reading these notes on Modules over PID. Exercise 67 (pg 24) asks to prove that: Problem. Let $A$ and $B$ be $n\times n$ matrices with complex entries. Then $A$ and $B$ are simultaneously ...
1
vote
2answers
74 views

An inequality on the rank of a block matrix

Let $\mathbb F$ be a field, and let $r_1, r_2, s_1, s_2$ be positive integers. Consider the matrix $$X:=\left[\begin{array}{cc} A & B \\ C & D \end{array} \right],$$ where $A \in \mathbb F^...
0
votes
2answers
450 views

Maximum and minimum of determinant of matrices with entries from $\{0,1\}$ or $\{-1,0,1\}$

Maximal and Minimal value of $\bf{3^{rd}}$ order determinant whose elements are from the set $\bf{\{0,1\}}$. Maximal and Minimal value of $\bf{3^{rd}}$ order determinant whose elements are from the ...
3
votes
1answer
318 views

$\dim C(AB)=\dim C(B)-\dim(\operatorname{Null}(A)\cap C(B))$

Let $A \in M_{n \times m}\left(F\right)$ and $B\in M_{m \times p}\left(F\right)$ for a field $F$. Prove: $\dim C(AB)=\dim C(B)-\dim(\operatorname{Null}(A)\cap C(B))$, where $C(X)$ denotes the column ...
2
votes
0answers
39 views

Eigenvalue perturbation of singular matrix

Consider a Hermitian matrix $\mathbf{A_0} \in \mathbb{C}^{N \times N}$ with one singularity, i.e. its eigenvalues in increasing order are: \begin{equation} 0 < \lambda_2 \leq \lambda_3 \leq \cdots \...
3
votes
1answer
67 views

Why is this matrix symmetric?

There is an example in the Convex Optimization lecture notes, Boyd. He just said in the lecture that the matrix which is underlined in red color is symmetric! How can we claim that when there is no ...
2
votes
0answers
21 views

Generalise Expression to slice Circulant Matrix

Suppose I have $4 \times 4 $ circulant matrix , $$A=A(0:3,0:3)=A(:,:)=\begin{bmatrix} 1 & 2 & 3 & 4 \\ 4 & 1 & 2 & 3 \\ 3 & 4 & 1 & 2 \\ 2 & 3 & 4 & 1 \...
0
votes
1answer
22 views

Is the condition number of unitary matrix always equal to 1?

I know that the 2-norm condition number $\kappa (\textbf U)={||\textbf U||_2}{||\textbf U^{-1}||_2}$ of a unitary matrix $\textbf U$ is always equal to 1. Is this true for all induced matrix norms, i....
0
votes
0answers
22 views

Entanglement of 3-qubit states

Given a separable 3-qubit state φ = φ0 ⊗ φ1 ⊗ φ2 with φi= ai0|0> + ai1|1>, |0>, |1> being the computational base. φ thus can be written as φ = b000|000> + b001|001> + b010|010> + b011|011> + ...
0
votes
0answers
15 views

Calculating the coefficients of a separable 2-qubit state

Given a separable 2-qubit state φ = φ0 ⊗ φ1 with φi= ai0|0> + ai1|1> φ thus can be written as φ = b00|00> + b01|01> + b10|10> + b11|11> with bij = a0ia1j. Now let some bij be given, i.e....
0
votes
3answers
134 views

What is the number of subspaces of a particular dimension?

If we have vector space $V$ with dimension $n$ then how many subspaces of $V$ with dimension $m<n$ are there? In my opinion the answer should be the number of ways to choose $m$ linearly ...
3
votes
1answer
129 views

Inverse of a matrix and its transpose

I'm trying to figure out why the calculation below works. I do know that $(A^T)^{-1} = (A^{-1})^T$. The matrix A = $\begin{pmatrix} 1 & -1 & 0 \\ 1 & 1 & -1\\ 1 & 2 & -1 ...
0
votes
1answer
47 views

Is there a linear transformation that sends v to (1,0)

Suppose we are given a linear vector field $v(x) = Ax$ on $\Bbb R^2$. Suppose $x_0$ is the point where this vector field does not vanish. Is it possible to find a linear transformation from $\Bbb R^2$ ...
0
votes
0answers
22 views

sending basissen

Lets say we have this $3\times3$ matrix: $$ \begin{bmatrix} 4 &−4 &12\\ 1& -1& 3\\ −1& 1 &−1 \end{bmatrix} $$ What is the algorithm to find a basis of $\Bbb R^3$ for which ...
-1
votes
0answers
12 views

Linear transformation and projection [on hold]

1 Suppose that W is a subspace of a finite-dimensional vector space V. (a) Prove that there exists a subspace W' and a function T:V→V such that T is a projection W along W'. (b) Give an example of a ...
0
votes
2answers
31 views

Linear equation in n variables with non negative solution

The problem is that given a positive integer y and n positive integers x1 , x2 , ... , xn does there exist non negative integers ...
2
votes
1answer
104 views

Why must a function be independent of coordinates?

What is the motivation for why a function should be independent of coordinates? In the case of a general manifold I kind of get why, since one (usually) defines a function $f$ as a map from the ...
2
votes
1answer
144 views

Laplace Challenge in One Examples, Is there any help?

this question is taken from 2014 exam on CE Entrance Exam, Question $32$ on the end of page $6$. Consider the Laplace equation of following polar coordination, $$\frac{1}{r}\frac{\partial}{\partial ...
4
votes
1answer
392 views

what is degree of minimal polynomial?

Let $V$ and $ W$ be finite dimensional vector space over $\mathbb R $ and let $T_1 : V \rightarrow V$ and $T_2 : W \rightarrow W$ be linear transformation whose minimal polynomial are $f_1 (x)= x^3+x^...
4
votes
2answers
97 views

What are the rules for taking derivatives in linear algebra?

I was reading through a paper on beamforming and came across an equation whose derivative I don't fully understand. A cost function is given as: $$ J(\mathbf{w}) = \mathbf{w}^HR\mathbf{w} +\lambda^*[...
1
vote
2answers
59 views

Compute $R^{2016}$ of a given counterclockwise rotation.

Write out the matrix $R$ of counterclockwise rotation by 30$^{\circ}$ in $\mathbb{R}^2$. Compute ${R}^{2016}$. Now this is an easy question to answer overall; 30 goes into 360 12 times and one twelfth ...
0
votes
0answers
28 views

problem solving in arithmetic

I've been given the following problem: The formula to find $Y$ is $Y=x_1+x_2+x_3+x_4-x_5$ The value of $Y$ is given as $100$. Now the question is: Is it possible to find without ambiguity $x_1$ ...
2
votes
1answer
30 views

Representation of an invertible square matrix as a product of elementary matrices.

Suppose $A$ is an invertible $2 \times 2$ matrix. What is the smallest integer $n$ such that $A$ is a product of $n$ elementary matrices? My guess is that at most 4 elementary matrices are ...
1
vote
0answers
48 views

Gradient descent: L2 norm regularization

So I've worked out Stochastic Gradient Descent to be the following formula approximately for Logistic Regression to be: $ w_{t+1} = w_t - \eta((\sigma({w_t}^Tx_i) - y_t)x_t) $ $p(\mathbf{y} = 1 | \...
2
votes
1answer
79 views

Does $\mathrm A \mathrm A^T \succeq x^2 \mathrm I$ imply $\frac{\mathrm A + \mathrm A^T}{2} \succeq x \mathrm I$?

Let $A $ be an $n \times n $ matrix such that $AA^T \geq x^2I, x\geq 0 $, which means that the matrix $AA^T-x^2I$ is positive semidefinite. Can we show that $(A+A^T)/2 \geq xI$? Thanks
3
votes
3answers
541 views

Space spanned by matrices

I have a set of $5 \times 5$ matrices, $M_1, M_2,\dots, M_{19}, M_{20}$. I want to try to find a basis from this set and also to find relationships between these matrices. This is how I think I ...
0
votes
1answer
49 views

$A^2$ is bounded $\implies$ $A$ is bounded?

Let $A_n$ be a sequence of $k \times k$ real matrices. Assume $A_n^2$ is bounded w.r.t some norm. Is $A_n$ also bounded? I was able to show this is true if $A_n$ are symmetric matrices (using SVD). ...
2
votes
0answers
28 views

If $AB = BA$ for $A,B \in \mathcal{L}(V,V)$, then $A$ and $B$ have these properties [duplicate]

There is a base such $A$ and $B$ are both upper triangular on these base, and if $A$ and $B$ are diagonalizable, then $A$ and $B$ are diagonalizable simultaneously. For the first I have no idea. To ...
1
vote
1answer
36 views

Algebraic number spaces

While studying about Vector spaces and subspaces I came across the following question:- $Q.$ Do $algebraic$ numbers form a subspace of the vector space $\Bbb R$? According to my knowledge of $...
4
votes
2answers
245 views

Union of a subspace and its orthogonal complement

The following statement seems to be true, but I am not sure: For any subspace $A$ of $\mathbb{R}^n$, there is a nonzero vector $\vec{x}$ such that $\vec{x}\in A\cup A^\bot$ and each entry of $x$ is ...
0
votes
1answer
40 views

Coin toss related problem

What is the minimum number of times a fair coin needs to be tossed so that the probability of getting at least two heads is at least 0.96? Is there any shortcut way to calculate this?
-5
votes
0answers
43 views

How to solve a function when given a graph? [on hold]

I'm not looking for the answer, but how to solve this myself. If there is a video I could be linked that would be very helpful. Thank you.
1
vote
0answers
18 views

Derivative of quadratic form involving singularity

This might be a silly question, but i have been really curious about the following: Consider the following function seen thru a single variable, say $\alpha$: \begin{equation} f(\alpha) = \mathbf{x}^...
5
votes
1answer
12k 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.
1
vote
1answer
61 views

Linear independence of standard basis vectors from Vandermonde style vectors

Is it true a statement that all $n$ dimensional vectors of the standard basis (e.g. $[1 \ 0 \ 0 \ ...]^T$, $[0 \ 1 \ 0 \ ...]^T$ etc ..) are linearly independent from the set of the $n-1$ vectors $...
0
votes
0answers
30 views

converting to row echelon form

Without swapping rows, transform the following augmented matrix into Row-Echelon form. $\left[\begin{array}{ccc|c}-1& 3& 2& -9\\ -2& 3& -2& -39\\ 1& -6& -5& 3\end{...
0
votes
0answers
27 views

Tensor product and dual of vector spaces

Consider $\mathbb{F}$ the algebraic closure of a finite field with characteristic $p>0$, and let $\mathbb{F}_q$ the unique subfield of $\mathbb{F}$ with $q=p^\alpha$ elements. So if we have $J$ ...
0
votes
1answer
25 views

Does a system of linear equation “equal ” to a matrix equation or it is just a trick to solve these equation?

Does a system of linear equation "equal" to a matrix form Ax=b or it is not equal(as if it just a "trick" to solve system of linear equation)?
1
vote
0answers
36 views

$V$ be a vector space , $T:V \to V$ be a linear operator , then is $(\ker(T) \cap R(T) ) \times R(T^2) \cong R(T)$? [duplicate]

Let $V$ be a vector space , $T:V \to V$ be a linear operator , then is it true that $(\ker(T) \cap R(T) ) \times R(T^2) \cong R(T)$ ? (note that the direct product is well-defined as both the spaces ...
2
votes
1answer
2k views

How to remove linearly dependent rows/cols

How would one remove linearly dependent rows/columns from a rank-deficient matrix. For example, (from wikipedia): $$ A = \begin{bmatrix} 2 & 4 & 1 & 3 \\ -1 & -2 & 1 &...
1
vote
1answer
41 views

Doubly infinite matrices $A=(a_{i,j})_{i,j=\infty}^{\infty}$

Let $A=(a_{i,j})_{i,j=\infty}^{\infty}$, where $$ \|A\|:=\sum_{r=-\infty}^{\infty}\sup_{j}|a_{j,j+r}|<\infty. $$ I want to show that for all matrices $\|AB\|\leq\|A\|\|B\|$. I obverse that $$ (AB)...
4
votes
2answers
359 views

Angle between two planes in four dimensions

Suppose I have two planes defined in 4D space, either in terms of vectors spanning the planes, $X = t_1 A_1 + t_2 B_2$ and $X = t_3 A_3 + t_4 B_4$ (where $X$, $A$'s, and $B$'s are vectors with four ...
0
votes
0answers
27 views

What would be a basis of $L^2(\Omega )$

Let $(\Omega ,\mathcal F,P)$ a probability space and $$L^2(\Omega )=\{random\ variable\ X\mid \mathbb E[X^2]<\infty \}$$ is a vector space. What would be a basis of $L^2(\Omega )$ ? I also know ...
0
votes
0answers
17 views

Associativity when solving matrix equation obtained from SVD

In this question, I asked how to solve a matrix equation $Ax=b$ using the Singular Value Decomposition (SVD) of $A$. The conclusion was rather trivial, given that $A$ is specified as square matrix. $$ ...