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 ...

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Proving if a transformation is linear.

I have a terrible understanding of Linear Algebra so I'm trying every resource out there that I can. Hopefully my questions won't come off as idiotic. Anyway, we have a transformation from $R^2$ to ...
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1answer
25 views

Find minimal x and y that creates 4

Hello I had this in my exam, I've never studied this and I am interested in knowing how to solve it, and what is the category of this type of question: For all positive numbers that creates $x^2 ...
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1answer
6 views

“cover the unit sphere by c-fine grid” to prove the vector length preserved by random projection?

The below figure is extracted from the paper http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=4031351 . I did not understand the techniques used in the proof, namely, 1."cover the unit ...
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15 views

Lipschitz constant of L2 difference

What is the Lipschitz constant of $$f(A)=||Ax||_2-||Ay||_2?$$ In particular, is it $||x-y||_2$, i.e. is it true that given $A,B,x,y$, the following inequality holds: $$|f(A)-f(B)|\leq ||x-y||_2 ...
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10 views

Convergence proof strategy

I've been asked to prove that the following iterations converge to the eigenvalues of SPD $A_0 \in \mathbb{R}^{n \times n}$ $A_0 = \begin{bmatrix}a & b\\ b & c \end{bmatrix}$ with $a \geq ...
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2answers
27 views

Change of basis to find coordinates

This was my previous post: http://math.stackexchange.com/posts/1243265 I changed my question quite drastically and I didn't feel I was asking the question correctly. I attempted with change of ...
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9 views

Diagonalization of an endomorphism

Let $K$ be a field of characteristic $0$ and $K[X,Y]$ be the polynomial ring in two variables. Consider the endomorphism $\Delta \in End_K (K[X,Y]$, $\Delta : f \mapsto X \cdot \delta_X (f) + Y ...
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12 views

Determining Counts of Discrete Objects Using Linear Algebra

I'm teaching myself linear algebra and was able to solve the following question using trial and error, but--how would one setup and solve a question like this using Linear Algebra? I have 32 bills ...
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1answer
20 views

Representation of a matrix as product of unitary matrices and diagonal matrix

Let $C=A+B$ where $A$ is a symmetric positive definite matrix and $B$ is a positive semi-definite skew symmetric matrix. Clearly $C$ is neither symmetric nor skew symmetric.Then is it possible to ...
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1answer
22 views

how to identify the subspace of vectors? [on hold]

Which of the following subsets of $\Bbb R^3$ are subspaces of $\Bbb R^3$? A. The $3\times 3$ matrices with all zeros in the second row B. The $3\times 3$ matrices whose entries are all integers C. ...
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21 views

How to find eigen vector for an eigen value in generalized eigen value problem

I have a generalised eigen value problem of the form $A$x = λ$B$x. I have computed the eigen value (say λ1) I am interested in using Eigen library(C++). However, because the library does not support ...
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1answer
18 views

Finding a linear map.

I have some problem with a question related to linear maps. I know the solution but I can not understand the reason behind it. For any polynomial $p∈P^2$ let: ...
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17 views

What is my error in this matrix / least squares derivation?

I'm doing a simple problem in linear algebra. It is clear that I have done something wrong, but I honestly can't see what it is. let, $y = Ax$, $y_{ls} = Ax_{ls}$ where A is skinny, and $x_{ls} = ...
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0answers
25 views

Proving the hat functions are linearly independent

$$ H_i(x) = \begin{cases} (x-x_{i-1})/(x_i-x_{i-1}), & x_{i-1}\le x\le x_i, \\ (x_{i+1}-x)/(x_{i+1}-x_i), & x_i\le x\le x_{i+1}, \\ 0, & \text{otherwise}. \end{cases} $$ How can I prove ...
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1answer
63 views

$A^{-1}$ has integer entries if and only if the ${\rm det}\ (A) =\pm 1$

So, $A$ is a nxn matrix with integer entried. The question is to prove that $A^{-1}$ has all integer entries if and only if ${\rm det}\ (A) =\pm 1$ I know that $A^{-1}= {\rm adj}(A)/{\rm det}(A)$ ...
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1answer
19 views

Bounded operator on $L^{2}(a,b)$

Let $p\in]1,\infty[$ and consider the mapping $$ T : L^{2}(-2,2) \to L^{2}(-2,2), \quad (Tf)(x):=xf(x)$$ I want to show that $T$ is bounded, $||Tf||_L \leq T ||f||_L $. So, $$ ||Tf||_L \leq ...
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1answer
39 views

Is the Inner Product a uniformly continuous function?

I know it's continuous but is it uniformly continuous?
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0answers
31 views

How can I solve this system of linear equations?

$$\begin{align} x+y(k^2-6)+z(4k+4)&=5k+3\\ -x+y(2k^2-6)+z(4k+4)&=6k+3 \end{align}$$ I must use matrices to find for which values of $k$ this system has: exactly one solution, ...
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2answers
22 views

Point reflection uniqueness

Suppose we have normed vector space V and mapping R from V to itself satisfying the following properties: 1) R has unique fixed point $~a\in V$ 2) $\forall x\in V ~~~~ |Rx-a|=|x-a|$ 3) $\forall x ...
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1answer
50 views

Playing around with Geometric series. Where is my algebraic mistake?

Playing around with geometric series I got confused. First, consider the following equation $$\sum_{i=0}^n 1^i = n+1$$ Now, replacing $1$ by $\frac{a}{a}$ gives $$\sum_{i=0}^n ...
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18 views

How to differentiate between $(\lambda_{0}-\lambda)^{k} \,\text{and } g(\lambda) \,\text{in } f_{A}(\lambda)$?

By definition, $\lambda_{0}$ has algebraic multiplicity $k$ if $\lambda_{0}$ is a root of $f_{A}(\lambda)=(\lambda_{0}-\lambda)^{k}g(\lambda)$. What am I missing from this? ...
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57 views

Linear Algebra Proof conformation

I am trying to prove this theorem in my book. Because it is provided without proof, please let me know what you think! $\mathbf{Theorem:}$ Let V be a finite , $n$ -dimensional vector space and let U ...
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Is the profinite completion of $SL_2(\widehat{\mathbb{Z}})$ isomorphic to $\widehat{SL_2(\mathbb{Z})}$?

Is the profinite completion of $SL_2(\widehat{\mathbb{Z}})$ isomorphic to $\widehat{SL_2(\mathbb{Z})}$ ? Does anything change if we replace $SL$ with $GL$?
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36 views

When is the product of two arbitray matrices symmetric?

Let $\mathbf{A}$ be a real $n \times m$ matrix. Let $\mathbf{B}$ be a real $m \times n$ matrix. How to solve the following matrix equation? $$\mathbf{A}\mathbf{B}=\mathbf{B^{t}}\mathbf{A^{t}}$$ ...
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3answers
49 views

Basic way to show for $n\times n$ matrices $A$ and $B$, that $(AB)^{-1} = (B^{-1})(A^{-1})$

In looking at matrix inverses, I know the following works (I is the identity matrix): If $AB$ are nxn matrices and are invertible, then $(AB)C = I$, and therefore $C = (AB)^{-1}$. I can show that ...
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1answer
29 views

Tutte matrix - Determinant

I'm trying to understand the proof of the "magic theorem" about the Tutte matrix which states: Let $T$ be the Tutte matrix of $G(V, E)$. Then, $$\det(T) = 0 \quad\Longleftrightarrow\quad G ...
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0answers
12 views

Computing covariance matrix in PCA

I am implementing PCA in matlab and I have to compute the covariance matrix. I am using 'cov' command from matlab to compute the covariance matrix. But it is very slow and takes a lot of time to ...
2
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0answers
30 views

Inequality with eigenvalues

Let matrix $ X $ is Hermitian and denote $ \lambda_1(X) \ge \lambda_2(X) \ge \ldots \ge \lambda_n(X) $ eigenvalues of matrix $ X $. Prove that $ \lambda_i(A + B) \le \lambda_i(A) + \lambda_1(B) $ I ...
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3answers
51 views

Find the eigenvalues of $A$. $A^2 = 1$ and $A\ne\pm1$

$A \in \mathbb{R}^{n\times n}$, with $A^2 = 1$ and $A\ne\pm1$ Show that the only eigenvalues of $A$ are $1$ and $-1$.
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1answer
25 views

Linear Algebra-invariant subspaces

Suppose $V$ is a real vector space and $T\in \mathcal L (V)$ has no (real) eigenvalues. Prove that every subspace of $V$ invariant under $T$ has even dimension.
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Finding all the invariant subspaces of a certain linear transformation.

Assuming I have given affine transformation $ \mathbb{R}^3\to \mathbb{R}^3 $ which has matrix representation $$ \left[\begin{array}{cccc} 3&2&-3&-10\\ 4&10&-12&-29\\ ...
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Differentiating a matrix

Let $$f(x) = \left[\begin{array}{ccccc} 6 &-5 &-2 &1 &7\cr -7 &0 &-2 &2 &-3\cr -3 &0 &0 &-9 &-8\cr x &6 &-3 &-3 &1\cr -3 &0 ...
3
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1answer
32 views

Let $trcA=0$.why $A=M+N$ where $M$ and $N$ are nilpotent matrices?

Let $A \in {M_n}$ and $trcA=0$.why $A=M+N$ where $M$ and $N$ are nilpotent matrices?
3
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1answer
27 views

Finding a kernel and an image of $T^2$

Let $T$ be a linear transformation $T: \mathbb{R}^4 \to \mathbb{R}^4$ that is defined by: $$T\begin{pmatrix}x\\y\\z\\u\end{pmatrix}=\begin{pmatrix}0\\z\\y\\x\end{pmatrix}$$ Find the kernel and image ...
3
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2answers
32 views

If $A$ can be written as a sum of nilpotent matrices why $trcA=0$?

Let $A \in {M_n}$. If $A$ can be written as a sum of two nilpotent matrices, why $trcA=0$?
3
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1answer
190 views

linear algebra questions about matrices

Given a a matrix $X$ where $$ X= \begin{bmatrix} 1 & 0 \\ 0 & p(x) \end{bmatrix} $$ where $p(x)$ is some polynomial of degree $3$ or 4 and different from $0$. I'm trying to ...
3
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2answers
82 views

Let $A,B \in {M_2}$ and $C=AB-BA$. Why is ${C^2} = \lambda I$ true?

Let $A,B \in {M_2}$ and $C=AB-BA$. Why does ${C^2} = \lambda I$?
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1answer
22 views

Examples of Parallel planes and others.

I need to draw this but I would like to add an extra mark there by putting some examples of these. They must be in a 3D cartesian plane. Any help? Three parallel planes Two parallel planes and a ...
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Estimate the upper bound of the spectral norm a block matrix

I actually want to estimate the upper bound of the following matrix: $\Phi(k,t) = \prod_{s=2}^{k-t+1} \left[\begin{array}{cc}a(k-s)\tilde{W}+(b(k-s)+2a(k-s))I_N & -b(k-s)\tilde{W}-b(k-s)I_N \\I_N ...
0
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1answer
13 views

Determinant with one parameter, how to deal with this?

Let $t\in \mathbb R$ be a parameter, and $$|A(t)|= \begin{vmatrix} a_{11}+t &a_{12}+t &\cdots &a_{1n}+t\\ a_{21}+t &a_{22}+t &\cdots &a_{2n}+t\\ \vdots &\vdots ...
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0answers
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Optimal Matching Distance

I'm stuck on problem II.5.9 from Bhatia's Matrix Analysis. The problem is as follows: Let $\{\lambda_1,\dots,\lambda_n\},\{\mu_1,\dots,\mu_n\}$ by two $n$-tuples of complex numbers. Let $$ ...
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1answer
15 views

Is $Q=V^TWV$ positive definite?

I have the real symmetric $k \times k$ matrix $Q = V^T W V$, where I know $V$ is a $n \times k$ orthogonal matrix (its columns are orthogonal) and $W$ is a $n \times n$ diagonal matrix with all its ...
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2answers
32 views

Prove boundedness of the matrix series

Suppose $A$ is a square matrix, such that all eigenvalues of $A$ has norm strictly less than $1$, can I say $\sum_{i=k_0}^kA^{k-i}$ is bounded for all large enough $k_0$ and $k$? From some other ...
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1answer
13 views

Principal Component analysis by eigenvalue decomposition.

I do know how to perform PCA by using SVD but I am unaware about how to use eigenvalue decomposition of X(transpose)*X matrix. I found a paper online which explains the approach to perform PCA by ...
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0answers
16 views

How to figure out whether PCA can be performed on a data set or not?

I do have idea on the way PCA works but I do not know how to figure out whether a high dimensional data set is suited for PCA compression. I googled for some algorithms but could find any. Are there ...
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2answers
17 views

Finding x'y' coordinates from xy coordinates with unit basis vectors

I wasn't really sure where to get started with this question as I don't fully understand what it's asking.. I can see that u1 is made up of i + j (or u2) and that x' is scaled for some scalar k from ...
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2answers
37 views

Prove the determinant is the product of its diagonal entries

Prove that the determinant of an upper triangular matrix is the product of its diagonal entries. What I have so far: We will prove this by induction for an $n$ $\times$ $n$ matrix. For the case of a ...
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28 views

Show that there is a vector $w$ in ${\rm ker}\ (T)$ such that $v=u+w$

Suppose $U$ and $V$ are vector spaces such that $T:U\rightarrow V$ is a linear map. Suppose also that $u$ and $v$ are vectors in $V$ such that $f(u)=f(v)$. Show that there is a vector $w \in \rm ...
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2answers
28 views

Linear algebra problem about projections

Let A,B be real matrices of order $n \geq 6$. Let $A + \alpha B$ be projection operator for any $\alpha \in \mathbb{R}$. True or false: if A is orthogonal projection, then $A \neq B$. ...
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0answers
24 views

Derivative of a trace function

Let $K$ be a Hermitian matrix, and $X$ be a positive one. What is the derivative of the trace function $$ \mbox{ Tr } X|e^{itK} - X|^3$$ with respect to $t$ at $t = 0$ ? There is a nice formula for ...