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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885 views

Power of a matrix, given its jordan form

Can someone please explain how to find the power of a matrix $A$, given $A=MJM^{-1}$ where the matrix $J$ is in the Jordan canonical form? Or else please explain how to find the powers of a matrix ...
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2answers
148 views

Proving that $A$ is diagonalizable

Let $A\in\mathbb{C}^{n\times n}$ be a $n$ by $n$ matrix such that $A^k = I$ for some natural number $k$. Find a nonzero annihilating polynomial of A and prove that A is diagonalizable. I will say ...
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1answer
38 views

Showing unitary similarity of these two matrices

Let $A \in B(H)$ for a Hilbert space $H$, and $\alpha \in \sigma_{p}(A)$, the point spectrum of $A$. Suppose ker$(\alpha I-A)$ is not a reducing subspace of $A$ then $A$ has the following matrix ...
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1answer
229 views

Finding eigenvalues and “eigenmatrices”.

On the space of $2\times 2$ matrices, let $T$ be the transformation that transposes every matrix. Find the eigenvalues and "eigenmatrices" for $A^T =\lambda A$. By taking determinants on the left and ...
3
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1answer
410 views

Intuition behind the definition of linear transformation

I have studied that given vector spaces $V_1$ and $V_2$, a function $T:V_1 \rightarrow V_2$ is called a linear transformation of $V_1$ into $V_2$, if following two properties are true for all $u, v ...
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1answer
58 views

Norm of functional associated to vector $p$-norm [duplicate]

I read that the norm of a linear functional $f:V\to K$, with $K=\mathbb{R}\lor K=\mathbb{C}$, associated to the $p$-norm $\|x\|=(\sum_{i=1}^n|x_i|^p)^{\frac{1}{p}}$, for $p>1$, is ...
2
votes
1answer
137 views

Geometric Product

I have a problem with the geometric product: In my book the unit trivector is defined like this: $(e_{1}e_{2})e_{3}=e_{1}e_{2}e_{3}$ But that would mean $(e_{1}e_{2})e_{3}= (e_{1} \wedge e_{2})\cdot ...
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1answer
39 views

Some simple matrix identities

I've recently been learning some linear algebra and I've isolated what seem to be some important matrix relations (often used tacitly). I would be most grateful if someone could just check that I have ...
2
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1answer
47 views

Question about a definition

There was a definition on my notebook. But sadly I cant read (...) part. What do we call $w_1,w_2,w_3...w_k$? Let V be a vector space on field F and $w_1,w_2, w_3..$ are subspaces of V. for any ...
5
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1answer
111 views

Can I turn $Ax=b$ into $Ax=0$?

For a system of equations $$ \begin{bmatrix}d_1 & d_2 & \dots & d_n \end{bmatrix} \begin{bmatrix}u_1\\u_2\\ \vdots \\ u_n \end{bmatrix} = d_{n+1} $$ where each $d$ is a column of ...
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2answers
82 views

Spanning Set Of $V$ After Linear Transformation Will Span $U$

let there be a Linear Transformation $T:V \rightarrow U$ and $B={v_1,...,v_n}$ a spanning set of $V$, so $C=T(v_1),...,T(v_n)$ will span $U$. Is it right because: 1. there is only one linear ...
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1answer
465 views

Minimizing an error function by deriving a system of linear equations

Consider the following formula: $$E(\mathbf{w}) = \frac{1}{2}\sum_{n=1}^{N}\{y(x_n,\mathbf{w})-t_n\}^2$$ where $\mathbf{w}$ is a vector of weights; $x_n$ and $t_n$ come from two vectors of length ...
0
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1answer
79 views

Eigenvalues of rank one matrix

For a given vector $v=(v_1,v_2,\dots, v_n)$ consider the matrix $B=v^tAv$ where $A=(a_{ij})_{i,j=1,\dots,n}$ with $a_{ij}=1$ i.e. $$ B=\begin{pmatrix} v_1v_1&v_1v_2& \dots & v_1v_n\\ ...
2
votes
1answer
46 views

Column/Row Space check

I have the following matrix: \begin{bmatrix} 1 & 2 & 0 & 1 & 0\\ 3 & 6& 1 & 6 & 1\\ 2 & 4 & -1 & -1 & -1\\ 4 & 8 & 0 & 4 ...
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0answers
221 views

Preparation for a Linear Algebra Class

I have just entered my Junior Year as a CS student. While I have already taken discrete math and Theory of Computation, and have not found myself needing any additional math skills thus far; I ...
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2answers
39 views

How does dot product work in matrix algebra?

I am working on a weighted minimization problem. Without the weights, the error function can be expressed as $e^T e$. With weights, $e$ first need to element-wise multiple by $w$, then the same ...
4
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1answer
174 views

Is the dot product valid for coordinate vectors of infinite length?

On Wikipedia, it says the dot product is valid for "any number of dimensions." Let's call it $n$. $$u\cdot v = |u||v|\cos(\theta)$$ Is this still true if we let $n$ go to infinity? EDIT: By ...
0
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1answer
103 views

How to remove vector x from (x'Ax)/(x'Bx)?

Are there any simple expression for the following scalar? $$a=(x'Ax)(x'Bx)^{-1}$$ where $x'=$transpose of $x$, $A,B\in\mathcal M_{n\times n}(\mathbb R)$ and $x\in\mathcal M_{n\times 1}(\mathbb R)$. ...
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3answers
129 views

Suppose $A$ is a 4x4 matrix such that $\det(A)=\frac{1}{64}$

Suppose A is a 4x4 matrix such that $\det(A)=\frac{1}{64}$ then $\det(4A^{-1})^T$ I created a 2x2 matrix $B$ and transposed it both had the same determinant I then found $\det(B)$ and ...
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0answers
222 views

Proving that number of codes with even weight is the same as number of codes with odd weight for a specific code book

Consider the $[n,n]$ code-book $C_0=\{0,1\}^n$ with $n$ being odd and the codes $c_i \in C_0=[c_1,c_2,...,c_{2^n}]$ being sorted in the ascending order of hamming weight (from $0$ to $n$). Now let's ...
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0answers
61 views

Density of Pythagorean triples

We define a Pythagorean triple as a triple $<a,b,c>$ such that $a,b,c\in \mathbb N$ and $a^2+b^2=c^2$. In order to avoid duplicates, we say that a triple $<a,b,c>$ is legit iff $b>a$. ...
3
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1answer
56 views

Sums of special vectors

Let $v$ be a vector obtained by taking a sum of $k$ vectors the of the form $(0,0,\ldots,0, -n, *,*,\ldots,*)$, where $"*"$ stands for either $0$ or $1$, and the position of the $-n$ entry can vary ...
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1answer
93 views

Find solutions to magic puzzle with sums

I need help to solve the folowing puzzle using linear algebra (matrix and Gauss-Jordan Method): (for example the second horinzontal line: w + w + w + z = 45 or the ...
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2answers
442 views

Linear algebra - Memorising proper definitions of homomorphism types

I am reading a book about linear algebra. On the basis of this book, I worked out the terminology below. Problem: To me, it looks like Wikipedia defines homomorphism differently. Apart from that: Do ...
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1answer
140 views

How to avoid complex value for square root of a symmetric matrix?

I want to find square root of a matrix $Z$ which is a symmetric matrix using eigen values. So I find the eigenvalues($A$) and eigenvectors($B$) of $Z$ and find $B A^{1/2} B$. But because of small ...
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2answers
345 views

Winning strategies in multidimensional tic-tac-toe

This question is a result of having too much free time years ago during military service. One of the many pastimes was playing tic-tac-toe in varying grid sizes and dimensions, and it lead me to a ...
2
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0answers
856 views

Dual norm of the matrix $L^1$ norm is infinity norm (and vice versa)

Recall that for a given norm $\|\cdot\|$ on $\mathbb{R}^n$, the dual norm is defined as a function $\|\cdot\|_*: \mathbb{R}^n \rightarrow \mathbb{R}$ with: $\|y\|_* = \max \limits_x \{x^Ty: \|x\|\le1 ...
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56 views

Build up a not diagonalizable linear map

I need an hint for this problem. Let be $M = \begin{bmatrix}2 & 1 \\ -2 & 0\end{bmatrix} \in M_2(\mathbb{K})$ and $H=\{A \in M_2(\mathbb{K}) : AM=MA \} $ Build up a linear map $f: ...
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1answer
41 views

Group Theory $Z_2$ representations

I am trying to understand some group theory. In the notes I am following, I am told: Recall the representations of $\mathcal{Z}_2$: Trivial: $\rho_0(e) = 1$, $\rho_0(a)$ = 1 (i) $\rho_1(e) = 1$, ...
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2answers
442 views

Calculate the unknown coordinates of a point $B (x_2,y_2)$ on a line with given distance from a known point $A(x_1,y_1)$

I have a line which represents a cross section. I have the coordinates of on its starting point. I need the coordinates of the end point of that cross section line. The distance between these two ...
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2answers
25 views

What does "$S\times S\rightarrow R$

In the textbook, Mathematical Methods and Algorighms for Signal Processing, Tood K. Moon, the $\mathbf{inner\;product}$ is defined it is a function $\langle\cdot,\cdot\rangle:S\times S\rightarrow R$ ...
2
votes
3answers
87 views

Vector spaces - Non-uniqueness of element with property of scalar-multiplicative identity element?

I am dabbling in vector spaces, thinking about the axioms on Wikipedia. Notably, $$1 \mathbf{v} = \mathbf{v},$$ i.e. identity element of scalar multiplication (IEOSM), attracted my attention. I am ...
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2answers
106 views

finding the dimension of a matrix, the sum of whose rows is zero

Let $V$ be a vector space of $n\times n$ matrices over $R$ and Let $W$ be subspaceof matrices with entries in each row adding upto zero.then the dimension is? n $\frac{n(n-1)}{2}$ $n(n-1)$ $n-1$ ...
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2answers
96 views

Can someone help me to prove this theorem from Axler's *Linear Algebra Done Right*?

If $p\in P(\Bbb{R})$ is a nonconstant polynomial, then $p$ has a unique factorization (except for the order of the factors) of the form ...
2
votes
1answer
281 views

When is the solution to a n initial value problem matrix differential equation invertible?

Suppose $A (t,s)$ a $n\times n$ matrix is the solution of the initial value problem below, where $B_s$ is also an $n\times n$ matrix, invertible for all $s$: $$\dfrac{d A(t,s)}{ds} = B_s A(t,s)$$ $$ ...
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0answers
66 views

prove that $\sum_{k=1}^\infty|x_k y_k|$ converges

Let $V$ be the space of real sequences $x_k$ so that $\sum_{k=1}^\infty x_k^2$ converges. Let $\langle x,y\rangle=\sum_{k=1}^\infty x_k y_k$ Prove that $\sum_{k=1}^\infty |x_k y_k|$ converges My ...
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1answer
494 views

Construct and apply a rotation matrix by doing the following

Create a 2x2 rotation matrix $A \ne I$. Determine, showing all work, the location of point $(3, 2)$ when it is rotated using the linear transformation generated by the matrix. Also, demonstrate, ...
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2answers
39 views

If $B^T$ consists of a basis of $\mathrm{im} (A)^\perp$, then $\mathrm{im}(A)=\ker (B)$?

Well basically, the question is in the title: Suppose we have $A\in\mathbb{R}^{n\times m}$ with rank $d$ and we fix a basis $(b_1,\ldots,b_{n-d})$ of $\mathrm{im}(A)^\perp$. Let ...
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2answers
167 views

Subspace of matrices AB = BA

I'm stuck with the following exercise: Let A be an $n\times n$ diagonal matrix with characteristic polynomial: $$\prod_{i=1}^{k}(x-c_{i})^{d_{i}}$$ where $c_{1},...,c_{k}$ are distinct. Let $W$ ...
3
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1answer
116 views

How to mathematically determine if the magnitude of a cross product is up/down(positive/negative?)?

So, I'm a newbie at complex vector math. I'm working on a 2D physics engine, and my issue is, with angular acceleration from torque, is it supposed to be positive or negative? I understand the right ...
9
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2answers
166 views

Solutions of $XA=XAX$.

All matrices are real and $n \times n$. The matrix $A$ is given. I am interested in solving $XA=XAX$. In particular, I would like some characterization of matrices that satisfy this equation. For ...
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2answers
45 views

Linear Transformation On Basis

What a Linear Transformation does on a basis? if the Linear Transformation is 1-1 and onto so every element of the basis goes to element of the basis of the other vector space? and what if it is not ...
2
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1answer
100 views

Proving result on spectral radius

How do I prove that $$\rho(A)=\inf\limits_{\text{operator norms}}\|A\|,$$ $\rho$ being the spectral radius, $A$ being a complex $n\times n$ matrix and operator norms being induced from vector norms by ...
2
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2answers
314 views

How to find eigenvalues of the following block circulant matrix

I have a block matrix of size PN x PN of the form: Where A and C are P x P matrices. I would like to find the eigenvalues of the matrix B, that is where
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1answer
47 views

angle between two vectors-given in matrix form

Let $u=\left\{\begin{pmatrix} 1&a&0\\0&1&0\\0&0&1\end{pmatrix} \begin{pmatrix}1\\1\\0\end{pmatrix}:a\epsilon R\right\}$ $v=\left\{\begin{pmatrix} ...
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1answer
47 views

Canonical form 2nd order PDE

I want to reduce the following equation to canonical form $yu_{xx} + 2(x+y)u_{xy} + 4xu_{yy} = 0$ for $x > y > 0$ I chose ɛ to be $x^2 - \frac{y^2}{2}$ and η to be $2x - y$ Then I found the ...
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0answers
150 views

solving a system of 2 equations with 4 variables

Can someone show me how to solve this system of equations? \begin{align} 4X+12Y-7Z-20W&=22 \\ 3X+9Y-5Z-28W&=30 \end{align} Does this mean that there are two free variables here?
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3answers
63 views

linear algebra-permutation [duplicate]

Given the permutation $$\sigma = \begin{pmatrix} 1&2&3&4&5\\3&1&2&5&4\end{pmatrix}$$ the matrix A is defined to be the one whose i-th column is the $\sigma(i)$-th ...
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3answers
65 views

Find the Eigenvector of a matrix

Find the eigenvectors of the matrix $$\displaystyle\begin{bmatrix} 0 &2 &3 \\ -2 &0 &5 \\ -3 &-5 &0 \end{bmatrix}.$$ So I start with $|A-\lambda I|=0$ ...
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1answer
41 views

Maximising a sum with respect to the unit ball.

Suppose that I have a vector $\boldsymbol{v} \in \mathbb{R}^d$, for some dimension $d>1$, and suppose I want to consider the sum $$ \begin{align*} \left(\sum_{k=1}^{d}v_k\right)^2. \end{align*} $$ ...