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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1answer
27 views

Distance between points

Suppose I have two matrices each containing coordinates of $m$ and $n$ points in 2 D. Is there an easy way using linear algebra to calculate the euclidean distance between all points (i.e., the ...
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0answers
24 views

A question in matrix norm

Let $I,A \in {M_n}$ and suppose $\left| {\left\| . \right\|} \right|$ be a matrix norm $\left| {\left\| I \right\|} \right| \ge 1$ and $\left| {\left\| A \right\|} \right| < 1$($I$ is identity ...
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3answers
70 views

$\frac{1}{{1 + {\left\| A \right\|} }} \le {\left\| {{{(I - A)}^{ - 1}}} \right\|}$

Let a matrix norm $ {\left\| . \right\|}$ have the property that $ {\left\| I \right\|} = 1$ and $ {\left\| A \right\|} < 1$. Why does the following inequality hold? $$\frac{1}{{1 + \left\| A ...
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0answers
15 views

How to make use of symmetric of sparse matrix to solve this kind of problem?

I have the following matrix to be solved: $$\left\{ \matrix{ {a_{11}}{x_1} + {a_{12}}{x_2} + \cdots + {a_{1n}}{x_n} = {y_1} \hfill \cr {a_{21}}{x_1} + {a_{22}}{x_2} + \cdots + {a_{2n}}{x_n} = ...
3
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1answer
43 views

Determinant proof using its properties

Prove without expanding: \begin{equation} \begin{vmatrix}bc&a^2&a^2\\b^2&ac&b^2\\c^2&c^2 & ab\end{vmatrix} = ...
4
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2answers
41 views

Matrix ring $M_2(\mathbb{C})$, $\mathbb{C}^2$ with $M_2(\mathbb{C})$-module structure.

Let $R$ be the $2 \times 2$ matrix ring $M_2(\mathbb{C})$. let $M = \mathbb{C}^2$ with its natural $R$-module structure (just given by the usual action of $2 \times 2$ matrices on $2$-dimensional ...
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1answer
23 views

Invertibility for a matrix that I don't know

I would like to know why $(e^{-At}-I)^{-1}$ is invertible when matrix A is Hurwitz.
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1answer
16 views

Dependence of product of matrix and a vector, on the rank of a Matrix

What is the significance of the rank of a matrix, say $A$, when I am multiplying a vector, say $x$, by $A$? In other words, let $x$ be a column vector of suitable dimension and let $rank (A)=m$. What ...
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0answers
41 views

How to find one matrix, which is subject to $B^3 = A$. How much is such matrices? [duplicate]

Here I have a problem with row echelon form. $$A := \begin{bmatrix}-6 & 3 & 7 \\ 0 & -1 & 0 \\ -14 & 12 & 15\end{bmatrix}$$
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1answer
51 views

A determinant coming out from the computation of a volume form

I am convinced that the following identity is true: \begin{equation} \det\begin{bmatrix} 1+a_1^2 & a_1 a_2 & a_1 a_3 & \ldots & a_1a_n \\ a_1a_2 & 1+a_2^2 & a_2a_3 & ...
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3answers
50 views

Can systems of 3 linear equations with 3 unknowns have more than one solution?

In each part,determine whether the given vector is a solution of the linear system \begin{align} 2x-4y-z&=1\\ x-3y+z&=1\\ 3x-5y-3z&=1 \end{align} (a) $(3,1,1)$ (b) $(3,-1,1)$ (c) ...
3
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3answers
51 views

A vectorspace over an infinite field is not a finite union of proper subspaces?

Show that if V is a vector space over an infinite field F, then V cannot be written as set-theoretic union of a finite number of proper subspaces.
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6answers
45 views

How to prove there's a vector $z \in \mathbb{R}^4$ orthogonal to two linearly independent vectors $x,y \in \mathbb{R}^4$?

Let $x, y \in \mathbb{R}^4$ with $\{x, y\}$ being linearly independent. Prove that there exists a non-zero vector $z$ that is orthogonal to both $x$ and $y$. Any hints on what to do after the ...
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3answers
42 views

How to solve these simultaneous equations using numerical methods?

How to solve these simultaneous equations for $\alpha$ and $\lambda$ using numerical methods? $\lambda * [(\frac{3}{4})^\frac{-1}{\alpha} - 1] = 11$ $\lambda * [(\frac{1}{4})^\frac{-1}{\alpha} - 1] ...
3
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3answers
18 views

Determinant of symplectic matrix

A $2n \times 2n$ matrix $S$ is symplectic, if $SJ_{2n}S^T=J_{2n}$ where \begin{equation} J_{2n} = \begin{bmatrix} 0 & I_n \\ -I_n & 0 \end{bmatrix}. \end{equation} My question is, how to ...
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0answers
8 views

How to modify Tikhonov regularization?

Consider a linear map $f: X \rightarrow Y$ and let $F$ is a matrix of $f$ and $b$ is one element of $Y$. Our goal is to obtain the element of $X$ corresponding to $b$. In ideal case we can get the ...
0
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1answer
25 views

Show that the matrix is a symmetric matrix

Let $T:V\to V$ be a symmetric linear map i.e $\langle Tx,y\rangle =\langle x,Ty\rangle $ .$V$ is a finite dimensional inner product space If $\{e_i:1\leq i\leq n\}$ is an orthonormal basis of $V$ ...
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1answer
555 views

Inverse of Symmetric Matrix Plus Diagonal Matrix if Square Matrix's Inverse Is Known

Let $A$ be an $n \times n$ symmetric matrix of rank $n$ with known inverse $A^{-1}$. Let $D$ be a diagonal matrix with the same dimensions and rank. What is the fastest way to compute $(A+D)^{-1}$? ...
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3answers
38 views

Let {v1, v2} be a basis for a subspace S of R 3 . If B = {w1, w2, w3} is a set of vectors in S, then B cannot be linearly independent.

Let $\{v_1, v_2\}$ be a basis for a subspace $S$ of $\Bbb R^3$ . If $\mathcal B = \{w_1, w_2, w_3\}$ is a set of vectors in $S$, then $\mathcal B$ cannot be linearly independent. I'm not sure how ...
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2answers
31 views

If Q is an orthogonal matrix, does it follow that $QDQ^T = Q^TDQ$?

Say A is a real, $n \times n$ symmetric matrix. Then it is orthogonally diagonalisable, with $A = QDQ^T = QDQ^{-1}$. Let's say we do not know that Q is symmetric (at first) - does the above hold?
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Is this always true about multiplication of matrices [on hold]

Let $\mathbf{A},\mathbf{B}$ be square matrices of size $n\times n$. Assume that $\mathbf{A}$ is symmetric and $\mathbf{B}$ is non-singular. Will $\mathbf{Y} = \mathbf{BAB}^T$ be always a symmetric (or ...
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1answer
306 views

MATLAB determining elementary matrices for LU decomposition

I am confused by this question I am studying for MATLAB practice.
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1answer
35 views

Is $\det(U_1\Lambda_1 U_1^t +U_2\Lambda_2 U_2^t +I)\le \det(\Lambda_1 +\Lambda_2 +I)$ correct?

I want to simplify or find an upper bound for the determinant $|K_1+K_2+I|$ where $I$ is identity matrix, $K_1$ and $K_2$ are positive semi-definite matrices of size $n$ and thus can be written as ...
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0answers
23 views

Is $\oplus_{i \in I} M_i$ necessarily projective?

If $\{M_i\}$ is a collection of projective modules over a ring $R$, then must the direct sum $\oplus_{i \in I} M_i$ necessarily be projective? I understand that each direct sum of two modules is ...
1
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1answer
13 views

Invertibility of Product implies invertibility of factors

Say $C=AB$ where $A,B,C$ are all $n\times n$ matrices. It's easy to show that if $A$ and $B$ are invertible then $C$ is invertible --> $C^{-1}=B^{-1}A^{-1}$. Does the converse hold? That is, if $C$ ...
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1answer
19 views

reordering the indices of a matrix

Let $A$ be an $n \times n$ matrix of rank r. Then by reordering the indices if necessary we can bring the matrix in the form $(\frac{A_1}{A_2})$ where $A_1$ is an $r \times n$ matrix, $A_2$ is an $n-r ...
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1answer
39 views

$\left\| {\left| {BA - I} \right|} \right\| < 1$ $ \Rightarrow $ $A$ and $B$ are both nonsingular

Let $A,B \in {M_n}$ satisfy the inequality $\left\| {\left| {BA - I} \right|} \right\| < 1$ and $\left\| {\left| . \right|} \right\|$ be a matrix norm on ${M_n}$.Why do $A$ and $B$ are both ...
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0answers
36 views

Coppersmith-Winograd algorithm

I'm interested in algorithms to compute matrix multiplications. Is the Coppersmith-Winograd algorithm similar to the Strassen algorithm ? I have two other questions: 1) Are the multiplications done ...
6
votes
5answers
162 views

Find an expression for $A^n = \left( \begin{array}{cc} 1 & 4 \\ 2 & 3 \end{array} \right)^n$

We want to find an expression for $A^n = \left( \begin{array}{cc} 1 & 4 \\ 2 & 3 \end{array} \right)^n$ for an arbitrary "n". I have tried writing out a few elements of the sequence as $n \to ...
2
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1answer
73 views

Question on linear algebra - Determinant multiplication.

Does anybody have a "non brute" force way to prove the following for non-singular matrices A, B: det(AB) = det(A) det(B)
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3answers
26 views

Use Vectors To Show Three Vertices Belong to a Right Triangle

The Full Question Theorems Used This is what I call theorem 1: My Work This problem has two major steps as far as I can see. First, I must show that these are points of a triangle(not ...
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1answer
105 views

Find a subspace of $\mathbb{R}^4$ for which $x^T*A*x$ = 0

Given a matrix $A$ find a two dimensional subspace $V \subset\mathbb{R}^4$ for which $\forall x \in V : x^TAx=0$ $$A = \begin{pmatrix}1&2&0&1\\ 2&3&1&1\\ 0&1&0&1\\ ...
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2answers
89 views

What are the “building blocks” of a vector?

Lets say I have a set of vectors $V$ that includes this vector: $$\begin{bmatrix}1\\2\\-1\end{bmatrix}$$ I interpret it as $x = 1, y = 2, z = -1$ (that being three dimensions for this vector). I know ...
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2answers
42 views

Show that $\mathcal{B}$ is a Basis for $V$

If $V= \{p(x) \in \mathbb{R}_3[x] : p(-1)=p(1)=0\}$, show that $\mathcal{B} = \{ 1 - x^2, x - x^3\}$ is a basis for $V$. Note: $\mathbb{R}_3[x]$ denotes polynomials with real coefficients of degree ...
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votes
1answer
26 views

Express a second-order cone (SOC) inequality as a linear matrix inequality (LMI)

For $y \in \mathbf{R}^n$ and $t \in \mathbf{R}$, show that: $$||y||_2 \leq t ~~\iff~~ F(y) \succeq 0$$ Where $\text{I}$ is the $n \times n$ identity matrix, and $$F(y) = \begin{pmatrix} t ...
1
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1answer
30 views

Find the projection of a vector onto a subspace of $\Bbb R^4$

I need to find the projection of $\vec b = (1,1,1,1)$ onto a subspace of $\Bbb R^4$ described as: $$V=\{(x,y,z,t)\,:\,x=y+t\ \hbox{and}\ 2x=y+z\}\ .$$ Thanks for any help i get guys.
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0answers
10 views

Maximizing Autoencoder Hidden Unit Function

Given \begin{align} a = f\left(\sum_{j=1}^{100} W_j x_j \right). \end{align} where $f$ is the sigmoid function, $W$ and $x$ are $100 \times 1$ matrices with the constrain \begin{align} ||x||^2 = ...
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2answers
39 views

QR decomposition proof

Let $A\in\mathbb{M}_{m\times n}(\mathbb{R})$ with $m>n$ and $rank(A)=n$ and take the decomposition $A=QR$ with $Q\in\mathbb{M}_{m\times n}(\mathbb{R})$ a orthogonal matrix and ...
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1answer
35 views

Is my algorithm correct? (Polar decomposition)

I cant seem to find my mistake. Consider this matrix $T = $\begin{bmatrix} 2 & 1 & 1 \\[0.3em] -1 & 2 & 0 \\[0.3em] 0 & 1 & -1 \end{bmatrix} I need ...
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0answers
32 views
+50

Understanding stability of fixed points in 2D maps.

I'm trying to understand the stability analysis for a map of the form $$(x_{n+1}, y_{n+1}) = A(x_n,y_n)$$ Where A is a 2x2 matrix - assumed to be diagonalisable and with distinct eigenvalues. I ...
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0answers
43 views

Can a matrix be similar to more than one matrix?

I have a little query about similar matrices I've been struggling with. Suppose I have a 5x5 diagonal matrix A with 5 distinct eigenvalues as entries in the main diagonal. The question is, to how ...
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0answers
20 views

Linear Algebra: Guidance on a Eigenvalue/Eigenbasis problem, please?

Here's the problem, but I only need some help with part C: http://i.imgur.com/UwRBGIO.png This is the information and answers from the back of the book: http://i.imgur.com/BFs2z2s.png I understand ...
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2answers
30 views

Finding orthogonal projections onto $1$ (co)-dimensional subspaces of $\mathbb R^n$

1)Consider the vector space $\mathbb{R}^n$ with usual inner product. And let S the subspace generated by $u\in \mathbb{R}^n,u\neq 0$. Find the orthogonal projection matrix $P$ onto the subspace ...
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1answer
29 views

Area Between Intersecting Lines - Elegant Solution?

I am running simulations, and the output will be a line y = mx+b. I am interested in the area below the line between x=0 and x=1. I am only interested in the area that is below the diagonal y = x. I ...
2
votes
3answers
25 views

Gradient of a line

The line L is a reflection of the line $2y + 3x =9$ in the $y-$ axis (I had to draw the graph on the grid previously) Find gradient of the line L How would I go about solving this?
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0answers
27 views

Reflection matrix and algebraic multiplicity

Let $Q\in\mathbb{M}_4(\mathbb{R})$ a reflection matrix onto $R(A)$ subspace, where $A\in\mathbb{M}_{4\times 3}(\mathbb{R})$ is defined by ...
1
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1answer
33 views

Minimum matching convolution

Let $\text{SPD}^n$ and $\text{PD}^n$ be the semi-positive and positive definite matrices in $\mathbb{R}^{n\times n}$, respectively. I want to find an $X\in \textrm{SPD}^n$ that minimizes $||X||$ ...
2
votes
1answer
21 views

Understanding the Replacement Theorem (Exchange Theorem)

I'm learning about Basis and Spans and now that's I've figured out what these are, I'm trying to understand the Replacement Theorem(also called the Exchange Theorem). The definition goes like this: ...
2
votes
4answers
30 views

Finding a matrix representation of the transpose transformation

Define $T : M_{n×n}(\mathbb{R}) → M_{n×n}(\mathbb{R})$ by $T(A) := A^t$. I know this transformation is linear and just takes a matrix and spits out it's transpose. I also know that the transpose is ...
1
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2answers
21 views

Show that the set is a basis for $S$.

Consider the subspace $S$ in $\Bbb R^3$, $S=\{(a,b,c)\mid a+b=c\}$. Show that the set $B= \{(1,0,1),(1,2,3)\}$ is a basis for $S$. I've started to set up a matrix, ...