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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4
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1answer
46 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 ...
0
votes
0answers
26 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
vote
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 $C$ ...
0
votes
1answer
20 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 ...
0
votes
1answer
41 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 ...
1
vote
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
167 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
votes
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)
1
vote
3answers
31 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 ...
2
votes
1answer
122 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\\ ...
1
vote
2answers
90 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 ...
0
votes
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 ...
0
votes
1answer
33 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
vote
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.
0
votes
0answers
13 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 = ...
0
votes
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 ...
2
votes
1answer
36 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 ...
1
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0answers
41 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 ...
-3
votes
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 ...
1
vote
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 ...
1
vote
1answer
34 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
30 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?
0
votes
0answers
29 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
vote
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
25 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
33 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
vote
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, ...
2
votes
1answer
26 views

Show that the subset $S$ in $\mathbb{R}_3$ is a subspace.

Show that the subset $S$ in $\mathbb{R}_3$ defined by $S=\{(a,b,c) \in \mathbb{R}_3 \text{ such that } a+b=c \}$ is a subspace. I'm having trouble adapting the definition of subspace with the part ...
3
votes
1answer
39 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{bmatrix} 1 & -1 & 0 \\ 1 & 1 & -1\\ 1 & 2 & -1 ...
1
vote
1answer
26 views

demonstrate that $v_3 \perp (v_1-v_2)$

$\bar{v}_1 \perp (\bar{v}_2-\bar{v}_3)$ and $\bar{v}_2 \perp (\bar{v}_3-\bar{v}_1)$ therefore $\bar{v}_3 \perp (\bar{v}_1-\bar{v}_2)$ By applying the dot product to $\bar{v}_1$ and ...
4
votes
3answers
67 views

Why does $\frac{1}{{\left\| {\left| {{A^{ - 1}}} \right|} \right\|}} \le \left\| {\left| B \right|} \right\|$?

Let $A,B \in {M_n}$ suppose that the following statements are true: $A$ is nonsingular, $A+B$ is singular, $\left\| {\left| . \right|} \right\|$ is matrix norm. Why is it true that: ...
1
vote
1answer
14 views

Is it possible to find a vector that is orthogonal to this set?

I have a set of four vectors in $\mathbb{R}^4$: $\{ \vec v_1, \vec v_2, \vec v_3, \vec v_4 \}$ The first three are linearly independent, but $ \vec v_4 $ is a linear combination of the others. Is it ...
2
votes
2answers
26 views

Algebric and geometric multiplicity and the way it affects the matrix

Given a matrix $A$. Suppose $A$ has $\lambda_1,\dots,\lambda_n$ eigenvalues each with $g_i$ geometric multiplicity and $r_1,\dots,r_n$ algebric multiplicity, $g_i\leq r_i$. Given this information ...
1
vote
3answers
45 views

How are these expressions equivalent?

I saw that $${y^2\over y^2+1} = 1 - \frac1{y^2+1}$$ but I can't see how, wolfram alpha agrees but I'm still not seeing it.
7
votes
1answer
78 views

$A$ is diagonalizable if $A^8+A^2=I$

Given a matrix $A\in M_{n}(\mathbb{C})$ such that $A^8+A^2=I$, prove that $A$ is diagonalizable. So let $p(x)=x^8+x^2-1$ and we know that $p(A)=0$. The next step would be to show that the algebric ...
4
votes
1answer
54 views

What operations can I do to simplify calculations of determinant?

My question is simple. Given an $n \times n$ matrix $A$, what operations can we do to the rows and columns of $A$ to make the calculation of its determinant easier? I know we can put it into row ...
1
vote
0answers
16 views

Polar decomposition varient

I have a factorisation to do, and I think that a varient of Polar decomposition will give me what I need, although I'm not sure of the exact form. I have \begin{equation*} \mathbf{y} = ...
0
votes
1answer
25 views

Doubt on Kantorovich inequality. Equivalence of inequalities.

To prove de Kantorovich inequality (for that we suppose the matrix A symmetric and definite positive) I need to demonstrate the next exercise: Proof that $$(x^TAx)(x^TA^{-1}x) \leq \frac{(\lambda_1 ...
0
votes
2answers
34 views

I'm struggling to find this transformation matrix

$T:\Bbb{P}_3 \to \Bbb{P}_3$ is a linear transformation such that: $$\begin{align} T\left(-2 x^2\right) &= 3 x^2 + 3 x \\ T(0.5 x + 4) &= -2 x^2 - 2 x - 3 \\ T\left(2 x^2 - 1\right) ...
1
vote
2answers
41 views

Rotation matrix

I'm finding different results for the 3D rotation matrix in the XY plane from different sources and I was hoping for someone to help clarify. In my "applications of vector calculus" book, the matrix ...
1
vote
1answer
58 views

Example of using the Hadamard's matrix to determine the superposition

I've came across those notes for Quantum computation from John Watrous. I am having troubles understanding the last example. We have those two vectors, or if I understood correctly, from now on ...
0
votes
2answers
23 views

methods of constructing a matrix from its null space span

I have a matrix of size $4\times3$ and its null-space span is $\{(1,2,3), (2,5,7)\}$. How can I find the original matrix? It is not obvious from the span which vectors are free.
0
votes
3answers
60 views

Self-Study Linear Algebra book for a complete understanding

I recently took an introductory class on linear algebra (covered solving linear systems, determinants, eigenvectors, diagonalization, some vector spaces, basis and combinations, transformations etc.) ...
2
votes
1answer
25 views

The relation between the algebraic dimensions of a vector space and its dual

Let $V$ be an (infinite dimensional) vector space over the field $\mathbb F (=\mathbb R$ or$ \mathbb C$). If $\alpha$ is the dimension of $V$, for some cardinal number $\alpha$, I want to know, what ...
0
votes
0answers
16 views

Find a generator for vectorial subspace

S = {$(a, b, c, d) ∈ C^4 : 2ia = b, c + d − ib = 0$} $c+d-i(2ia)=0$ $c+d+2a=0$ $c=-d-2a$ $(a,2ia,-2a-d,d)=a(1,2i,-2,0)+d(0,0,-1,1)$ Is this solution correct?
-1
votes
1answer
27 views

I have to show $a_{nn} \neq 0$. [on hold]

Let $D$ be an algebraic division ring with center $F$. $A,B$ are upper triangular matrices in $M_n(D)$. let $ A=\begin{pmatrix} a_{11}&a_{12} & \ldots &a_{1n}\\ 0 & a_{22} & ...
0
votes
0answers
29 views

If $T^{k}=0$ of some $k$, then $T^n=0$, where $n$ is dimension [duplicate]

Let $V$ an $n$-dimensional vector space and $T$ a linear operator on $V$. Suppose that there is some positive integer $k$ such that $T^{k}=0$. Prove that. $T^{n}=0$
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votes
2answers
29 views

Linear operator of infinite dimension

Let $T: V\rightarrow V$ a linear operator with finite dimension. If exists a linear operator $U: V\rightarrow V$ such that $TU=I$, prove that $T$ is invertible. Prove that if the ...
7
votes
4answers
3k views

Symmetric matrix is always diagonalizable?

I'm reading my linear algebra textbook and there are two sentences that make me confused. (1) Symmetric matrix $A$ can be factored into $A=Q\lambda Q^{T}$ where $Q$ is orthogonal matrix : ...