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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How do I restrict k to ensure my matrix has exactly 3 distinct eigenvalues?

$$A=\begin{bmatrix}-1&-1&0\\-12&3&-1\\k&0&0\end{bmatrix}$$ How do I restrict $k$ to ensure that my matrix has 3 distinct real eigenvalues? I tried going about it the long way ...
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0answers
13 views

Endomorphism commutes with its adjugate

Let $R$ be a commutative ring, $M$ a free $R$-module of rank $n$ and $f \in \rm{End}(M)$. The adjugate $f^\sharp$ of $f$ is defined by the equalities $$ f^\sharp(x) \wedge y = x \wedge ...
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2answers
42 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
38 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 ...
3
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1answer
45 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} = ...
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2answers
45 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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0answers
16 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} = ...
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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 ...
1
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3answers
74 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
42 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
24 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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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 $V$ ...
3
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3answers
19 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 ...
0
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6answers
51 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 ...
1
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2answers
32 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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3answers
69 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) ...
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0answers
26 views

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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0answers
25 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
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 ...
0
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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 ...
0
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3answers
39 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 ...
1
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3answers
27 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 ...
4
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1answer
36 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
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1answer
28 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 ...
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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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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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1answer
31 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
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 ...
2
votes
3answers
26 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?
2
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1answer
22 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: ...
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0answers
11 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
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, ...
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0answers
50 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 ...
0
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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 ...
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4answers
31 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 ...
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1answer
24 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
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1answer
37 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 ...
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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 ...
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2answers
24 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 ...
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1answer
40 views
1
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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.
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0answers
15 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
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1answer
109 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
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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 ...
7
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1answer
77 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 ...
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2answers
22 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.
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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) ...
0
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3answers
58 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.) ...