Linear algebra is concerned with vector spaces of all dimensions and linear transformations between them. This includes: systems of linear equations, basis, dimension, subspaces, matrices, determinant, trace, eigenvalues and eigenvectors, diagonalization, Jordan form, etc. For questions specifically ...

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118 views

Gershgorin interval of an eigenvalue and the largest coordinate of the corresponding eigenvector

Let $A=(a_{ij})$ be a $n\times n$ -- symmetric matrix with positive diagonal entries. The smallest eigenvalue, $\lambda_1$, is simple, and the corresponding unit eigenvector has all coordinates, ...
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51 views

A Nonzero Alternating Bilinear Form on the Space $P_1(F)$ Over $F$

Can anybody think of an example of a nonzero alternating bilinear form on the space $P_1(F)$ over $F$. $F$ is a general field like $\mathbb{R}$ or $\mathbb{C}$. $P_1(F)$ is the set of all ...
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41 views

Is $\varphi:x \mapsto A\cdot x$ an orthogonal projection for M

I got the transformation $\varphi:x \mapsto A\cdot x$ and the matrix $M = \begin{pmatrix} 1 & 0 \\ -1 & 0 \end{pmatrix}$. I have to check whether $\varphi$ is the orthogonal projection for ...
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29 views

Is $\phi^T_tP_t^{-1}\phi_t\to 0$ when $P_{t+1}=\sum_{k=0}^t\phi_k\phi_k^T+P_0$?

Let $\phi_t\in\mathbb{R}^n$, $\forall t\geq0$, and $\sup_t\|\phi_t\|_2^2\leq M<\infty$(euclidean norm). Define $n\times n$ positive definitive matrices as follow, ...
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193 views

I'm having trouble finding this matrix $T$ relative to $\mathcal B$ and the standard basis $\mathcal E$ for $\mathbb R^2$

This was a homework assignment, but unfortunately it was the last homework assignment of the semester so I never got feedback and I'm just reviewing it for a final. I'm supposed to let $\mathcal ...
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67 views

$\mathbb{R}^n$ and $\mathbb{Q}^n$: On the Nature of Solutions

I would just like to ask a simple question about solutions of non-homogenous linear equations both in $\mathbb{R}^n$ and $\mathbb{Q}^n$: What does it mean that a system of non-homogenous linear ...
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2answers
285 views

How to expand equation inside the L2-norm?

I want expand an L2-norm with some matrix operation inside. Assume I have a regression $Y=X\beta+\epsilon$. I want to solve (meaning expand), $$\displaystyle\|Y-X\beta \|_{2}^2$$ Should I do: 1) ...
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63 views

Prove using an example that there is no plane on R3 that contains every group of 4 points

Well, this is a homewrok question (which I know I should not be asking, but I cannot find an answer to this anywhere): The exercise is as follows: i) Find the equation of the plane of R3 that ...
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141 views

Decomposition of vectorspace and subspaces invariant under a linear operator

Are the following claims true or false? (1) Let $T$ be a linear operator on a finite dimensional vector space $V$ and let $V=W_1 \bigoplus W_2$ where $W_1$ and $W_2$ are $T$-invariant subspaces of ...
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35 views

Question between symmetric matrix and transforming to other bases?

If $A$ is symmetric and $A^{10} = 0$, then $A$ must be $0$. I was thinking that $A$ must have as many eigenvalues as it does rank, and from that statement, one of its eigenvalues must be $0$, but ...
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326 views

Determinant of a matrix with variables in it

Assuming that $z \neq 0$, compute the determinant $d_n(z) = \det D_n \left(1, z, 1 - \frac{1}{z^2} \right)$, where $z$ is a complex variable. In particular, compute the value $d_n(\sqrt{2})$. ...
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70 views

Intuition on matrix multiplication and algorithms

Yesterday, I was watching Strang's lectures on Matrix multiplication. He mentioned five different ways of looking at the multiplication $\mathbf{AB} = \mathbf{C}.$ Classic way (Row of $\mathbf{A} ...
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1answer
48 views

Number of characteristic polynomial?

By theorem (it takes too much space to write their proof so I just show it's conclusion), characteristic polynomial of a matirx $A \in M_{2*2}(\mathbb{Z}_{2})$ is ...
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57 views

What to take from representation of $S_d$?

I am reading about group representations, and books I read all contain the representation theory for symmetric groups $S_d$. However none of them presents the material in a friendly way. After reading ...
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1answer
38 views

What formula would plot a line with a 3% year to year reduction from an initial goal?

The Requirement I have some pretty simple logic written in ActionScript that plots points on a graph. The intent is to produce a line that demonstrates a 3% reduction year to year from the initial ...
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1answer
101 views

Linear Algebra — Block Matrix Inversion

Please excuse my formatting... $X=\left(\matrix{A & B\\C & D}\right)$ where $A,B,C,D$ are all $n\times n$ matrices. Assuming that all stated inverses exist show that ...
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263 views

Find the position vector of the point R that is closest to the origin on the plane a'x + b'y + c'z = e

(a) Write down the Cartesian equation for the plane through the point Q(1,0,0) with normal n = -$ \sqrt{6} $ i - $ \sqrt{2} $ j - k and compute the distance of the origin from this plane. (b) Let ax ...
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71 views

Vandermonde question

I'm studying time series analysis and in my book I came a cross with the following proof (The proof is actually the last page, but I posted as much information as possible on the problem): I have ...
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1answer
182 views

Proving $(\operatorname{ker}T)^{\perp}\subseteq \operatorname{Im} T^{*}$

Let V be a finite inner product space with $T:V\to V$ a linear transformation. How can I prove that, $(\operatorname{ker}T)^{\perp}\subseteq \operatorname{Im}T^{*}$ ? Edit: My purpose is to ...
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36 views

Subspace decomposition of the rational canonical form

Let $T:V\to V$ be a linear operator on a $k$-vector space $V$. Is there a nice intuitive description of what the subspaces corresponding to the blocks of the rational canonical form of $T$ are? The ...
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70 views

How to simplify a product of matrices

Let $A$ be an invertible $2\times2$ matrix and $Y,Z$ be $2\times1$ matrices, all with real elements. Then clearly $x=Y^{t}AZ$ is a real number, where $Y^{t}$ denotes the transpose of $Y$. Assuming ...
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108 views

Linear Algebra finding points with certain conditions

a) Consider an equilateral triangle in the i-j-plane with one vertex at the origin, and a second vertex with position vector i. Let $u = \textit{a}\textbf{i}+\textit{b}\textbf{j}$ be the position ...
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79 views

Exponential function of a Matrix

Here is the last part of a problem: For which $2\times 2$ orthogonal matrices does $$ e^A = I + A+ \frac{A^2}{2!}+ \cdots $$ converge to an orthogonal matrix. We need to show that if $A^*A=AA^*=I$ ...
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55 views

Approximating the solution of an overdetermined system help

I'm trying to find a solution for an overdetermined system in the form Mv = w by projecting w (the solution) onto the span of the columns of M and then setting Mv equal to this projection. ...
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24 views

How do i prove that "every monic polynomial is the characteristic polynomial of some matrix? [duplicate]

Let $F$ be a field and $f(X)\in F[X]$ be a monic polynomial such that $deg(f(X))=n$. How do i prove that there exists $A\in\mathscr{M}_n(F)$ such that $\det(XI_n - A)=f(X)$? I checked the solution ...
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50 views

Solution of a Quadratic Optimization Problem

Let $\mathbf{A_1}$ and $\mathbf{A_2}$ be two given $N\times N$ hermitian matrices. Then how do I solve the problem, \begin{align} ...
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222 views

Showing no non-trivial t-invariant subspace has a t-invariant complement.

The question is from Hoffman and Kunze Let T be a linear operator on a finite-dimensional vector space V. Suppose that: (a) the minimal polynomial for T is a power of an irreducible ...
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35 views

Recursive formulae involving a linear operator

Given a basis $e_{1}$, $e_{2}$ in the plane, define the linear operator $F$ as $F(e_{1})=3e_{1}+e_{2}$ and $F(e_{2})=e_{2}$. Furthermore, define the sequence $u_{1},u_{2},\dots$ of vectors in the ...
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74 views

Least Squares Approximation

Given: $\begin{bmatrix} 1 & -2\\ -2 & 4 \end{bmatrix} \cdot \begin{bmatrix} x\\ y \end{bmatrix} = \begin{bmatrix} 1\\ 5 \end{bmatrix}$ How do I find the least squares approximation and ...
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35 views

Is the polar decomposition useful in the real case as well?

I'm reading Roman's Advanced Linear Algebra p.252, where he talks about the Polar Decomposition. He states the theorem only for the case of $V$ a complex inner product space. Wikipedia also states the ...
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136 views

Improve Upper Triangular Matrix Computation for Gauss Jordan method?

I am solving Simultaneous Equations using the Gauss Jordan method. I am having a problem in computing the Upper triangular Matrix with sufficient accuracy for no of variables >50. Some of the elements ...
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96 views

Diagonalizing the sum of a matrix and a multiple of the identity matrix

Suppose we have a matrix $A = B+\lambda I$, where $B\in \mathbb{R}^{n\times n}$, $I$ is the identity matrix and $\lambda\in \mathbb{R}$. If I know the eigenvalues and eigenvectors of $B$, what can I ...
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1answer
48 views

Wronskian of $y, y_1,\dots, y_n$

Let $y_1,\dots, y_n$ be linearly independent functions in $C^\infty$. For each $y \in C^\infty$, define $T(y) \in C^\infty$ by $$[T(y)](t)=\begin{vmatrix} y(t) & y_1(t) & \cdots & y_n(t)\\ ...
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49 views

Diagonalization of a matrix $A \in GL(n,\mathbb{C})$

I am trying to show that $$ \frac{d}{dt} \log \det A_t = Tr (A^{-1}_tA'_t) $$ where $A_t \in GL(n,\mathbb{C})$ and $A'_t = \frac{d}{dt}A_t$. I think I can show why this is the case if $A$ is ...
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381 views

irreducible, diagonally dominant matrix

I am facing a problem for irreducible,diagonally dominant matrices. How to prove that irreducible, diagonally dominant matrix is invertible? Please help me in this problem.
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197 views

Linearly dependent eigenvectors when diagonlising a matrix

If I wanted to diagonalise an $n \times n$ matrix $A$. Let $P$ be the matrix of eigenvectors. Why is it that I need columns of $P$ to be linearly independent? If I had two equal eigenvalues and ...
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68 views

Contour Integral of a Definite Positive Matrix

I need to prove the following proposition: Proposition: Let assume that $H\left( \lambda\right) \in\mathbb{R}^{\left( n-1\right) \times\left( n-1\right) }$ is a matrix function such that ...
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78 views

Does this matrix have any properties?

The matrix is: $\left( \begin{array}{cc} \sin{\theta} & \cos{\theta} \\ \cos{\theta} & \sin{\theta} \\ \end{array} \right) $ I'm interested in its effect on points in the first quadrant, ...
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58 views

Definition of a countable direct sum of subspaces of a Banach space

Let $X$ be a separable Banach space and $K\subseteq X$ a subspace. Let $\{H_i\}_{i\in I}$ be a countable collection of subspaces of $X$. Is it correct that $K=\bigoplus H_i$ iff every element $k\in K$ ...
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1answer
87 views

Combinatorial identity. Using echelon matrices.

Determine the exponents $e_i$ s.t. the following identity is correct. $$\sum\limits_{i=0}^k q^{e_i} {\binom mi}_q {\binom{n}{k-i}}_q = {\binom{n+m}{k}}_q$$ Note: When $q=1$ the equation reduces to a ...
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72 views

Solving a system of four equations

If I have a system of four equations and i need to get ratio $E/A$ out of it. How can i do it? $\mathcal K$, $\mathcal L$ and $d$ are conctants, $A$ is independant variable where $B,C$ and $D$ are ...
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46 views

What's the meaning of matrix $A*\mathrm{orth}(A^T)$?

Could anyone give an explanation on the meaning of a matrix $A * \mathrm{orth}(A^{T})$, where $\textrm{orth}(A)$ is the matlab function, which compute the range of matrix $A$? $*$ is matrix ...
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1answer
57 views

Diagonalization over rings and the dimension of the cokernel of an endomorphism

So, I'm trying to prove the following: Let $\mathcal{O}$ be a DVR, $M$ a finite-rank free $\mathcal{O}$-module, and $\varphi \colon M \to M$. Then $\dim_k (M/\varphi(M)) < \infty ...
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39 views

Parallelogram law, dot product [duplicate]

Prove that if $||\cdot||$ satisfies $||u-v||^2 + ||u+v||^2 = 2(||u||^2 + ||v||^2)$ , then $u \cdot v = \frac{1}{2} (||u+v||^2 - ||u||^2 - ||v||^2)$ is dot product and $||u||^2 = u \cdot u$. I've ...
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105 views

Help find a proof : $ \lambda $ is $f$'s eigenvalue then $f|_{V_{\lambda}} $ has Jordan's basis

Could you help me find a fairly simple proof of the following theorem? $f: V \rightarrow V, \ \ \dim V < \infty, \ \ \lambda$ is $f$'s eigenvalue $\Rightarrow \ \ f|_{V_{\lambda}}: \ V_{\lambda} ...
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41 views

Combining elimination matrices

I am trying to combine several elimination steps into one matrix: more specifically I try to come up with a 3 by 3 matrix that first subtracts row 1 from row 2, subtract row 1 from row 3 and then ...
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128 views

DFT in complex form and Trigonometric Polynomial Interpolation: why different dimensions of basis vectors set?

I'm trying to figure out, how coefficients of Discrete Fourier Transform in complex form are converted into coefficients of Trigonometric Polynomials Interpolation: Say, I have a function vector with ...
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49 views

Wrong answer on elementary diophantine equation - why?

Solve the equation and show all possible, non-negative values for X and Y: $5X+4Y=60$ So I wanted to do it like that: $$5X+4Y=60\leftrightarrow0X+4Y=0 \pmod5$$, thus $4Y=5k$ where $k\in Z$. ...
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49 views

Linear Algebra: Linear transformation and eigenvalues [duplicate]

Hi could some one please help. I am having problems proving this. Let $A$ be an $n \times n$ matrix with complex entries and let $f (t) =\det(A - tI)$ be its characteristic polynomial. Prove ...
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220 views

deteminant of a block skew-symmetric matrix

If I have a matrix if the form \begin{pmatrix} A & -B \\ B & A \end{pmatrix} how do i turn it into something like \begin{pmatrix} X & Y \\ 0 & Z \end{pmatrix} so the determinant is ...