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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Describe the smallest subspace of $M_{2\times 2}$ that contains matrices…

Describe the smallest subspace of $M_{2\times 2}$ that contains matrices ...
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$P, Q$ are symmetric square. If $P \geq Q > 0$ then show $P^{-1} \leq Q^{-1}$ [duplicate]

Possible Duplicate: inequalty concering inverses of positive definite matrix Is $B^{-1}-A^{-1}$ a positive definite matrix? $P, Q$ are symmetric square. If $P \geq Q > 0$ then show ...
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587 views

Diagonalizing a Unitary Matrix

I'm trying to diagonalize the following unitary matrix: $\frac {1}{\sqrt{5}}\begin{pmatrix} 1 &2 \\ 2i &-i \end{pmatrix}$ My approach is to find the eigenvalues and eigenvectors in the usual ...
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259 views

Let $M_{2\times 2}$ be the vector space of all $2\times 2$ matrices. Show that the set of non-singular matrices is NOT a subspace.

I am working on problems from my textbook. However, I am lost as to how to show this. A. Let $M_{2\times 2}$ be the vector space of all $2\times 2$ matrices. Show that the set of non-singular ...
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233 views

When does the adjacency matrix of a graph have an eigenvalue zero?

When does the adjacency matrix $A$ of an undirected graph $G$ not have full rank? Is there any intuition to understanding this?
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Re-calculate solution after altering some elements in a linear system

Problem I have a linear system: $$ Mx = b $$ $M$ is like a Band Matrix. And assume I have a solution $x_{init}$ at beginning. There will be some operations which are going to alter some elements in ...
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203 views

self-adjoint operator and unitary orthogonal matrix

Please offer a solution to the following problem. It was offered in class by my professor as an additional exercise to try on one's own. Let $V$ be the inner product space, and assume that $\alpha ...
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34 views

$\alpha$ affine iff graph is affine subspace

I am just checking different analogous of $\alpha:V \longrightarrow W$ being affine. I have problems with this one: $\alpha:V \longrightarrow W$ affine $\iff$ $G_\alpha=\{(v,\alpha(v): v\in V)\}$ is ...
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518 views

Prove subspace of finite dimensional vector space is finite dimensional

Prove that any subspace, W, of a finite-dimensional vector space V must also be finite dimensional.
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248 views

Cross product as result of projections

The cross product between two vectors in $\Bbb{R}^3$ (call them a and b) is denoted a $\times$ b and the result is a vector in $\Bbb{R}^3$ orthogonal to the first two. There are a variety of ways of ...
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472 views

quadratic and bilinear forms

Does a quadratic form always come from symmetric bilinear form ? We know when $q(x)=b(x,x)$ where $q$ is a quadratic form and $b$ is a symmetric bilinear form. But when we just take a bilinear form ...
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37 views

Dependence of $|K|$ and the union of the subspaces.

If $V$ is a $K$-vector space with $|K|\ge n$ and $V_1, \ldots, V_n$ are subspaces of $V$ such that $V=V_1 \cup \cdots \cup V_n$, then $V=V_i$ for some $i$. Is there a counterexample that for $|K|< ...
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Are there programs out there that try to derive linear recurrence given a string of numbers?

Wolfram isn't helping me much so I am curious if there are other programs out there. I don't know what degree it is, but I have a series of numbers and I'd like to determine the linear recurrence ...
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Proving A is Invertible if $A + A^2 = I$

I'm trying to prove A is invertible by proving there is an $A'$, for $AA' = I$ So I got to this stage $A(I + A) = I$ Now I determine that $A' = I + A$, and from that I get $AA' = I$, I wanted to ...
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Interiors of spherical caps intersect if and only if …

A spherical cap is defined by $C(x_1,\alpha_1)=\{y \in S^{n-1} : x\cdot y \geq \cos(\alpha)\}$ and $\alpha_i\in [0, \pi]$ and $x_i \in S^{n-1}$ ( $x_1 \cdot x_2$ refers to the inner product of ...
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256 views

Adjoint operator on a finite dimensional space

Let $V$ be a finite dimensional inner product space and $T: V \to V$ be a linear operator. Show that the range of $T^*$ is the ortogonal complement of the null space of $T$. Since $V$ is finite ...
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92 views

Property of convex combinations.

If $x = \sum_{i = 1}^n a_i y_i$ with $a_i > 0$, $\sum_{i = 1}^n a_i = 1$ and $|x| \geq |y_i|$ why is it true that $x = y_i$ for all $i = 1, \ldots, n$? I can see that $|x| \leq \sum_{i = 1}^n a_i ...
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Let $AB$ be a diagonal matrix. Is A invertible?

Let $AB \in K^{n\times n}$, $AB$ is a diagonal matrix and the elements on the diagonal are non-zero. Is $A$ invertible? Since $AB$ is a diagonal matrix and the elements on the diagonal are non-zero, ...
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62 views

elementary congruence statement proof

it is me again :), i am trying to prove this statement of congruence, the statement is as follows: $a \equiv b \mod m \Longrightarrow a^{k} \equiv b^{k} \mod m $ for all $ k \in \mathbb{N}$ i ...
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425 views

Gaussian elimination mod k

We have this assignment in programming class, but I believe posting it in math will make more sense. So we're supposed to write a program that takes $n$ equations with each $n$ coefficients, ...
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congruence theorem prove

i am trying to prove this statement of congruence which is not really hard, but i am stumbling across the simple step. statement: $a \equiv b \mod m \land c \equiv d \mod m \Longrightarrow ac ...
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354 views

A question about inner products on abstract vector spaces

I have been reading some materials and, for the n-th time in my life, there was a definition of an inner product as a function $V \times V \rightarrow F$, where $V$ is an abstract vector space and $F$ ...
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644 views

If $A$ is singular, is $A^3+A^2+A$ singular?

Suppose that $A$ is singular, is $A^3 + A^2 + A$ singular as well?
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Given $P : R^n → R^n$ is a linear transformation. Show that there is an integer $k$ such that $R(P^k)=R(P^{k+1})=R(P^{k+2})=…$

Given $P : R^n → R^n$ is a linear transformation. Show that there is an integer $k$ such that $R(P^k)=R(P^{k+1})=R(P^{k+2})=...$($R(P)$ denotes the range of $P$.)
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If $Q$ is an orthogonal, then is $Q+\frac{1}{2}I$ invertible?

If $Q$ is an orthogonal matrix, then the matrix has orthonormal columns. I asked this question to my friend and he says: Let $Q= -\frac{1}{2}I$, then it is orthogonal, and $Q+\frac{1}{2}I$ is zero, ...
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General definition of a line

In the book on Linear Algebra that I am using, the author defines a line in an arbitrary vector space $V$, given direction $ 0 \neq d \in V $ and passing through $ p \in V$ as $ l(p;d)= \lbrace v\in ...
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80 views

How does the unit normal to a hyperplane change with the vectors that define it?

Let $v_1, \dots, v_{n-1}$ be linearly independent vectors in $\mathbb{R}^n$. Their span defines a hyperplane; let $u$ be the unit normal vector to this hyperplane. Now suppose we change $v_{11}$ ...
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137 views

In terms of complexity, is there a quicker way of checking if a matrix is nonsingular than computing the determinant?

To repeat the question, let $A$ be a square matrix. We wish to determine if $A$ is nonsingular, that is, invertible. One way is compute its determinant and check if it is nonzero. However, if $A$ is ...
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decomposition of singular value decomposition

Hi: Suppose I have a matrix $Y$ is that is $p\times q$ and of rank $r$. Then I read a paper that said that the SVD can be decomposed as $Y=U_0 \Sigma_0 V^*_0 + U_1 \Sigma_1 V*_1$ where $U_0$ and ...
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365 views

Proof relating to Orthogonal Complement

a.) Show that if $A=A^T$ is a symmetric matrix, then $A\mathbf{x}=\mathbf{b}$ has a solution iff b is orthogonal to $\ker A$. b.) Prove that if $K$ is a positive semi-definite matrix and ...
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182 views

Convergence of CG method

I have a question like how can we mathematically prove that for a general matrix Conjugate Gradient method will always converge within n steps in exact arithmetic ? where n is the size of the matrix. ...
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124 views

Principal Components of matrix with duplicate eigenvalues

If you have a matrix: \begin{pmatrix} 2 & 0 & 0 \\ 0 & 4 & 0 \\ 0 & 0 & 4 \\ \end{pmatrix} The eigenvalues are $2,4,4$. The eigenvectors are $(0,0,1)$, $(0,1,0)$ and ...
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Properties of the unit normal to a partially-rotated hyperplane

Let $A$ be an $n-1 \times n$ matrix. The span of the rows of $A$ define a hyperplane in $\mathbb{R}^n$; let $u$ be the unit normal to this hyperplane. Now, let $x \in \mathbb{R}^{n-1}$, and replace ...
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Trace minimization with constraints

For positive semi-definite matrices $A,B$, how can I find an $X$ that minimizes $\text{Trace}(AX^TBX$) under 'either' one of these constraints: a) Sum of squares of Euclidean-distances between pairs ...
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Is $ED=DF⇒E=F$ (matrices) correct?

I am to prove or find a counterexample for the following problem: Let $A, B, C, D, F \in GL(n, \mathbb{R})$. If $D^{-1}(A+B+C)D=F$ is correct then $A=F-B-C$ is correct. I have not found a ...
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Unitary matrix proof

Prove that unitary matrix $U$ satisfies $|\det U| = 1$, but $\det U$ is different from $\det U^{H}$. How can I prove these two statements? I guess I should use the fact that every column of unitary ...
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What's the relationship between Cauchy-Schwarz Inequality and Extended Cauchy-Schwarz Inequality?

I'm learning multivariate analysis. Cauchy-Schwarz Inequality plays an important role in several multivariate techniques. Cauchy-Schwarz Inequality:Let b and d be any two p $\times$ 1 ...
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65 views

Find an invertible matrix $B$ such that all eigenvectors of $B$ are scalar multiples of a given vector.

Let $u = \left( {\begin{array}{*{20}{c}} 1 \\ 0 \\ 0 \\ \end{array}} \right)$. Find an invertible matrix $B$ such that all eigenvectors of $B$ are scalar multiples of $u$. My ...
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What is the intuition for using definiteness to compare matrices?

If $a$ and $b$ are two numbers on the real line, we compare $a$ and $b$ by knowing which of them comes first as we move from $-\infty$ to $\infty$ on the real line. However when $A$ and $B$ are ...
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What is the dimension of subspace $\{\vec{u}\in\Bbb{R}^n:\vec{n}^T\vec{u}=\vec{0}\}$?

The subspace in question: $V=\{ \vec{u} \in \Bbb{R}^n : \vec{n}^T\vec{u}=\vec{0} \}$ I am assuming that $\vec{u} = \begin{bmatrix}x_0 \\ x_1 \\ \vdots \\ x_2\end{bmatrix}$. The dimension of a vector ...
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Find the smallest positive integer $k$ such that $A^k=0$ where $A$ is a nonzero matrix

Let $B = \left( {\begin{array}{*{20}{c}} \lambda & 1 & 0 \\ 0 & \lambda & 1 \\ 0 & 0 & \lambda \\ \end{array}} \right)$ where $\lambda \ne0$.(i) Find the ...
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Dimension of a set of symmetric $n\times n$ matrices with a certain characteristic polynomial

Let $M_{n \times n}$ be the set of all $n\times n$ symmetric matrices such that the characteristic polynomial of each $A\in M_{n\times n}$ is of the form $$t^n+t^{n−2}+a_{n−3}t^{n−3}+⋯+a_1t+a_0.$$ ...
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60 views

Gradient of a scalar with vector

Does this expression $\frac{da^{\intercal}a}{da}$ evaluates to $2*a$ or $2*a^{\intercal}$ ? Here $a$ is column vector. Derivation steps would be appreciated.
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A is 5×5 matrix, all of whose entries are 1

A is 5×5 matrix, all of whose entries are 1, then (a) A is not diagonalizable (b) A is idempotent (c) A is nilpotent (d) The minimal polynomial and the characteristics polynomial of A are ...
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82 views

Dimension of least squares solution

How is it possible that the matrix equation $\textbf{Ax} = \textbf{b}$ has a least squares solution that is a line through the origin? Isn't it the point closest to the vector $\textbf{x}$ in the ...
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Completing squares by symplectic transformations

A quadratic polynomial of $2n$ variables is given as $$ H = \sum_{i,j=1}^{2n} A_{ij} x_i x_j = x^T A x, $$ where $A$ is a symmetric matrix. I am looking for a symplectic transformation of these ...
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Given a matrix $A \in M_n(\mathbb{R})$, can we find two orthogonal matrix satisfy that$O_1AO_2$ is a diagonal matrix

Assume $A\in M_n(\mathbb{R})$ and $\det(A)\not=0$, is there existing two orthogonal matrix $O_1$,$O_2$ that satisfy $$O_1AO_2=\begin{pmatrix}\lambda_1 & & & \cr & \lambda_2 & ...
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107 views

Let$(a,b,c)$ be a nonzero vector in the row space of a $3\times 3$ matrix $B$. Show that the nullspace of $B$ is a subset of the plane $ax+by+cz=0$

Let $(a$ $b$ $c)$ be a nonzero vector that belongs to the row space of a $3\times 3$ matrix $B$. Show that the nullspace of $B$ is a subset of the plane $ax + by + cz = 0.$ My thoughts so ...
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192 views

Orthogonal matrix and unitary matrix

Orthgonal matirx Q : Suppose that $A=Q\lambda Q^{T}$ iff $A=A^{T}$ (A is symmetric) It means A's eigenvectors are orthogonal and unit length. By the property of Hermitian martrix " If $A=A^{H}$, ...
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175 views

Value of Vandermonde type determinant

Let $x_1,...,x_n $ are distinct real numbers. Is it a formula for the Vandermonde type determinant $V(x_1, \cdots,x_n)$ whose last column is $x_1^k,\ \cdots,\ x_n^k$, where $k \geq n$, instead of ...