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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Eigenvector corresponding to eigenvalue $ 1 $ of a stochastic matrix

I am trying to justify fact $ 5 $ in this link which states that if $ A $ is a column stochastic matrix, then $ A $ has eigenvalue $ 1 $ and a unique eigenvector such that all entries are either ...
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First Order Difference Equations - Using Eigenvectors/Values

I was reading some notes and there was the following section: Start with a given vector $\vec{u}_0$. We can create a sequence of vectors in which each new vector is $A$ times the previous vector: $$\...
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1answer
37 views

I'm looking for a matrix $M$ with $\det(M)=a^2+b^2+c^2+d^2$

In order to show that $(a^2+b^2+c^2+d^2)(A^2+B^2+C^2+D^2)= \alpha^2+\beta^2+\gamma^2+\delta^2$ with $a,b,c,d,A,B,C,D,\alpha,\beta,\gamma,\delta \in \mathbb Z$. I would like to find a matrix with ...
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Rank and null space of a particular block matrix.

Let $D_1, D_2 \in \mathbb{R}^{N \times N}$ be diagonal matrices with diagonals that are linearly independent vectors. Let $A, B \in \mathbb{R}^{N \times N}$ be rank-deficient matries. Define $S = \...
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297 views

To show a set of projectors summing to the identity implies mutually orthogonal projectors

The general setting is the study of positive operator measures in quantum mechanics http://en.wikipedia.org/wiki/Positive_operator-valued_measure instead of the projector operator measures. Going ...
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Logic of Elementary Row Operations to Create Equivalent Systems

Can anyone explain why the 3rd operation applied on a system creates an equivalent system with the same solution. Elementary Row Operations. 1. Interchange two rows. 2. Multiply a row with a ...
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3answers
57 views

What is the meaning of subtracting from the identity matrix?

If I subtract the matrix $A$ from the identity matrix $I$, $I - A$, is there a meaning to the resulting matrix perhaps given some conditions like invertibility or symmetry? For example, $$ \begin{...
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I hope you resolve the question with surrounding solution method [on hold]

That we know that: $$(i-\sqrt 3)^x-(i+\sqrt 3)^y=2^{xy}$$ Find the value of: $x+y$ .
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Does $\forall v ( T_1 v = 0 \lor T_2 v = 0 \lor \dots \lor T_n v =0 )$ imply $T_1 = 0 \lor T_2 = 0 \lor \dots \lor T_n = 0$?

Let $V$ and $W$ be vector spaces and $T_1$, $T_2$, $\dots$, $T_n$ be linear transformations from $V$ to $W$, such that for every $v$ in $V$, either $T_1 v = 0$, $T_2 v = 0$, $\dots$ or $T_n v = 0$. ...
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2answers
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The trace functional and its scalar multiples [duplicate]

I am trying to solve the following problem: Show that the trace functional on $n \times n$ matrices is unique in the following sense. If $W$ is the space of $n \times n$ matrices over the field $F$ ...
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24 views

Describe all solutions of Ax = 0 (2)

Let $A = \begin{bmatrix}1&-5&-3&2\\4&-20&-12&8\end{bmatrix}$ Describe all solutions of $Ax = 0$ $x = x_2 \begin{bmatrix}\\\\\end{bmatrix} + x_3 \begin{bmatrix}\\\\\...
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stopping criteria for power-iteration to find rank-1 matrix

I start with B=I, A positive matrix, and compute B=(BA)/norm(B) by iterating until B is sufficiently close to rank-1 matrix. What is a good stopping criterion for this algorithm? There's Birkhoff ...
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3answers
57 views

Prove that for any diagonalizable matrix $A$, $A^n$ is diagonalizable and also $aA^m+bA^n$

Suppose that A is a diagonalizable matrix. 1) Prove that $A^n$ is diagonalizable 2) Prove that $aA^n + b A^m$ is diagnalizable, for every $a,b\in\mathbb{K}$ I thank you any help or hint you can ...
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1answer
47 views

Matrix addition and eigen values/vectors

If I start with matrix A given by $A = \begin{bmatrix}a & b \\ c & d \end{bmatrix}$ and I express it as a sum $A = \begin{bmatrix} w & x \\ y & z ...
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1answer
92 views

Solving 3x3 Matrix Q using Nonlinear Least Squares or Cholesky Decomposition

I am trying to solve a system of equations using Cholesky decomposition. I would like to solve for the 3x3 matrix Q given: $\hat{i_f}^t Q Q^t \hat{i_f} = 1 $ $ \hat{j_f}^t Q Q^t \hat{j_f} = 1 $ $...
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1answer
40 views

If $Ax = O$ has only one solutions, the columns of A span R?

I've been doing some excersices about inner product and I found something interesting but I don't know if my approach is correct at all. Supose that ${v_{1}, v_{2}, ..., v_{n}}$ is a base for a ...
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2answers
46 views

Describe all solutions of Ax = 0

Let $A = \begin{bmatrix}1&-5&3&-3&-4&-2\\0&0&1&1&0&-5\\0&0&0&0&1&-3\\0&0&0&0&0&0\end{bmatrix}$ Describe all ...
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393 views

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

Show that if $V$ is a vector space over an infinite field $\mathbb{F}$, then $V$ cannot be written as set-theoretic union of a finite number of proper subspaces.
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Expressing the orthogonal projections on a linear operator $T$'s eigenspaces as polynomials in $T$

In the inner product space $\mathbb{C}^{2}$ with its standard inner product, let $$ T\begin{pmatrix} x\\y \end{pmatrix} = \begin{pmatrix} 3x+4y\\-4x+3y \end{pmatrix} $$ a linear operator. Express the ...
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What kind of $n^{th}$ order polynomials are solvable by a square matrix with integer entries?

Consider a polynomial (monic for simplicity): $$x^n+a_1x^{n-1}+\dots+a_{n-1}x+a_n=0$$ Here we assume $x$ is a complex number. $a_k$ are integers. Now consider the corresponding matrix polynomial: $...
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1answer
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Proof Relationship between System Solution and Matrix Rank [on hold]

Can anyone prove this theorem? Suppose a system of m equations in n variables is consistent, and the rank of a augmented matrix is r. (1)The set of solutions involves exactly n-r parameters. (2)If ...
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1answer
21 views

Find a Jordan basis for the endomorphism $g:M_2(R)\longrightarrow M_2(R)$ such that…

Find a Jordan basis for the endomorphism $g:M_2(R)\longrightarrow M_2(R)$ such that $M(g,B) = \begin{pmatrix} 2&0&3&0\\ 1&2&0&3\\0&0&2&0\\ 0&0&1&2 \...
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427 views

How to find matrix of orthogonal projection from gram-schmidt orthogonalization

I'm having a little difficulty understanding Gram-Schmidt orthogonalization. I have a problem to apply Gram-Schmidt orthogonalization to the system of vectors $(1,1,1)^T, (1,2,1)^T$ then write the ...
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3answers
112 views

Prove that there are not two matrices 2x2 such that: $AB-BA=I_2$

I tried this question by multiplying explicitly the matrices but I think I'm not getting anything, so I think, well let's suppose false so $C(AB-BA)=C$ and find a contradiction but also I'm not ...
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1answer
52 views

A question about orthogonality

Let $\mathcal{A}$ be a unital $*$-algebra over $\mathbb{C}$ and let $a,b\in\mathcal{A}$ be projections, that is, $a=a^*=a^2$ and $b=b^*=b^2$. If $a+b=1$, then $ab=0$. This follows from - \begin{align*...
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35 views

Polar set of orthogonal matrices set is nuclear norm ball

Reltated problems: Show that the dual norm of spectral norm is Nuclear norm. Proof that nuclear norm is convex. The set of orthogonal matrices is defined as: $$\mathcal{O}(n) = \{X\in \...
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1answer
56 views

Trying to find 2 equalities

STRICTLY: I do not need coding help. I just need to know why 2 equations won't come to a single equality I'm trying to use a conditional statement made of 4 equations to see if it is possible to take ...
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3answers
24 views

Expressing a vector v as a linear combination of x and y

Express the vector $v = \begin{bmatrix}49\\0\end{bmatrix}$ as a linear combination of $x = \begin{bmatrix}6\\5\end{bmatrix}$ and $y = \begin{bmatrix}-5\\4\end{bmatrix}$ $v = $ ____ $x + $ ______$...
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1answer
28 views

Linear Independence of Secant-squared or tangent-squared

Due to a result from Chowla, we know that the set $\cot(2\pi k/n)$, such that $\gcd(k, n) = 1$, is linearly independent over the Rationals. Do we have any similar results for $\sec^2(2\pi k/n)$ or $\...
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3answers
63 views

Finding complex solution to $X^2 = A$

Let $A=\begin{pmatrix}2&3\\4&-2\end{pmatrix}$. (i) Find an invertible matrix $P$ such that $P^{-1}AP$ is diagonal. (ii) Find $A^n$ (for positive integers $n$). (iii) Find four (...
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3answers
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Finding a and b in a matrix

Find $a$ and $b$ such that $\begin{bmatrix}-11\\9\\-12\end{bmatrix} = a \begin{bmatrix}1\\-3\\3\end{bmatrix} + b \begin{bmatrix}7\\3\\0\end{bmatrix}$ I think it's trivial that $a = -4$, which is ...
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1answer
26 views

Inner product (real or complex), sequence of real numbers

Suppose that $\{v_1, v_2, \dots, v_n\}$ is a basis for a vector space $V$ with inner product $\langle\cdot, \cdot\rangle$ (real or complex). Prove that for each sequence of $n$ real numbers $r_1, r_2, ...
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1answer
208 views

A zookeeper wants to give an animal 42 mg of vitamin A and 65 mg of vitamin D per day.

He has two supplements: the first contains $10\%$ vitamin A and $25\%$ vitamin D; the second contains $20\%$ vitamin A and $25\%$ vitamin D. How much of each supplement should he give the animal each ...
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2answers
47 views

Algebra, linear transformation, minimal polynomial [on hold]

Let $T : M_{n×n}(\Bbb F) \to M_{n×n}(\Bbb F)$ the linear transformation defined by $T (A) = AB$, for some matrix $B \in M_{n×n}(\Bbb F)$ fixed. Show that the minimal polynomial of $T$ coincides with ...
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1answer
74 views

Tricky norm-inequality $\|x\|_p \le n^{\frac{1}{p}- \frac{1}{r}} \|x\|_r.$ for $p \in (0,1)$

For $r>p \ge 1$ one can show that in $\mathbb{C}^n$ we have $$\|x\|_p \le n^{\frac{1}{p}- \frac{1}{r}} \|x\|_r.$$ My question is now: Does this also hold for $1 \ge r>p>0$? Obviously we ...
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2answers
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How to find the restrictions of side length on an obtuse triangle

Question: In Triangle ABC, the angle ∠ABC is an obtuse angle. The Side AB is 1cm, and the side BC is 3cm. Side AC is (3x+10)/(x+3) cm Find the restriction(s) on x. I have tried a few different ...
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2answers
43 views

How to describe range of a linear transformation?

I'm self studying Linear Algebra from Hoffman Kunze, and I've come upon this problem. With complex number $z=x+iy$, $$T(z)=\begin{pmatrix} x-7y & 5y \\ -10y & x+7y \\ \end{pmatrix}$$ is ...
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Given $A = \Sigma\lambda uu^H$. $A = -A^H$. Prove $\lambda$ is imaginary

Given $A = \Sigma\lambda uu^H$. and $A = -A^H$. Prove $\lambda$ is pure imaginary. (Btw, $u$ are orthonormal vector, don't know how to write here in math-stackexchange with the ^) I've two proofs I'...
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Auslander-Reiten theory: exercise $23.b$ of 'Elements of the Representation Theory of Associative Algebras'

I am solving exercise $23.b$ of chapter IV of 'Elements of the representation theory of associative algebras' by Assem, Simson and Skowronski. The question is the following: Consider the following ...
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1answer
29 views

Lipschitz continuity of $\sqrt{A}$

Let $U \subset\mathbb{R}^n$ be an open set, $\mathbb{S}^n$ be the set of all $n\times n$ symmetric real matrices, $A:U\to \mathbb{S}^n$ be a uniformly Lipschitz continuous function. Suppose $\exists ...
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1answer
41 views

How to find the matrix base change in this vector space?

Good Morning. I'm trying to solve this exercise: I took the quotient vector space formed by $\operatorname V=\mathbb{R}^5/\langle(3,2,4,-2,5)\rangle$. After starting the vector $x=(3,2,4,-2,5)$ I ...
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If a vector subspace is in a union of other subspaces, then it's contained in one of them [duplicate]

Problem: Let $V$ be a finite dimensional vector space and $V_1,\ldots,V_n\subset V$ vector subspaces. Show that if $W\subset V$ is a vector subspace and $$W\subset V_1\cup\cdots\cup V_n,$$ then $...
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168 views

A four square matrix and $A^5$(a raised to the power of 5)$=0$. Then $A^4=?$

A four square matrix and $A^5$(a raised to the power of 5)$=0$. Then $A^4=$ $I$(identity matrix) $-I$ $0$ $A$ My attempt: You can use the characteristic equation $$ A^2-Tr(A)A+I_2\det{A}=O_2,$$ ...
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Find the eigenvalues the block matrix $M=\begin{bmatrix}A+2D & A \\ A & D \end{bmatrix}$

Let $A$ be any square matrix with eigenvalues $\lambda_1,\lambda_2,\cdots,\lambda_n$ and $D$ is a diagonal matrix with entries $d_1,d_2,\cdots,d_n$, then how can one find the eigenvalues of the ...
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1answer
132 views

Finding the Orthogonal Complement to a subspace

So suppose I have a vector space, $V$ which is all continuous functions on $[0,1]$. Additionally, we have an inner product over $V$ where $\langle f,g \rangle = \int_{0}^{1}f(x)g(x)dx$. Now suppose I ...
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Relationship between geometric multiplicity, algebraic multiplicity and left and right eigenvectors of a matrix

The following statement is from the book Matrix Analysis by Horn and Johnson. An eigenvalue λ with geometric multiplicity 1 can have algebraic multiplicity 2 or more, but this can happen only if ...
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1answer
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Help solving the equation [on hold]

I'm stuck and don't know what to do next to solve this equation. Any hints? $y(x_2−x_1)−y_1(x_2−x_1)=x(y_2−y_1)−x_1(y_2−y_1)$
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2answers
21 views

Which one is equation of tangent

Is equation of tangent plane $z=f(x_{0},y_{0})+f_{x}(x_{0},y_{0})(x-x_{0})+f_{y}(x_{0},y_{0})(y-y_{0} ) $ or $z=f_{x}(x_{0},y_{0})(x-x_{0})+f_{y}(x_{0},y_{0})(y-y_{0} ) $ In my book I found ...
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1answer
44 views

Properties of transfer matrices and their traces

I'm having difficulties understanding some arguments in my statistical mechanics lecture and would like to make them more rigorous by proving some properties. For the Ising model on a lattice we ...
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2answers
298 views

Algebraic multiplicity = geometric multiplicity?

I was wondering if algebraic multiplicity was equal to the geometric multiplicity. If the matrix (of size $n\times n$) is diagonalisable, i.e. the characteristic polynomial is of the form $$p(x)=(x-\...