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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Simultaneous diagonalizability of commuting unitary operators

I'm trying to prove the following: If $S\colon V\to V$ and $T\colon V\to V$ are unitary linear transformations on unitary space $V$ ($\dim V=n$, $n$ is finite), such that $ST=TS$, then they have ...
4
votes
2answers
167 views

Prove that if $g(t)$ is relatively prime to the characteristic polynomial of $A$, then $g(A)$ is invertible

I'd like to write to you two problems that I tried to solve, I'm not sure of the solution of the first. Let $A\in M_n (F)$ be a matrix and $g(t)\in F(t)$ a polynomial, $P_A(t)$- the characteristic ...
7
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2answers
5k views

Why are all nonzero eigenvalues of the skew-symmetric matrices pure imaginary?

Assume that $A$ is an $n\times n$ skew-symmetric real matrix, i.e. $$A^T=-A.$$ Since $\det(A-\lambda I)=\det(A^T-\lambda I)$, $A$ and $A^T$ have the same eigenvalues. On the other hand, $A^T$ and ...
2
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1answer
121 views

Principal Component Analysis problem

I'm not sure this is the right place but here I go: I have a database of 300 picture in high-resolution. I want to compute the PCA on this database and so far here is what I do: - reshape every image ...
3
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1answer
159 views

$A\in M_{n}(C)$ and $A^*=-A$ and $A^4=I$

Let $A\in M_{n}(C)$ be a matrix such that $A^*=-A$ and $A^4=I$. I need to prove that the eigenvalues of A are between $-i$ to $i$ and that $A^2+I=0$ I didn't get to any smart conclusion. Thanks
12
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1answer
335 views

Is this similarity between trees and vector space bases just a coincidence?

A vector space basis is a set of vectors that span the space and is linearly independent. It is well-known that for finite dimensional vector spaces this is equivalent to: The set is minimal with ...
2
votes
1answer
286 views

Help with a proof - Dimension of a family of commuting operators over a vector space of finite dimension

I am working on a proof of the following problem and I am not sure if I am making the appropriate reduction to a smaller space. Let $F$ be a subspace of the vector space $\mathbb{C}^{4\times4}$ such ...
2
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1answer
726 views

Determining the Jordan form of a matrix given the invariant factors

I am trying to recover the Jordan normal form of a matrix given a list of invariant factors and was wondering if I am proceeding correctly in constructing the Jordan blocks. Let $F = \mathbb{C}$ and ...
0
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1answer
101 views

Solution to a system

I'm looking for a general method to find at least one solution to the system below $$a_0 = a_1x_1 + \cdots + a_nx_n$$ $$b_0 = b_1x_1 + \cdots + b_nx_n$$ $$c_0 = c_1x_1 + \cdots + c_nx_n$$ $$d_0 = ...
9
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1answer
463 views

Detecting symmetric matrices of the form (low-rank + diagonal matrix)

Let $\Sigma$ be a symmetric positive definite matrix of dimensions $n \times n$. Is there a numerically robust way of checking whether it can be decomposed as $\Sigma = \mathcal{D} + v^t.v$ where $v$ ...
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1answer
120 views

Comparing rank-deficient matrices

I have two $3\times 3$ matrices each of rank 2. How can I check that they are equivalent? What definition of equivalence is there in this case?
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2answers
1k views

Minimal polynomial and diagonalizable matrix

Let $B,C$ square matrices above a field,and $D$ Rectangle matrix in the correct size above the same field. $A=\begin{pmatrix} B &D \\ 0& C \end{pmatrix}$ I need to prove that if $A$ is ...
5
votes
2answers
1k views

Thoroughly understand the concepts and formulas in Linear Algebra

In linear algebra(I'm the beginner so far), I feel the concepts and ideas are based on on another, like layer by layer, and there're many relations among the properties. For example, Matrix ...
9
votes
1answer
1k views

Does equality of characteristic polynomials guarantee equivalence of matrices?

I have a qualifying exam coming up in a couple days and I am just trying to understand some pathological examples I have in my notes. I will list a similar problem which I know the solution to and ...
3
votes
1answer
614 views

Irreducible characteristic polynomial of a linear transformation

I am trying to understand why the linear transformation $T$ corresponding to the companion matrix of the minimal polynomial of an irreducible polynomial over $\mathbb{Q}[x]$ (for instance $x^3-x-1$) ...
5
votes
2answers
384 views

Why are these examples striking?

The question is from an exercise in Gilbert Strang's Linear Algebra and its Applications. The powers $A^k$ approach zero if all $|\lambda_i|<1$, and they blow up if any $|\lambda_i|>1$. ...
5
votes
1answer
149 views

Can the following probabilistic argument about eigenvalues be made rigorous?

Consider the following $n \times n$ matrix $$ \left( \begin{matrix} 1/2 & 1/2 & 0 & 0 & 0 & 0 \\ 1/2 & 0 & 1/2 & 0 & 0 & 0 \\ 0 & 1/2 & 0 & 1/2 ...
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2answers
241 views

Why the eigenvectors of the Laplacian of a Ring graph are sinusoids?

The eigenvectors of the Laplacian of a Ring graph with $n$ vertices are: $x_k(u) = \sin(2\pi ku/n)$ and $y_k(u) = \cos(2\pi ku/n)$ for $1\leq k \leq n/2$. The explanation according to Spielman's ...
0
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1answer
165 views

Minimal polynomial for a bijective linear transformation

Let $V$ be a vector space over a field $F$ which is not algebraically closed and $T:V \rightarrow V$ be a linear transformation. I am just trying to understand if injectiviity can be used to study ...
2
votes
1answer
161 views

Left ideals of the ring of endomorphisms

Let $X$ be a finite dimensional vector space over $K$, where $K=R$ or $K=C$, let $P=\operatorname{End}_K X$ be the ring of all endomorphisms of the space $X$, and let $I$ be a left ideal of $P$. Is ...
3
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1answer
98 views

Under what circumstances can $ALA^{-1}+BLB^{-1}=2 XLX^{-1}$ be solved for the linear operator $X$ only depending on $A$ and $B$ but not on $L$?

For $A,B,L$ linear operators, when is there a linear operator $X\{A,B\}$ such that $$ALA^{-1}+BLB^{-1}=2 XLX^{-1}$$ can be solved independently for all $L$ only depending on $A$ and $B$?
4
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2answers
123 views

About positive-definiteness of a matrix Q depending on matrices H and P

$H + H^T$ is a positive definite matrix and $P$ is also a positive definite matrix. Will $Q = PH + H^TP$ be a positive definite matrix? In my calculations, it is not positive definite. But I read a ...
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4answers
297 views

about $\operatorname{SO}(2)$ group

When $$ A =\begin{pmatrix} a & b \\ c & d \end{pmatrix},\quad a,b,c,d \in \mathbb{R}, \quad A^2 -2aA + I = 0 $$ Is $ A \in \operatorname{SO}(2)\; $ if $A^n \in SO(2)\;$ for some $n \in ...
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1answer
64 views

Curious about how to work out a formula to solve a simple scenario

I have 838 rows of data which each have a calculated column showing me the duration from creation to resolution. The average of all those rows duration is 63 hours. I want to say if I move x rows up ...
0
votes
1answer
920 views

Finding distance between a vector and a subspace

Let the inner product of $V=M_3(\mathbb{R})$ to be $\langle A,B\rangle =\mathrm{trace} (A^tB)$. I need to fine the distance between $X= \begin{pmatrix} 2 & 1 &0 \\ 0&-1 &2 \\ ...
2
votes
2answers
5k views

Simultaneous diagonalization

Let $V$ be a vector space from a finite dimension and let $T,S$ linear diagonalizable transformations from $V$ to itself. I need to prove that: a. If $TS=ST$ so every eigenspace $V_\lambda$ of $S$ ...
0
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3answers
416 views

Similarity of an invertible matrix to a diagonal matrix

I am trying to solve some sample test questions and am looking for shortcut to a problem. Question: True or False "Every invertible matrix $A \in \mathbb{Q}^{n \times n}$ is similar to a diagonal ...
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1answer
159 views

Counterexample involving the minimal polynomial of a linear operator

Let $F$ be a field and let $V = F^{4\times4}$ be the vector space of $4x4$ matrices over $F$. For $A \in F^{4\times4}$, define $T_A : V \rightarrow V$ by $T_A(B) = AB$ for each $B \in V$. ...
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votes
3answers
21k views

How to tell if a set of vectors spans a space?

I want to know if the set $\{(1, 1, 1), (3, 2, 1), (1, 1, 0), (1, 0, 0)\}$ spans $\mathbb{R}^3$. I know that if it spans $\mathbb{R}^3$, then for any $x, y, z, \in \mathbb{R}$, there exist $c_1, c_2, ...
0
votes
0answers
111 views

Number of diagonals perpendicular to given diagonal of the unit n-cube

I think I know how to do this problem but just want to check that my reasoning is correct. "Let $U$ denote the unit cube in $\mathbb{R}^n$, and let $D$ be a given diagonal of $U$. How many other ...
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3answers
2k views

Converting this recursive function into a non-recursive equation

I am trying to convert the following recursive function to a non-recursive equation: $$ f(n) = \begin{cases} 0,&\text{if n = 0;}\newline 2 \times f(n -1) + 1,&\text{otherwise.} ...
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3answers
200 views

$A\in M_n(\mathbb{R})$ symmetric matrix , $A-\lambda I$ is Positive-definite matrix- prove: $\det A\geq a^n $

Let $a>1$ and $A\in M_n(\mathbb{R})$ symmetric matrix such that $A-\lambda I$ is Positive-definite matrix (All eigenvalues $> 0$) for every $\lambda <a$. I need to prove that $\det A\geq a^n ...
3
votes
3answers
1k views

$A^{T}A$ positive definite then A is invertible?

Say if $A$ is an $n \times n$ matrix, why is it that if $A^{T}A$ is positive definite, the matrix $A$ is then invertible? All I know is $A^{T}A$ gives a symmetric matrix but what does $A^{T}A$ is ...
0
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1answer
185 views

writing a subgroup of the affine group as a semidirect product

I read the following: The group $\mathbb Z +...+\mathbb Z=\mathbb Z^n$ of covering translations and $S_n$ acting on $\mathbb R^n$ by permuting coordinates, both lie in $aff(\mathbb R^n)$ the group ...
0
votes
1answer
113 views

Find greatest value of an angle

I am not sure how to go about the last bit of this problem. With respect to an origin $O$, the points $P$ and $Q$ have variable position vectors $p$ and $q$ respectively, given by, $$ ...
4
votes
2answers
167 views

Why does this method for solving matrix equations work?

I have this assignment: Given: $A = \begin{pmatrix} 2 & 4 \\ 0 & 3 \end{pmatrix}$ $C = \begin {pmatrix} -1 & 2 \\ -6 & 3 \end{pmatrix}$ Find all B that satisfy $AB = ...
3
votes
1answer
542 views

Polynomials and Linear Operators

Let $p$, $q$, and $r$ be polynomials such that $p(x) = q(x)r(x)$, and let $T$ be a linear operator on a vector space $V$. Is there a simple way to show that $p(T) = q(T)r(T)$ ?
2
votes
1answer
246 views

How to get $p(A)=0$ without Cayley-Hamilton theorem when A is diagonalizable?

Suppose $A$ is a $n\times n$ real matrix, and the characteristic polynomial is $p(\lambda)=\det(A-\lambda I_n).$ By the Cayley-Hamilton theorem, $p(A)=0$. Here is my question: Assume that ...
2
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1answer
103 views

On the sets of injective/surjective linear mappings between Euclidean spaces

Denote by $\mathcal L'(\mathrm R^n,\mathrm R^m)$ and $\mathcal L_\prime (\mathrm R^n,\mathrm R^m)$ the subsets formed by the surjective and the injective mappings, respectively, of the normed ...
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2answers
297 views

What's the “geometry” in “geometric multiplicity”?

The geometric multiplicity of an eigenvalue is defined as the dimension of the associated eigenspace, i.e. number of linearly independent eigenvectors with that eigenvalue. Here are my questions: ...
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1answer
140 views

Vandermonde and curve interpolation

I hesitate here because of an understanding with a calculation problems. I want to calculate an interpolation using the Vandermonde matrix. see: http://en.wikipedia.org/wiki/Vandermonde_matrix My ...
0
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3answers
70 views

Proving that if $\vec{k}$ is a least solutionof $A\vec{x}=\vec{b}$, then $c\vec{k}$ is for $A\vec{x}=c\vec{b}$

Say if $\vec{k}$ is a least squares solution to $A\vec{x}=\vec{b}$ such that $A\vec{x}=\vec{b}$ has no solutions. My guts feel tell me that for all constants $c\vec{k}$, where $c\in\mathbb{R}$, it is ...
2
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1answer
293 views

Best way to sketch a proof of the primary decomposition theorem for F[x]-modules

Hi I am trying to find if there is a more simple way to prove the following theoreom without going through the two page proof in the Linear Algebra textbook I have been using. Let $V$ be a ...
1
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1answer
110 views

Rank of a proper sub-module

I am looking for a counterexample to a simple question about proper sub-modules. The book I am reading mentions the following theorem but implys that there are pathological examples related to the ...
0
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1answer
1k views

How can I find equivalent Euler angles?

I have a rotation over time represented as a series of Euler angles (heading, pitch and bank). I'd like to represent the individual rotation curves as continuously as possible. An anyone help me ...
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2answers
932 views

What are the rules for basic algebra when modulo real numbers are involved

That is real numbers modulo an integer. I'm just interested in shuffling around +-*/ operations. If a concrete example helps here's my current problem. (I'm from a programming background so there's ...
0
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2answers
321 views

Recursive matrices raised to the kth power

For a $n\times n$ matrix $A$ that has independent eigenvectors, I want to raise the power of $A$ recursively like $A^{1}\vec{u_{0}}=\vec{u_{1}}$ and then to find out $\vec{u_{k}}$, I could use ...
0
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1answer
374 views

How many orthogonal matrices are there over a given finite ring or field?

I want to know how many $2\times 2$ orthogonal matrices exist over the ring $\mathbb{Z}_n$ or the field $\mathbf{F}_p$. And how many $2\times 1$ orthogonal vectors exist over the ring $\mathbb{Z}_n$ ...
3
votes
4answers
336 views

Linear Transformation

I am new to linear algebra so I apologize beforehand for those of you who are math wizards. I need to know and understand why or why not for the following question: T or F; $T (x,y) = (2x+5y,-x+2)$ ...
3
votes
2answers
666 views

What if the only eigenvectors of $A$ are multiples of $x=(1,0,0)^T$?

The question is from an exercise in Gilbert Strang's Linear Algebra and its Applications: Suppose the only eigenvectors of $A$ are multiples of $x=(1,0,0)$. True or false: (a) $A$ is not ...