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 ...

learn more… | top users | synonyms

3
votes
2answers
504 views

Dimension of $GL(n, \mathbb{R})$

Why is the group $GL(n, \mathbb{R})$ of dimension $n^{2}$?
2
votes
1answer
833 views

Finding determinant of an infinite matrix

I am trying to find the simplest way to get an expression of the determinant of the following infinite matrix as m tends to infinity. $$ \left[\begin{array}{cccccc} 1 & a_{1} & 0 & ...
0
votes
1answer
121 views

Linear transformation over two vector spaces

We know linear transformation $T$ over two vector spaces $V,W$ and the rank $r$ (dimension of image $T$) of $T$ . We also know matrix representation $M$ of $T$. Why rank of $M$ is $r$?
2
votes
3answers
1k views

(good) numerical inversion of an almost singular matrix: is it possible?

Ok, so I know that if I have a system that looks like Ax=b it is foolish to solve it by solving for the inverse of A and one should instead use something like ...
2
votes
1answer
87 views

Bounding the rank of matrix after adding one row and one column

I have a matrix \begin{align} Y = \left( \begin{array}{cc} c & \mathbf{b}^{\top} \newline \mathbf{b} & X \end{array} \right) \end{align} Both $X$ and $Y$ are positive semi-definite ...
2
votes
1answer
94 views

Why is this matrix positive?

I run into the following fact: Let $X$ be $n \times n$ positive semidefinite matrix, and let $a$ be an $n$ dimensional vector consisting of diagonal entries of $X$ (i.e. $a_i = X_{ii}$). Suppose ...
1
vote
1answer
797 views

Matrix Transformation Onto?

A linear transformation $T\colon\mathbb{R}^3\to\mathbb{R}^2$ whose matrix is $$\left(\begin{array}{ccc} 1 & 3 & 3\\ 2 & 6 & -3.5+k \end{array}\right)$$ is onto if and only if ...
7
votes
2answers
855 views

Can a basis for a vector space be made up of matrices instead of vectors?

I'm sorry if this is a silly question. I'm new to the notion of bases and all the examples I've dealt with before have involved sets of vectors containing real numbers. This has led me to assume that ...
0
votes
0answers
48 views

Maintaining the line with the 2D iterands

Suppose a linear system is given $$AX=B,$$ where $A\in\mathbb{R}^{n\times n}$ is a symmetric strictly diagonal matrix, and $X, B\in\mathbb{R}^{n\times 2}$. Therefore, the 2D Jacobi iterative solver is ...
4
votes
4answers
225 views

Eigenvalues of a matrix

Let $A$ be a square matrix of order, say, $4$. Consider the matrix $$B=\left( \begin{array}{ccc}A &I \\ I & A\end{array} \right)$$ where $I$ is the identity matrix of order $4$. Let ...
0
votes
1answer
322 views

Determine the steady state from a discrete dynamic system with only the eigenvalue of the diagonalized transition matrix.

For a discrete dynamical system I know that the transition matrix A is diagonalizable with the eigenvalues of 0.1, 0.2, and 0.3. The question asks what I can say about the long term behaviour of the ...
2
votes
1answer
50 views

The form of the states on an algebra of $n\times n$ matrices with complex entries

I have a question concerning $n \times n$ matrices. Denote by $M_n(\mathbb{C})$ the algebra of all $n \times n$ matrices with complex entries. Let $\phi$ be a state on $M_n(\mathbb{C})$ i.e. a linear ...
0
votes
1answer
992 views

Some theorem about block matrix determinants with symmetric inner matrices?

I could do this problem with bruteforce but I think there must be some elegant theorem that helps to calculate the determinant with the block matrix (here having symmetric matrices inside) such as: ...
1
vote
2answers
1k views

Determinant of Large Matrix with Gauss rule?

$$A=\begin{pmatrix} 1 & -1 & 0 & 2 \\ 2 & 1 & 0 & 0 \\ 1 & 1 & 2 & 2 \\ 0 & 0 & 1 & 1 \\ \end{pmatrix}$$ With the lower determinant method, I ...
2
votes
2answers
569 views

Prove 2 diagonal matrices of size n are similar.

I have the following two matrices: A = \begin{pmatrix}1&0&...&0&0\\ 0&2&...&0&0\\ 0&0&...&n-1&0 \\ 0&0&...&0&n\end{pmatrix} B = ...
0
votes
2answers
400 views

proof that a continuous additive homomorphism $\mathbb{R}^n\to\mathbb{R}^m$ is $\mathbb{R}$-linear

How can we prove that an additive homomorphism $ \Phi \colon \mathbb{R}^{n}\to \mathbb{R}^{m} $ is $\mathbb{R}$-linear. i.e. satisfies $ \Phi (rv)=r \Phi (v)$ for $ r\in \mathbb{R} $ and $ v \in ...
1
vote
2answers
248 views

eigenvector proof

Let $V$ be a vector space and $T : V\to V$ a linear transformation with the property that $T(W)\subset W$ for every subspace $W$ of $V$. How can we prove that there is an element $\lambda$ in the ...
3
votes
1answer
161 views

A is diagonalizable

Let $ f(x)= x^{n}-nx+1 $ and let $A$ be an $ n \times n $ matrix with characteristic polynomial $f$. I am going to prove that if $n> 2$ then $A$ is diagonalizable over the complex numbers. If ...
1
vote
2answers
58 views

Bounding $\Vert M^{-1} \Vert$ and eigenvalues of $M$.

We work in $\mathbb{R}^n$. Let $M$ be an $n\times n$ matrix with $$ x^TMx \geq k\Vert x\Vert^2 $$ for all $x \in \mathbb{R}^n$, where $k>0$. I want to show that $\Vert M^{-1} \Vert \leq ...
2
votes
1answer
115 views

A simple question about determinant

Let $a_i=[a_{i1},a_{i2},\ldots,a_{in}]\in \mathbb{R}^n$, for $i=1,\ldots,n-1$. How to prove that $$ \sum_{i_1,\ldots,i_{n-1}=1}^n \varepsilon_{i,i_1,\ldots,i_{n-1}} a_{1,i_1} a_{2,i_2}\cdots ...
3
votes
2answers
154 views

When is $V$ a direct sum of a subspace $S$ and the set $(V-S)\cup\{\mathbf{0}\}$?

In general, for a vector space V, $\dim(V)< \infty$, with subspace $S \subseteq V$ and the set $W :=(V-S) \cup \{ 0\}$ we find that $V \neq S \oplus W$. For example, if $V=\mathbb{R}^2$ and ...
2
votes
1answer
168 views

Justification of an Algebraic Manipulation

While reading a proof i came across this step which i could not understand. The chunk below is part of bigger expression, but in the interest of reducing noise i am just posting the sub expression if ...
1
vote
1answer
578 views

Simultaneous Iteration, Convergence to Eigenvectors

I have a question about the simultaneous iteration. I am currently working for an exam and I do not understand this step (taken from Numerical Linear Algebra from Trefethen/Bau): For the power ...
2
votes
2answers
89 views

Is the quotient of two (identically behaved) linear recurrent sequences still recurrent? Still linear recurrent?

Let $(u_n)$ and $(v_n)$ be two linear recurrent sequences, satisfying the same recurrence relation. Suppose that $v_n$ is nonzero for all large enough $n$, then we can set $w_n=\frac{u_n}{v_n}$. I ...
6
votes
0answers
112 views

Piecewise Affine Bijections of $\mathbb{R}^n$

I have a min-max function $f:\mathbb{R}^n\to\mathbb{R}^n$ of the form $$f(x) = \min_{i=1,\dots,n}\max_{j=1,\dots,n}(\alpha_{ij}^Tx + \beta_{ij})\quad\text{where each } \alpha_{ij}\in ...
-2
votes
2answers
818 views

Solving linear equation using maple

Use Maple to find the general solution to the following system of linear equations: $$\begin{eqnarray} 2x1 + x2 - x3 + 3x4 = 2\\ x1 + 2x2 - x4 = -1\\ 3x1 + 2x2 - 2x3 + x4 = 1\\ ...
1
vote
2answers
122 views

need help with proof of simple theorem - linear algebra / analysis

i'm stalled in my attempt to prove that $(1-a^2)^{\frac{1}{2}} - (1-b^2)^{\frac{1}{2}}$ goes to zero faster than $|A-B|$, where A,B are vectors in $\mathbb{R}^n$ with $a=|A| \leq 1$ and $b=|B|\leq 1$. ...
2
votes
2answers
217 views

derivative of a map of vector space of matrices

Question: Let $A_{n\times n}$ be the vector space of all real $n\times n$ matrices. If I define a map $$g:A_{n\times n}\rightarrow A_{n\times n}$$ such that: $$g\left ( X \right )=X^{2}$$ In ...
0
votes
1answer
65 views

eigenvalue sign of $M - \lambda_{k} I$

Let $M$ be a symmetric $n\times n$ tri-diagonal matrix, with positive values in its main diagonal. and let $\mathbf{1} \in R^n$ be the vector of all 1, such $M \mathbf{1} = 0$ Suppose $M$ has ...
3
votes
2answers
460 views

Conjugacy classes in a matrix group

Consider the matrix group $PGL_{2}(\mathbb{F}_{q})$ for $q$ odd. Why is it that $\begin{pmatrix} -1 & 0\\ 0 & 1\end{pmatrix}$ has $q(q + 1)/2$ elements in its conjugacy class while ...
1
vote
2answers
1k views

Inner Product Spaces - Triangle Inequality

I have to show that: For an inner product space $V$, $\|x + y \| = \|x\| + \|y\|$, for all $x$, $y \in V$ if and only if one of the vectors $x$ or $y$ is a scalar multiple of the other. I am ...
3
votes
2answers
359 views

Why is the group action on the vector space of polynomials naturally a left action?

When seeking irreducible representations of a group (for example $\text{SL}(2,\mathbb{C})$ or $\text{SU}(2)$), one meets the following construction. Let $V$ be the space of polynomials in two ...
1
vote
1answer
76 views

questions about Wronskians.

Let $u_1, \ldots, u_n, u, v$ be functions. If $W(u, u_1, \ldots, u_n)=W(v, u_1, \ldots, u_n)$, is $u=v$? Here $W(u_1, u_2, u_3)$ for example is defined by $$ \pmatrix{ u_1& u_1'& ...
0
votes
2answers
679 views

Self - adjoint and Unitary operator

For W be a finite dimensional subspace of an inner product space V. Given V is the direct sum of W and its orthogonal complement W'. For a map U defined on V as U (v + v') = v - v' , for all v in W , ...
3
votes
1answer
699 views

Unitary and Upper Triangular Matrix

I am trying to prove that a matrix that is both "unitary and upper triangular" must be a diagonal matrix. I am thinking that the fact that columns of all unitary matrices form an orthonormal basis ...
5
votes
0answers
221 views

Finding conjugacy classes of $PGL_{2}(\mathbb{F}_{q})$

Assume $q$ is odd. How does one go about finding the conjugacy classes of $PGL_{2}(\mathbb{F}_{q})$? I know that for $GL_{2}(\mathbb{F}_{q})$, one can consider the possible Jordan Normal Forms of the ...
1
vote
3answers
117 views

Question about norms of a matrix when exchanging two of its rows

Assume I exchange two rows of a square complex $n\times n $ matrix. Are the Euclidean norm and the Hilbert-Schmidt norm of the new matrix (obtained from the first one by exchanging two of its rows) ...
1
vote
1answer
364 views

Mixture problem? Repeatedly extract/replace X units of fluid (with fluid #2) to attain X% mixture ratio?

Let's say I have a 100 liter container of water. I extract 10 liters of water, and replace it with 10 liters of juice. Now, the container is now 10% juice / 90% water. I can keep repeating this to ...
5
votes
1answer
408 views

Hilbert Schmidt Norm-Rank-inequality

Problem: Let $A_{n.n}$ be square complex matrix. Prove the following: $$\left \| A \right \|=\left \| A \right \|_{HS}\Leftrightarrow rank(A)\leqslant 1$$. Where $\left \| . \right \|_{HS} $ is the ...
1
vote
2answers
354 views

Reverse Cuthill McKee Ordering and Solution of systems of Linear Equations

I just learned about the RCM. I am trying to solve a problem that is a result of fluid dynamics and chemistry so I have a very large sparse matrix. I also learned reducing the bandwidth would ...
2
votes
2answers
80 views

Finding a set of similarity representatives for a collection of $2\times 2$ complex matrices

Here is the problem statement: Find a subset $Y$ of $X:=\{A \in \text{Mat}_{2\times 2}(\mathbb{C})\ |\ A^4=A\}$ so that the following occur: If $A$ $\in$ $X$ then $\exists$ $B$ $\in$ $Y$ such that ...
1
vote
1answer
163 views

Proof on the inequality involving matrix splitting and trace operator

Suppose positive definite matrices $V, B, D\in\mathbb{R}^{n\times n}$ are given, where $D$ only contains diagonal entries of $V$, i.e., $D=diag(V)$, and $X, G\in\mathbb{R}^{n\times 2}$. Could the ...
0
votes
1answer
170 views

SVD Question on MIT OCW Linear Algebra Lecture 29

Apparently Gilbert Strang Linear Algebra lecture 29 does SVD on $A$ being calculated using $$ A^TA = V \Sigma^T \Sigma V^{T} $$ and $$ AA^T = U\Sigma \Sigma^TU^T $$ Example is $$ A = ...
2
votes
3answers
146 views

Explanation to the details of the proof that $F[x]$ is not finite-dimensional.

I have several questions concerning the proof. I don't think I quite understand the details and motivation of the proof. Here is the proof given by our professor. The space of polynomials $F[x]$ is ...
1
vote
1answer
146 views

Null Space and Range of Particular kind of Operator on Hilbert Space

Let $H$ be the real separable Hilbert space with orthonormal basis $\{e_n\}$ and consider the operator $T:H \times H \to H \times H$ given by $$T(\sum a_ne_n, \sum b_ne_n) = \sum A_n(a_ne_n, ...
5
votes
0answers
134 views

Isomorphism of representations of the symmetric group

This might be a silly question, but I don't understand why the solution to the following problem implies the result: Let $A = \mathbb{C}S_d$ and let $c_{\lambda}$ denote the Young symmetrizer (with ...
0
votes
1answer
812 views

Hermitian matrices that commute

My question is: If $A$ and $B$ are two Hermitian matrices, and $AB$ is also a Hermitian matrix, then how do prove that both $A$ and $B$ are diagonalizable through the same unitary matrix (i.e the ...
2
votes
1answer
185 views

Is there a closed-form solution to this linear algebra problem?

$A$ and $B$ are non-negative symmetric matrices, whose entries sum to 1.0. Each of these matrices has $\frac{N^2-N}{2}+N-1$ degrees of freedom. $D$ is the diagonal matrix defined as follows (in ...
9
votes
3answers
4k views

Get Transformation Matrix from Points

I have built a little C# application that allows visualization of perpective transformations with a matrix, in 2D XYW space. Now I would like to be able to calculate the matrix from the four corners ...
0
votes
1answer
222 views

Invertible complex square matrix

Here is a small question: I was reading a problem in a textbook where the question is: Prove that $A$ is invertible? (where $A$ is a complex square matrix). In the solution: the author proved that ...