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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Orthogonal diagonalisation of a 4x4 matrix

Can somebody help me to orthogonally diagonalise the matrix $\begin{bmatrix}0 & 0 & 0 & 1\\0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & ...
2
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2answers
17 views

Show a $T:V \to V$ exists such that $T$ maps subspaces to each other, under a few general conditions

Let $V = \mathbb{R^n}$ and let $U_1,U_2,W_1,W_2 \subset V$ be subspaces of $V$ of dimension $d$ such that $\dim(U_1 \cap W_1) = \dim(U_2 \cap W_2) = k,$ where $k \leq d \leq n.$ Prove that there ...
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Given block matrix $M$, show determinant relationship between $M$ and the block elements of $M.$

Given that $M = \begin{pmatrix} A & B \\ C &D \end{pmatrix}$ and $M^{-1} = \begin{pmatrix} P & Q \\ R & S \end{pmatrix},$ where $A, B,\dots$ are $k \times k$ matrices, show that ...
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0answers
11 views

Existence of an inner product under which all elements of a finite subgroup of $GL(V)$ are isometries

I'm currently hopelessly stuck on an exercise in linear algebra and I could use some hints. Let $V$ be a finite-dimensional vector space over the reals and let $G$ be a finite subgroup of ...
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1answer
20 views

Prove that there is a base of $\mathbb R^4$ made of eigenvectors of matrix $A$

Matrix of linear operator $\mathcal A$:$\mathbb R^4$ $\rightarrow$ $\mathbb R^4$ is $$A= \begin{bmatrix} 1 & 1 & 1 & 1 \\ 1 & 1 & -1 & -1\\ 1 ...
5
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1answer
54 views

What are vector norms used for?

I'm currently working with a computer science problem that requires me to build vectors that can return their own norms. Based on Wolfram Alpha's description, I think I have an idea of how this is ...
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2answers
50 views

How to find an upper triangular with $\ U^2 = I $ which gives $\ U $ is its own inverse

It's obvious that $\ I $(identity matrix) of any size $\ N $ satisfies$\ I^2 = I$ so that $\ I $ is its own inverse. However if we consider $\ N = 2$ and attempt to find such a triangular matrix $\ ...
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1answer
15 views

conjugate matrix $A^*=A$.

Let $A\in M_n$ and $A^*$ is conjugate of $A$. Suppose that $A=A^*$ prove that $det(A)\in R$. My work: $$det(A) = det(A^*) = det(A^{t \ conjudate.})$$ now i want to say that $$det(A^{t \ conjudate.}) ...
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1answer
13 views

Finding the non-unique inverse dot-product

I have an equation $\vec{x} \cdot \vec{y} = K$ over $\mathbb{R}^n$. I want to solve for $\vec{y}$ in terms of parameters. I tried taking the pseudo-inverse of $\vec{x}$, but I end up with a larger ...
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3answers
34 views

Rank of a linear transformation?

I'm given a linear trasnformation: $T:M_2\rightarrow M^{\:}_2$ which is defined such as $T\left(X\right)=AX$, where $A$ is: $$A=\begin{pmatrix}1&-2\\ -2&4\end{pmatrix}$$ Find the rank of T? ...
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1answer
12 views

What is the effect of applying toeplitz matrix on both sides of + semi-def matrix

I have a positive semi-def matrix $\mathbf{A}$ that has an eigen value decomposition $\mathbf{A}=\mathbf{V\Lambda V^T}$. I have a real toeplitz matrix $\mathbf{Q}$. Can I say anything definite about ...
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1answer
10 views

LP: add extra costs in the objective function for every variable which is not equal to $0$

I am trying to optimise an LP problem but extra costs should be added if a variable is larger than $0$. For example, if we have the following objective function: $$\text{minimize} \qquad 2X_1 + 3X_2 ...
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1answer
18 views

Confusion with some theorems leading to the canonical decomposition of an operator

Let $\Bbb K$ be a field and $R = \Bbb K[X]$ be the ring of polynomials with coefficients in $\Bbb K$. Let $\cal L_{\Bbb K}$$(V)$ denote, for a finite dimensional $\Bbb K$-vector space $V$, the set of ...
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1answer
40 views

Expected number of times to get arbitrary arrangement of coins

I'm thinking about a question: We consider tossing coins repeatedly. Using $+1$ to denote front and $-1$ back, given a positive interger $m$ and $\sigma=(\sigma_1,\dots,\sigma_m)$ where ...
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2answers
26 views

Prove/Disprove question on matrix vector multiplication and linear independence [on hold]

If $\left\{Bv_1, \ldots , Bv_k\right\}$ is a linearly independent set in $\mathbb{R}^k$ where $B$ is a $k \times n$ matrix, then $\left\{v_1, \ldots ,v_k\right\}$ is a linearly independent set in ...
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4answers
98 views

Compute $\det{T}$ where $T(X)=AX+XA$

Consider the linear transformation $T:V\to V$ given by $T(X) = AX + XA$, where $$A = \begin{pmatrix}1&1&0\\0&2&0\\0&0&-1 \end{pmatrix}.$$ Compute the determinant $\det ...
3
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1answer
41 views

If a linear transformation $T$ has $z^n$ as the minimal polynomial, there is a vector $v$ such that $v, Tv,\dots, T^{n-1}v$ are linearly independent

Let $T: V \to V$ with the minimal polynomial $z^n$. Prove that there's a vector $v$ such that $v, Tv, T^2v, ..., T^{n-1}v$ are linearly independent. The way I did it orginally was not allowed. No ...
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2answers
58 views

Let $V$ be a $K$-vector space, $f: V \rightarrow V$ a linear map. Under some condition, show that $v, f(v),…,f^n(v)$ are linearly independent.

Let $V$ be a $K$-vector space, $f: V \rightarrow V$ a linear map. Let $v \in V$. May a number $n ≥ 0$ exist, so that: $f^n(v) \not= 0$ and $ f^{n+1}(v) = 0$. Show that $v, f(v),...,f^n(v)$ are ...
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1answer
38 views

Visualization of the dual space of a vector space

I am wondering what the motivation was for defining a dual space of a vector space, and how to visualize the dual space. I'm asking since it doesn't seem to me to be intuitive to deal with such a ...
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1answer
27 views

Bound for symmetric part of matrix

Let $A $ be an $n \times n $ matrix such that $AA^T \geq x^2I, x\geq 0 $, which means that the matrix $AA^T-x^2I$ is positive semidefinite. Can we show that $(A+A^T)/2 \geq xI$? Thanks
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3answers
220 views

Eigenvalues of Matrix vs Eigenvalues of Operator

I'm having some trouble reconciling the concept of eigenvalues of operators with eigenvalues of matrices: Say you have an $n\times n$ matrix $A$. It represents a linear operator $T:V\to V$ with ...
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1answer
18 views

A fact about symmetric matrices and square roots

Is it true that if $A$ is symmetric then any square root is symmetric? I can't prove this using basic symbolic computation, so what if we insist that $A$ is diagonalizable, or even positive definite?
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1answer
16 views

Continuous dependence of matrix elements

I've stumbled upon several solution of linear algebra problems which use notion of "continuous dependence" of matrix polynomials on matrix elements. For instance (translated, so any inaccuracies are ...
2
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2answers
65 views

Prove that $ 0 \leq B^T(C+B\mathcal{I}_V^{-1}B^T)^{-1}B \leq \mathcal{I}_V $

In Walter Zulehner's article Nonstandard norms and robust estimates for saddle point problems page 546 at the very top, he writes that $$ 0 \leq B^T(C+B\mathcal{I}_V^{-1}B^T)^{-1}B \leq \mathcal{I}_V ...
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1answer
8 views

Find an orthogonal base of a bilinear form on a field of characteristic 2

Let $K$ be a Field of characteristic $2$. On $V=K^2$ the symmetric bilinearform $\beta (x,y) = x_1y_2+x_2y_1 $ is defined. Now i have to either find an orthogonal base of $V$ or show that such a ...
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1answer
50 views

Problems with Apostol's calculus

I am self teaching myself so I couldn't ask any teacher but here are somethings in Chapter 1 that I can't understand. In the book Area is defined axiomatically but some parts of it are not just making ...
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1answer
18 views

Are the following statements true for elementary row operations?

I was wondering if the following statements about ERO's (elementary row operations) are mathematically correct. Any help and insight is appreciated. Any additional facts to the table is also welcomed. ...
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1answer
26 views

Finding a polynomial to satisfy a matrix equation

Is there a canonical way of finding a polynomial $p$ such that $$ p\left(\begin{bmatrix} 1& 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 ...
135
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1answer
6k views

How does one prove the matrix inequality $\det\left(6(A^3+B^3+C^3)+I_{n}\right)\ge 5^n\det(A^2+B^2+C^2)$?

Question: let matrices $A,B,C\in M_{n}(C)$ be Hermitian and Positive definite matrices, such that:$$A+B+C=I_{n}$$ Show that: $$\det\left(6(A^3+B^3+C^3)+I_{n}\right)\ge 5^n\det(A^2+B^2+C^2)$$ ...
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1answer
22 views

Determinant comparison about skew-symmetric matrices

Suppose $S$ is a real skew-symmetric matrix, show that $\det(I+S) \geq 1$, where equality holds iff $S=0$. My idea is to define a function $f(t)=\det(I+tS)$, for a fixed $S \neq 0$, and then show ...
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3answers
676 views

Positive symmetric matrices and positive-definiteness

Is a symmetric real matrix with diagonal entries strictly greater than $1$ and off-diagonal entries positive but strictly less than $1$ necessarily positive-semidefinite?
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2answers
2k views

Stochastic matrices?

Let $A$ be a symmetric stochastic matrix, such that the sum over the columns, for each row, is 1, and all elements are positive. $A$ dimensions are $n \times n$ Let's say that $B$ is a matrix which ...
3
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2answers
2k views

$A^TA$ is always a symmetric matrix?

Through experience I've seen that the following statement holds true: "$A^TA$ is always a symmetric matrix?", where $A$ is any matrix. However can this statement be proven/falsefied?
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1answer
3k views

Help with proving that the transpose of the product of any number of matrices is equal to the product of their transposes in reverse

Specifically I am trying to show that (An)T = (AT)n where A is an mxm square matrix and n is a positive integer. This is where I'm stuck: To prove the theorem I would like to show that ((An)T)ij = ...
5
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2answers
50 views

Numerical Calculation of Eigenvalues of a large real Symmetric tridiagonal matrix

If I have an $N \times N$ matrix where every entry is zero except for along the super-diagonal and sub-diagonal, where the each entry is the conjugate of the last, like the following $5 \times 5$ ...
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0answers
13 views

Boundary of $\sum_{j}x_j(x_j-x_i)$ for $x_i \in[0,1]$

Does $\sum_{j}x_j(x_j-x_i)$ for $x_i\in[0,1]$ and $0\le i,j\le N-1$ have a upper and lower boundary? And how to calculate them? Thanks!
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2answers
45 views

What is the geometrical interpretation of determinant of a matrix in general? [duplicate]

My question is simple (and maybe I am wrong asking this question even) what is the geometrical interpretation of determinant of a matrix in general ? I could not think anything.
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1answer
21 views

Calculate the matrix from minimal polynomial and eigenspaces

I need to find a matrix $A \in M(6,\mathbb C)$ that satisfies following: $e_1+e_2+e_3\in \ker(L_A-3\cdot \operatorname{id})^3 \setminus \ker(L_A-3*id)^2$ $\operatorname{span}(e_1+e_4,e_5+e_6)= ...
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0answers
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Perturbation of the principal eigenvector of a PSD matrix

Setting: I have a $n \times n$ PSD matrix $A$ and $\tilde{A}=A+E$ be its symmetric perturbation such that $\|E\|_2=\epsilon.$ Let $(\lambda,u)$ be the principal eigenvalue, eigenvector pair of $A$ and ...
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2answers
27 views

number of solutions of system of two equations, two unknowns (Matrix)

How can we find that when a system of two equations, two unknowns has Infinite Solutions. I want a solution with matrix. I know this method (which is not with matrix): $ax + by = c$ $a'x+ b'y = c'$ ...
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2answers
29 views

What does the sum of subsets of a vector space mean?

On page $57$ of Second edition of Hoffman Kunze, the authors write Definition If $S_1, S_2, \dots, S_k$ are subsets of a vector space $V$, the set of all sums $$\alpha_1 + \alpha_2 + \dots+ ...
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2answers
11k views

Fast way to calculate Eigen of 2x2 matrix using a formula

I found this site: http://www.math.harvard.edu/archive/21b_fall_04/exhibits/2dmatrices/index.html Which shows a very fast and simple way to get Eigen vectors for a 2x2 matrix. While harvard is quite ...
2
votes
3answers
29k views

Image and Kernel of a Matrix Transformation

So I had a couple of questions about a matrix problem. What I'm given is... Consider a linear transformation $T: \mathbb R^5 \to \mathbb R^4$ defined by $T( \overrightarrow{x} )=A\overrightarrow{x}$, ...
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1answer
14 views

Solving for the particular solution of a system of differential equations

Consider the IVP $\vec{y}'= \begin{bmatrix}0 & -1\\-1 & 0\end{bmatrix}\vec{y} + \begin{bmatrix}t \\e^{2t}\end{bmatrix}$ $\vec{y}(0) = \begin{bmatrix}1 \\1\end{bmatrix}$ The complementary ...
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0answers
16 views

Are there explicit formulas for the eigenvalues and eigenvectors of a generic 4x4 density matrix?

I have a 4x4 density matrix all of whose elements are nonzero. Its form is $$\begin{pmatrix} a & b & c & d \\ b^* & e & f & g \\ c^* & f^* & h & j \\ ...
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1answer
16 views

“All phase plane solution points remain stationary as $t$ increases”?

Consider the linear system $y′(t)=A\vec{y}(t)$, where $A$ is a real $2\times2$ constant matrix with repeated eigenvalues. All phase plane solution points remain stationary as $t$ increases. I ...
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0answers
35 views

On Modular exponents.

If $g^z=h\bmod p$ where $z$ is the only unknown then is it possible to find in polynomial time $g^{z^t}\bmod p$ at some $t\neq 0$ or $1\bmod(p-1)$?
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4answers
45 views

How come the matrix $I - A$ where $\|A\| < 1$ is nonsingular? [on hold]

As the title says, why is the matrix $I - A,$ where $I$ is the identity and $\|A\| < 1$, nonsingular?
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1answer
14 views

Verify if linear combination of vectors is in lattice

Let $\mathbf{a}, \mathbf{b}$ and $\mathbf{c}$ be vectors in $\Bbb{R}^3$. How do I verify if there is a linear combination of them that belongs in the lattice $\mathcal{L}(B)$ where $B = \{(1,1,1)\}$?
0
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2answers
178 views

How to calculate eigen values and evectors of Jordan Block matrix

If there is a Jordan block matrix with $A(i,i) = a$ for all $i=1$ to $n$, $A(i,i+1)=b$ for all $i=1$ to $n-1$ and $A(j,k)=0$ otherwise. What will be eigenvalue and eigenvector of $J$? To calculate ...