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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2
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2answers
80 views

trace of $(A^4-A^3)$

$A$ be a $2\times 2 $ complex matrix with $\det A=-6$ and trace(A)=1, I need to find the trace of $(A^4-A^3)$ so $A^2-A-6I=0$ so $(A^4-A^3)=6A^2$ how to proceed next?
0
votes
1answer
104 views

Surjection/Injection in Product of Linear Transformation

I wish somebody could help me with this one. Let $S: \mathbb{R}^3\to \mathbb{R}^4$ and $T: \mathbb{R}^4\to \mathbb{R}^3$ be linear transformation such that $T*S$ is the identity map of $R^3$. Then ...
2
votes
1answer
65 views

If $A$ is nilpotent, $f_A$ defined by $f_A(X) = XA-AX$ is also nilpotent

I am trying to solve this problem: Let $A \in M_n(\mathbb C)$ be a nilpotent matrix. Define a linear map $f_A:M_n(\mathbb C) \rightarrow M_2(\mathbb C)$ by $f_A(X) = XA-AX$. Prove that the ...
2
votes
1answer
58 views

Linearly independent for cosinus

Let $\{\lambda_k\}_{k=1}^n$ be a sequence of real numbers. How can I show that $\{\cos \lambda_k x\}_{k=1}^n$ are linearly independent in $C(-1,1)$ where $\lambda_k \neq \lambda_j ~~\forall~~ k\neq ...
1
vote
0answers
38 views

vector summation with constrains

In $R^N$ space, there are $m$ vectors $v_1, v_2, \cdots, v_m, v_i\in R^N$. The direction of $v_i$ can be inverted, i.e. $v_i=-v_i$. The objective is to get a new vector $v=\sum_{i=1}^{m}v_i$ whose ...
5
votes
1answer
186 views

Positive semidefinite matrix problem

This is a simple question, at least, looks like. Let $x\in\mathbb{R}^n$ and consider the matrix $C$ such that $C_{ij}=|x_i|+|x_j|-|x_i-x_j|$, show that $C$ is positive semidefinitive. I could prove ...
1
vote
2answers
515 views

The possible set of eigenvalues of a $4\times 4$ skew symmetric, orthogonal matrix

The possible set of eigenvalues of a $4\times 4$ Real skew symmetric, orthogonal matrix is $1.\{\pm i\}$ $2.\{\pm i,\pm 1\}$ $3.\{\pm 1\}$ $4.\{\pm i,0\}$ As it is real skew ...
3
votes
1answer
503 views

approximating diagonal of inverse sum of low rank and diagonal matrices

I was wondering if there is any theorem or algorithm to approximate the diagonal elements of the inverse of sum of low rank symmetric positive semi-definite and non-negative diagonal matrix. Let me ...
0
votes
1answer
93 views

Simple Projection Proof

Let $V = U \oplus W$, and define $P_{U,W} \in L(V)$ where $P_{U,W}$ denotes the projection onto $U$ with null space $W$. I am trying to verify three properties and would like some feedback and help. ...
1
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0answers
69 views

Is conversion to a quadratic form possible?

Is it possible to represent the expression $(Ax+b)^TS(Ax+b)$ in the form of a quadratic form $(x+e)^TF(x+e) + K$ for some $e$, $F$ and $K$, and matrix $A$ with dimensions $m\times n$, $m>n$. Note: ...
4
votes
1answer
1k views

Pseudo inverse of a product of two matrices with different rank

Let $V$ be an $n \times n$ symmetric, positive definite matrix (of rank $n$). Let $X$ be an $n \times p$ matrix of rank $p$. Define $A^- = (A^\top A)^{-1} A^\top$ as the pseudo inverse of $A$ when ...
2
votes
1answer
103 views

How to force unitary Euclidean norm in a complex matrix by multiplication with a diagonal matrix

I need to solve the following problem: Suppose a non-sparse, non-singular complex matrix $\mathbf{P}$. If I want to force all rows in $\mathbf{P}$ to present unitary Euclidean norms by multiplying ...
3
votes
1answer
1k views

Relation between eigenvectors of covariance matrix and right Singular vectors of SVD, Diagonal matrix

I have a $m \times n$ data matrix $X$, ($m$ instances and $n$ features) on which I calculate the Covariance matrix $C$ and perform eigenvalue decomposition. so $C=W \Sigma W'$ where $W$ are the ...
4
votes
0answers
56 views

Subgroup consisting of unipotent elements centralizes a flag

Is there a reasonably elementary and short proof that a subgroup of consisting of unipotent matrices over a field centralizes a flag? By elementary, I mean accessible to students who have had a ...
0
votes
1answer
175 views

Inverse of a sum of matrices using SVD

Given $$ A^TA = U\Sigma U^T \,\,\,\text{(SVD)} $$ Where, by definition of SVD, $U$ is an orthogonal matrix and $\Sigma$ is a diagonal matrix. Suppose I want to compute $$ (A^TA + \gamma D)^{-1} $$ ...
4
votes
3answers
83 views

Proving an inequality in Eigenvalues of a matrix

If $\lambda_1$, $\lambda_2$, $\lambda_3$ are the eigenvalues of the matrix : $$ \begin{pmatrix} 26 & -2 & 2 \\ 2 & 21 & 4 \\ 4 & 2 & 28 \\ ...
1
vote
1answer
130 views

Nilpotent matrices and similarity.

True or false: Let $A,B\in M_5(\mathbb{C})$. $A$ and $B$ are similar iff they have same nilpotent index. One direction is clear, but the second is not true, right? Thank you!
2
votes
5answers
129 views

Eigenvectors problem

So I have a question which I do not know how to solve. Let $A,B \in M_{n}(\mathbb{C})$ and let $AB=BA$. I have to prove that there exists a common eigenvector for both $A$ and $B$. How? I have no ...
0
votes
2answers
60 views

what is the solution if it exist?

In linear algebra we know that if $A$ (is an invertible matrix) is given, then the solution for the system $Ax=b$ is $x=A^{-1}b$ for any $b$. Can any one tell me what will be the solution $x$ (if it ...
2
votes
1answer
32 views

Confusion related to matrix multiplication

I am having this simple confusion.Lets consider a multivariate gaussian distribution with mean $\mu$ and precision matrix $K$. Then the exponential term is $$(x-\mu)' K (x-\mu)$$ If I open the above ...
0
votes
3answers
71 views

Determinant of eigenvalues

Is this matrix considered real with non real eigenvalues? $\begin{bmatrix} \cos x&-5\\ 5&\cos x\\ \end{bmatrix}$ When I made this matrix up, at the moment it looked like a real matrix with ...
0
votes
2answers
1k views

Finding reflection transformation matrix

I have two 3 dimensional points. $A [x_1, y_1, z_1]$ and $B [x_2, y_2, z_2]$. I need to find a transformation matrix which when multiplied to $A$ will give me $B$ and when multiplied by $B$ give me ...
3
votes
2answers
3k views

Strictly diagonally dominant matrices are non singular

I try to find a good proof for invertibility of strictly diagonally dominant matrices (defined by $|m_{ii}|>\sum_{j\ne i}|m_{ij}|$). There is a proof of this in this paper but I'm wondering ...
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2answers
36 views

hermitian transformations problems

So I have a question which I do not know how to solve. $V$ is a inner product space over $\mathbb{R}$ $S,T:V \rightarrow V $ are a linear maps we know that $SS^{*}=S^{*}S$ and that $TT^{*}=T^{*}T$ ...
3
votes
3answers
90 views

An uncountable subset in $\Bbb{R}^n$ in which each $n$ elemented subset is a base

Using Zorn's Lemma I can show that there is an uncountable subset in $\Bbb{R}^n$ such that any $n$-elemented subset is independent. Can you construct such a set ??
2
votes
1answer
303 views

How can I calculate the correct rotation output when input needs modification?

I am making a GPS for vehicles in a game and I need them to turn accordingly to the rotation between the ending point and starting point. I have calculated the rotation, and before I can apply the ...
0
votes
4answers
457 views

Dual basis for $\{1,x,x^2,x^3\}$ [closed]

How one can find dual basis for $\{1,x,x^2,x^3\}$? Thank you.
1
vote
1answer
97 views

Proof involving Images and Kernels

Let $A:V \rightarrow V$ be a linear map. Prove: $$\operatorname{Im}A \cap \operatorname{Ker}A=\{0 \} \Rightarrow \forall n \in \mathbb{N}:\operatorname{Ker}A^{n}=\operatorname{Ker}A$$ Here's my work: ...
9
votes
1answer
309 views

A conjecture about vector space

Let $V$ be a $(r+1)$-dimensional vector space, and $p$ be a positive integer and $1\leq p\leq r-1$. Let $$X=\{v_1,\cdots,v_{2r+1-p}\}\subseteq V$$ be a finite set containing $(2r+1-p)$ different ...
3
votes
2answers
293 views

Suppose $T^2$ is diagonalizable and $\ker{T}=\{0\}$, and every eigenvalue of $T^2$ is nonnegative. Show that $T$ is diagonalizable.

Suppose $T^2$ is diagonalizable and $\ker{T}=\{0\}$, and every eigenvalue of $T^2$ is nonnegative. Show that $T$ is diagonalizable. Of course $T$ is an operator on $V$. It seems to me that if I take ...
5
votes
2answers
299 views

Why is a projection matrix symmetric?

I am looking for an intuitive reason for a projection matrix of an orthogonal projection to be symmetric. The algebraic proof is straightforward yet somewhat unsatisfactory. Take for example another ...
0
votes
3answers
846 views

The definition of “span” and related theorem.

In wikipedia and the most of the linear algebra texts, The definition of the span is following as. "Given a vector space V over a field K, the span of a set S of vectors (not necessarily finite) is ...
1
vote
0answers
798 views

What are these physicists talking about? Dyadic green function?

I am interested in a mathematical explanation(in the sense that you say for example: is it a mapping from A to B) what a dyadic green function and the unit dyad actually is? I am reading this as ...
2
votes
1answer
157 views

Cyclic error correcting code

Notation: I denote the field with $2$ elements by $\mathbb{F}_2$. For a vector $u\in\mathbb{F}_2^m$, I write $w(u)$ for the Hamming weight of $u$ (the number of components equal to $1$ in $u$). ...
3
votes
1answer
181 views

How to simplify the characteristic polynomial of a given matrix?

Reading through this paper I've come across a statement that I don't follow, could someone give some pointers/hints? Let $A$ be the $2n\times 2n$ matrix given by $$A=(I_n\otimes F)+(G\otimes ...
0
votes
1answer
102 views

Question about diagonalizable matrix

I have a question regarding a part in a book of Pattern Recognition I'm reading. In appendix of the book it is said that a positive definite symmetric matrix $A$ can be diagonalized by the ...
4
votes
2answers
93 views

Visual interpretation of cv + (1 - c)w

Question - Draw the line of all combinations that has $c\mathbf{v} + d\mathbf{w}$ and $c + d = 1$. Solution - All combinations with $c + d = 1$ are on the line that passes through $\mathbf{v}$ and ...
1
vote
1answer
220 views

How to prove a matrix A and a linear operator T, where T is left multiplication by A, have the same characteristic values.

I'm having a hard time putting this into a proof. It's posed as a question in Section 6.2 of Hoffman and Kunze, asking whether it is true or not. It seems obviously true, since if T is left ...
3
votes
2answers
298 views

Dimension of Ker(T)

For a positive integer $n > 1$, let $ T: \mathbb{R}^{n\times n} \to \mathbb{R}$ be the linear transformation defined by $T(A) = \operatorname{Tr} {(A)}$, where $A$ is an $n \times n$ matrix with ...
3
votes
2answers
3k views

What is the relationship between the null space and the column space?

Just looking at some tutorial videos, I'm noticing somewhat of a trend... but it wasn't spelled out explicitly, so I'd like to verify if this theory of mine is correct... (forgive my horribly un-exact ...
2
votes
6answers
446 views

What exactly does linear dependence and linear independence imply?

I have a very hard time remembering which is which between linear independence and linear dependence... that is, if I am asked to specify whether a set of vectors are linearly dependent or ...
3
votes
2answers
156 views

Problem with matrix and vector norms

I already try to multiply by a orthogonal matrix in both sides, multiply by $q$ and $d$, factor, expand...nothing works. This problem comes from Demmel's book, Applied Numerical Linear Algebra. Let ...
2
votes
0answers
41 views

Projection between subspaces of $\mathbb{R}^d$

Suppose I have two $k$ dimensional subspaces of $\mathbb{R}^d$, which I call $A$ and $B$. Let $\text{proj}_S (x)$ denote the projection of an $x \in \mathbb{R}^d$ onto subspace $S$. Now is given that ...
5
votes
6answers
238 views

If $A\vec{v}=\lambda\vec{v}$, then does $A=\lambda$?

When my professor started teaching eigenvectors and eigenvalues the other day, the very first thing I noticed was the fact that $A\vec{v}=\lambda\vec{v}$ (assuming that the equation is satisfied under ...
0
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1answer
99 views

Is this still considered $\{0\}$

Is the following solution to the matrix a zero subspace? (Assume that the last column of zeros is the constant portion of the matrix) I'm working on some kernel problems, and if a linear ...
-1
votes
1answer
44 views

Linear map of plane [closed]

Consider linear map of plane which transform line $x=1$ in herself. Show: $1$ is eigenvalue of this linear map $(0,1)$ is eigenvector of this linear map
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vote
1answer
38 views

If $T:\mathbb{R}^2\to \mathbb{R}^2$ and $T(1, -1)^T=(0,1)^T, \space T(1,1)^T=(1,0)^T$, find $T(1, -7)^T.$

If $T:\mathbb{R}^2\to \mathbb{R}^2$ and $T(1, -1)^T=(0,1)^T, \space T(1,1)^T=(1,0)^T$, find $T(1, -7)^T.$ where $T$ is a linear transformation. I assume I have to use the property of linear ...
1
vote
1answer
63 views

Finding $\operatorname{rank} T$ where $T(u)=(u,v)w$

Let $V=\mathbb{C}^{n \times n}$. Defining $T(u)=(u,v)w$ for $v,w\in V$, what is $\operatorname{rank} T$? Thank you!
1
vote
1answer
203 views

LU Decomposition of a matrix $A$.

I'm supposed to compute the L,U factors s.t. A = LU. A = \begin{bmatrix} 0.1 & 1 \\ 1 & 1 \\ \end{bmatrix} I get solutions that require a permutation matrix but can't seem to solve one ...
0
votes
1answer
194 views

Is there fast approximation of the n-th power of diagonalizable matrix A?

My thoughts on the subject. Because of diagonalizability $A$ can be written as $A = PDP^{-1}$ and then $A^n = PD^nP^{-1}$ here $P$ is matrix of eigenvectors and $D$ is diagonal matrix with ...