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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1answer
187 views

Congruent and Similar Matrices

I tried to solve the following question: Find 2 matrices A and B in M2(C) such that A is similar to B but not congruent. Find 2 matrices A and B in M2(C) such that A is congruent to B but not ...
4
votes
1answer
2k views

Understanding how to find a basis for the row space/column space of some matrix A.

I just need some verification on finding the basis for column spaces and row spaces. If I'm given a matrix A and asked to find a basis for the row space, is the following method correct? -Reduce to ...
2
votes
0answers
106 views

relation between minimal polynomial and jordan normal form

I just solved some exercises on minimal polynomials and i remember that there is a relation between the minimal polynomial and the jordan normal form. But my question is the following : knowing the ...
0
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5answers
69 views

Eigenvalues and eigenvectors - help

I was reading my lecture notes on using matrices to solve ODEs and came across this and am having trouble understanding it: (Please note that the two eigen values that are calculated are 2 and 0 ...
3
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3answers
1k views

how to extend a basis

This is a very elementary question but I can't find the answer in my book at the moment. If I have, for example, two vectors $v_1$ and $v_2$ in $\mathbb R^5$ and knowing that they are linear ...
2
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8answers
420 views

Do matrices have a “to the power of” operator?

Well I was sure that saying "$A^3$" (where $A$ is an $n\times n$ matrix) is nonsense. Sure one could do $(A\cdot A) A$ But that contains different operators etc. So what did my prof mean by the ...
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2answers
431 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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3answers
81 views

A basic doubt on linear dependence and basis vectors

I see that linear independence/dependence is defined for a finite set of vectors in books. But, basis vectors are always independent and they need not be finite. Is the definition consistent ?
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2answers
48 views

A basic question on subspace

The reason subspace was introduced that we want to concentrate on a reduced subset of a vector space for which still it is a vector space under those operations. Then there should be a notion of ...
4
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2answers
192 views

Hermitian matrices and great circles

I am considering parameterised curves in an $n$-dimensional complex vector space, given by the solution to the discrete Schrödinger equation: $$ |\psi\rangle(t) = e^{-iHt}|\psi_0\rangle, $$ Where $H$ ...
3
votes
2answers
295 views

Fundamental theorem of algebra: a proof for undergrads?

The fundamental theorem of algebra is the statement that a complex polynomial of positive degree has at least one root. I do not know complex analysis but I searched for proofs of the statement and ...
3
votes
1answer
228 views

On minimizing matrix norm

Given $m\geq n \geq k$ and $A \in M^{m\times n}(\mathbb{C})$, the problem asks to evaluate $$\inf_{\textrm{rank}(B)\leq k}||A-B||$$ and gives condition on $A$ making the above minimum is taken by ...
2
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1answer
63 views

Linear Algebra; computational problems

I dug this problem up from an old exam. I am not asking how to solve them, but I want to get a "feel" for the problem. I could technically solve this brutally,I just want to develop some problem ...
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0answers
35 views

Algebraic Issue

I am a totally no good at mathematics, And that's why I don't know if the question already exist. X1=1000 Y1=800 A1=700 B1=500 X2=1200 Y2=1000 A2=? B2=? PS: X1/Y1=X2/Y2, A1/B1=A2/B2 Actually I ...
2
votes
1answer
448 views

Computing orthogonal projection onto range space of a given matrix

I have matrix $A =\begin{bmatrix} 2 & -2 & 0 & 0 & 0 & 0 \\ -2 & 2 & 0 & 0 & 0 & 0 \\ 0 & 0 & 2 & -2 & 0 & 0 \\ 0 & 0 & -2 ...
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0answers
60 views

What is the significance of the matrix in the LAPACK logo?

This is the LAPACK linear algebra library logo: What is the significance of this matrix?
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votes
4answers
176 views

if rank(A)=trace(A)=1 than A is a projection

Let A be a complex square matrix. Suppose that rank(A)=trace(A)=1. Prove that A is a projection. I have no idea how to prove it, so will be thankful for any help.
4
votes
1answer
104 views

If AB is a projection then BA is a projection

Given two complex matrices $A$ and $B$, and knowing $AB$ is a projection, prove or provide a counterexample that $BA$ is a projection. Stuck with this question, need help. According to the matrix ...
3
votes
1answer
53 views

system of matrix equations

I have the equation $ \mathbf{x}^T A \mathbf{x} = b $, where $b$ is a scalar, $\mathbf{x}$ a vector of size $M$, and A a matrix of size $M\times M$. $b$ and $\mathbf{x}$ are given. How many such ...
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1answer
53 views

reflection representation of isometry

I am reading the book Naive Lie Theory It proves that any isometry of $R^n$ that fixed the origin O is the product of at most n reflections in hyperplanes through O. The proof is elementary and by ...
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2answers
151 views

modulo 2 linear equation algorithm

Given is a set of modulo 2 linear equations. I'm looking for a performant algorithm that solves these linear equations. The Row Reduction to the ...
2
votes
1answer
167 views

What does it mean/imply that all my singular values are ones?

Suppose I apply SVD (singular value decomposition) on some real-valued matrix $M$, that is, $M = USV^T$. Now, if $S$ is an identity matrix, what does it mean? Does $M$ have some special properties? ...
3
votes
2answers
66 views

action of $O(n,\mathbb{R})$ on ${S}^{n-1}$

Is the action of $O(n,\mathbb{R})$ on ${S}^{n-1}$ transitive? I think this is true as orthogonal matrices are supposed to rotate and keep the length fixed, but how do I prove this? EDIT: Based on ...
0
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1answer
74 views

If I have two orthonormal vectors in $\mathbb R^3$…

If I have two orthonormal vectors in $\mathbb{R}^3$, $\mathbf{u}$ & $\mathbf{v}$ (so both unit length). and another vector $\mathbf{w}$ that is orthogonal to $\mathbf{u}$ (not to $\mathbf{v}$ ...
3
votes
3answers
84 views

Proving $(f^*)^*=f$

I solved it like this : $$\langle (f^*)^*(v),w \rangle=\langle v,f^*(w)\rangle=\langle f(v),w\rangle$$ My lecture notes gave a proof with some more steps. Now i'm not sure, maybe i messed something. ...
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2answers
179 views

linear transformation, ker(T) and im(T) - question from final exam

Assume $T:V\to V$ is a linear transformation, $\mathrm{dim} V = n$. Let $v$ be a vector of $V$ such that for $1\leq k\leq n : v, T(v), \dots , T^{k-1}(v)$ : they are all NOT zero, but $T^k(v) = 0 $. ...
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vote
1answer
297 views

Normal distributed rotation matrix in 3D

How can I compute normally distributed 3D rotation matrices with Mathematica? For 2D matrices I would sample a normal distributed angle and directly create a rotation matrix with: ...
1
vote
1answer
122 views

Solve covariance matrix of multivariate gaussian [duplicate]

This is a practical, and basic question. I have a multivariate Gaussian in $M$ dimensions with center $\mu$ (known, lets assume $0$) and some points $p$ where I have the value of $$ \ln(L)= ...
2
votes
1answer
101 views

Prove that $C^n(\mathbb{R})$ is a subspace using induction.

Let $V$ be the set of all functions $f:\mathbb{R}\to\mathbb{R}$. Prove by induction that $C^n(\mathbb{R})$ is a subspace of $V$. I feel that this could be shown directly without much issue using the ...
0
votes
1answer
81 views

How to solve matrix differentiation

How to solve: $${\frac{\partial}{\partial{\vec{\mu}^T}}\left\{\frac{1}{a}\sum^n_{i=1}z_i(\vec{y_i}-\vec{\mu})^TB^{-1}\right\}}$$ where $a$ and $z_i$ are elements, $\vec{y_i}$ and $\vec{\mu}$ are ...
3
votes
2answers
92 views

Significance of order in linear algebra

This question has been bothering me for some time now. All theorems in my linear algebra notes use ordered lists of vectors. For example consider the theorem here: A spanning list of vectors may ...
4
votes
1answer
58 views

When polynomial is power

$P(x)$ ia a polynomial with real coefficients, and $k>1$ is an integer. For any $n\in\Bbb Z$, we have $P(n)=m^k$ for some $m\in\Bbb Z$. Show that there exists a real coefficients polynomial ...
4
votes
1answer
78 views

A is symmetric iff A=P-Q, where P,Q are positive definite matrices

Show that an $n \times n$ real matrix $A$ is symmetric iff $A$ can be written as $$A=P-Q$$ where $P$ and $Q$ are some $n \times n$ positive definite matrices. Can there be anything said similarly ...
8
votes
1answer
325 views

Making non-singular matrices singular

What is the minimum value of $k$ such that every non-singular $n\times n$ real matrices can be made singular by switching EXACTLY $k$ entries with ZERO ?
5
votes
1answer
142 views

Subspace spanned by eigenvectors is a subalgebra

Let $L$ be a Lie algebra over an algebraically closed field and let $x\in L$. Prove that the subspace of $L$ spanned by the eigenvectors of $\operatorname{ad}x$ is a subalgebra. Suppose the ...
2
votes
2answers
51 views

Stumped with Matrices

(a) How do we find $A^{-1}$? (b) If $XA=B$, how do we use (a) to find $X$? Any guidance would be much appreciated!
2
votes
0answers
72 views

Changing a simplex grid to an orthogonal grid.

Well I'm on my way in learning noise, a computer algorithm that's used to create real life structures and textures, etc. The noise I'm trying to learn is Simplex Noise, I already have it in the ...
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vote
0answers
38 views

Fourier transform of a sequence by a matrix

Let $n$ be a positive integer and $H$ the Hilbert space $\ell^2(\mathbb Z^n,\mathbb C^n)$. For $u\in H$, denote by $\mathcal{F}(u)$ the Fourier transform of $u$, defined by $\displaystyle ...
4
votes
2answers
351 views

If the determinant and the trace of a matrix are the same, then can we show they have the same eigenvalues?

If $\det A = \det B$ and $\operatorname{tr}A=\operatorname{tr} B$, then can we show $A$ and $B$ have the same eigenvalues?
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vote
2answers
40 views

Finding a basis for $U=\{A\in\mathbb{M}_{22}\mid A^T=-A\}$.

Find a basis for $U=\{A\in\mathbb{M}_{22}\mid A^T=-A\}$. $\mathbb{M}_{22}$ denotes the set of all $2 \times 2$ matrices. This question appeared on an examination I wrote yesterday. Does a basis ...
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vote
1answer
40 views

Direct sum and similarity question

Let $A,C$ be $n\times n$ matrices and $B,D$ be $m\times m$ matrices. Assume $A\oplus B$ is similar to $C\oplus D$. Then, are $A&C$ similar and $B&D$ similar respectively? If they are, how ...
2
votes
2answers
286 views

How to find the degrees between 2 vectors when I have $\arccos$ just in radian mode?

I'm trying to write in java a function which finds the angles, in degrees, between 2 vectors, according to the follow equation - $$\cos{\theta} = \frac{\vec{u} \cdot ...
0
votes
2answers
674 views

Definition of a Field of Characteristic $n$?

Let $V$ be a vector space over a field of characteristic not equal to $2$. Prove that $\{u, v\}$ is linearly independent with $u, v$ being distinct if and only if $\{u+v, v-v\}$ is linearly ...
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0answers
45 views

Is there any simple way to write down permutations set-theoretically?

Let $\{J_{n_1}(\lambda_1),...,J_{n_m}(\lambda_m)\}$ and $\{J_{l_1}(\mu_1),...,J_{l_k}(\mu_k)\}$ be finite sequences of Jordan blocks with entries in a field $F$. Let $A\triangleq ...
2
votes
1answer
46 views

Why in this proof we get $\alpha \geq 0$?

I've solved the following problem: "Let $u,v \in \mathbb{R}^n$ with $u \neq 0$ be such that $|u+v|=|u|+|v|$ (euclidean norm), show that there's $\alpha \in \mathbb{R}$ with $\alpha \geq 0$ such that ...
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0answers
49 views

A separation lemma in a real vector space

A lattice $N$ is a free $\mathbb{Z}$-module of finite rank. Let $V$ be the real vector space $N\otimes_\mathbb{Z} \mathbb{R}.$ A cone is a set $\sigma = \{ r_1 v_1 + \ldots + r_k v_k \in V : r_i\geq 0 ...
3
votes
1answer
96 views

Eigenvalues of $\operatorname{ad}x$

Let $x\in \operatorname{gl}(n,F)$ have $n$ distinct eigenvalues $a_1,\ldots,a_n$ in $F$. Prove that the eigenvalues of $\text{ad }x$ are precisely the $n^2$ scalars $a_i-a_j$ ($1\leq i,j\leq n$), ...
2
votes
1answer
205 views

Make a matrix invertible

Suppose that $ A $ is $n \times n $ matrix with a 1-dimensional null-space. Show that we can choose vectors $u$ and $v$ so that the linear transformation \begin{equation} B = A + u \otimes v^t ...
6
votes
1answer
51 views

Is this proof that the vectors are colinear correct?

I was solving the following exercise: "Let $x,y \in \mathbb{R}^n$ be nonzero such that if $z$ is orthogonal to $x$ then $z$ is orthogonal to $y$. Prove that $x$ and $y$ are colinear". My idea was: ...
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votes
3answers
86 views

Finding the rank of a certain general matrix

If I have an $m \times n$ matrix $A$ and an $n \times m$ matrix $B$ such that $AB=I_m$, how do I go about calculating the rank of $A$ and the rank of $B$? Any clues would be much appreciated!