For any topic related to matrices. This includes: systems of linear equations, eigenvalues and eigenvectors (diagonalization, triangularization), determinant, trace, characteristic polynomial, adjugate, transpose, Jordan normal form, matrix algorithms (e.g. LU, Gauss elimination, SVD, QR), ...

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1answer
41 views

Nilpotent degree $2$ 'families' of $4\times 4$ matrices

$\def\b{\begin{bmatrix}}\def\e{\end{bmatrix}}$ Are $\b0&0&a&b\\0&0&c&d\\0&0&0&0\\0&0&0&0\e$ and it's transpose, $a,b,c,d\in \Bbb C$ the only nilpotent ...
7
votes
1answer
193 views

Extending a Chebyshev-polynomial determinant identity

The following $n\times n$ determinant identity appears as eq. 19 on Mathworld's entry for the Chebyshev polynomials of the second kind: $$U_n(x)=\det{A_n(x)}\equiv \begin{vmatrix}2 x& 1 & 0 ...
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3answers
34 views

Finding a basis for all $2\times2$ matrices A such that…

Finding a basis for all $2\times2$ matrices A such that $$\left[ \begin{matrix} 1 \hspace{5pt} 2 \\ 0 \hspace{5pt} 3 \end{matrix} \right]A=\left[ \begin{matrix} 0 \hspace{5pt} 0\\ 0 \hspace{5pt} 0 ...
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0answers
18 views

General Form of $R$-matrix for Gram-Schmidt

I got this particular definition of Gram-Schmidt in class today, and wanted to run it by some folks, especially the definition of the change-of-basis matrix $R$: "Let $V$ be a subspace of ...
2
votes
1answer
42 views

In terms of matrices: $\forall v\in V,\phi(\phi(v))=0$

$\phi: V\to V$( a linear operator here) How to interpret $\forall v\in V,\phi(\phi(v))=0$ in terms of matrices? Can I have some hint? I suppose $\phi(V)= \begin{bmatrix} ...
0
votes
1answer
56 views

Solving for an unknown matrix

Solve for $X$: $$ \left[ \begin{array}{cc} 2 & 5 \\ 0 & 0 \\ \end{array} \right ] - X \left [ \begin{array}{cc} -7 & 8 \\ 7 & -7 \\ \end{array} \right ] \;\;=\;\; I $$ $X$ is an ...
4
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4answers
755 views

Is the identity matrix the only matrix which is its own inverse?

I just gave a proof for this question. Here's my follow up question: Let $A \in \ \mathbb{M}_n(\mathbb{F})$ where F is a field and there exists $n\in N$ where $A^n$= I. In the case where n=1,2, ...
0
votes
2answers
108 views

Given a Transformation Matrix $T$, find $T$ relative to a new basis $\beta$

$T(a_1,a_2,a_3) = (3a_1+a_2,a_1+a_3,a_1-a_3)$. $(a_1,a_2,a_3)^T$ is written with regards to the standard basis. We can figure out $T$ in matrix form by calculating $T(a_1),T(a_2), T(a_3)$. That's ...
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1answer
48 views

Linear Algebra: Compute Area of Parallelogram

I have this one Linear Algebra question that is asking me to compute the area of a parallelogram defined by 4 vectors. Here is the question: Let $\vec{u}=\begin{bmatrix}a\\b\end{bmatrix}$ and ...
1
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1answer
33 views

2d projection of a 3d image

I am having a problem where I have a $2$D object which can move in $3$-dimensional space about a fixed point (the origin). I want to rotate this object using Euler angles and axes of rotation. If ...
0
votes
0answers
35 views

Given the linear system Ax=b; where A has m rows and n columns, proof the following statements

1.-if rk(A)=m<=n (A has full row rank), then the system has at least 1 solution 2.-if rk(A)=n<=m (A has full column rank), then the system has at most 1 solution 3.-if rk(A)=n=m (A has full ...
0
votes
1answer
55 views

Invertible matrices property

I'm wondering about this property : $\forall A \ \in \ \mathbb{M}_n(\mathbb{R})$ if there is $k \ \in \ \mathbb{N}^{*}$ such as $A^k=I_n$ then $A$ is invertible. I think this assertion is true I do ...
2
votes
1answer
56 views

Show that $A^TA$ has at least one positive eigenvalue if $A$ is not all-zero

I need some help on showing that $A^TA$ has at least one positive eigenvalue if $A$ is not all-zero. $A$ is rectangular and can have dependent columns in general. I can show that it cannot have ...
0
votes
1answer
41 views

what is the convex hull of the rank k psd matrix

Given the set $\{X|0\preceq X , rank(X)=k\}$. What is the convex hull (convex envelope) of this nonconvex set? If we further require $X=VV^T$, where $V^TV=I$, $V$ has the size $n\times k$. Then the ...
1
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0answers
46 views

Integration of multivariate Gaussians with cross terms

I'm stuck with the following integral: $I=\int ... \int exp\Big(-\frac{1}{2} \sum \limits_{t=1}^{n} x_{t}^T{\Sigma_{x}}^{-1} x_{t}+\sum \limits_{t=1}^{n} x_{t}^T{\Sigma_{x}}^{-1} z_{t} -\frac{1}{2} ...
1
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0answers
40 views

How to construct solutions for a set of polynomial matrix inequalities

How can one find solutions to the set of (polynomial) matrix inequalities $$M \succ 0,\quad A_i^TMA_i \preceq c\cdot M,\quad\forall i=1,\dots,m$$ where $M=M^T\in\mathbb{R}^{n\times n}$ and $A_i ...
0
votes
1answer
45 views

Building matrix expressions for product of sum, isolating vector of constants

This identity to build the matrix expression for the expression below is pretty straightforward: $$ \left.\sum\limits_{j=1}^M \left( a_j \cdot f_{i,j} \right) \;\right|_{i=1}^N = ...
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votes
2answers
47 views

Linear algebra - distance between a plane and line

I have: Plane: $2x+2y-z=1$ Line: $(1,1,0)+t(-1,-1,2)$ How to get the point $p$ which is in the plan and how to know the distance ? i know the normal vector is $(2,2-1)$. Any help will be ...
0
votes
1answer
16 views

Matrix, Quardratic form and boundedness

Let $A$ be an $n \times n$ symmetric matrix. Is this set $\{x \in \mathbb{R}^n: x^TAx=1\}$ bounded?($x^T$ stands for transpose of the vector $x$)
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0answers
27 views

the rank of QR decomposition

I saw this in a paper, where one has a QR decomposition $C=QR$ ($C\in R^{m\times r}$, $Q\in R^{m\times r}$ is column orthogonal, $R\in R^{r\times r}$, $m>r$). However, under the condition that the ...
1
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1answer
72 views

Find the matrix given the determinant

Is there a general method to find a 3x3, or 2x2 matrices, given the determinant? I want to do a project with my students when we start to study Systems of Equations. It would be interesting if the ...
2
votes
1answer
58 views

Orthogonal Diagonalization of a matrix

I am having problem in diagonalization of \begin{bmatrix}-1&-1&-4\\-1&-4&-1\\-4&-1&-1\end{bmatrix} This is symmetric so it must be orthogonally diagonalizable. The eigen values ...
2
votes
1answer
42 views

Eigenvalues of a matrix based on rank

I'm trying to answer this question: Let $A$ be an $n \times n$ matrix with rank $n-1$. Furthermore, let $Q$ be an orthogonal matrix. Name an eigenvalue of $Q^T(A-1I)Q$. I know that $Q^T=Q^{-1}$ and ...
0
votes
2answers
70 views

How to determine if a set of vectors is a basis for a subspace?

So I have a homework question which I am not sure if I am answering correctly. The questions is as follows. Determine whether the set is a basis for $\mathcal{R}^3$. If the set isn't a basis, ...
0
votes
1answer
22 views

Matrix product equals O

I'm stuck in this question : \begin{bmatrix} x & 4 & -1 \end{bmatrix} \begin{bmatrix} 2 & 1 & 0\\ 1 & 0 & 2\\ 0 & 2 & 4 \end{bmatrix} \begin{bmatrix} x\\ 4\\ 1 ...
1
vote
1answer
38 views

Eigenvalue and Eigenvector of special matrices

Find atleast one eigenvalue and eigenvector of the following matrices: (i) a $N \times N$ unit matrix ( a matrix with all entries as $1$) (ii) a $N \times N$ matrix obtained by deducting an identity ...
0
votes
1answer
31 views

marix proof by induction

Given the matrix $A$ \begin{pmatrix} 2 & 1 \\ 0 & 1 \end{pmatrix} use proof by induction to show that $A^n$ for $n=1,2..$. is \begin{pmatrix} 2^n & 2^n - 1 \\ 0 & 1 ...
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2answers
34 views

Curious about some (basic?) linear algebra notation

I was reading an old linear algebra textbook today, and I was actually having some trouble understanding the notation given in a problem. Here is what it said (or something similar): Consider the $n ...
0
votes
1answer
37 views

What are these matrix identities called?

The first one seems to be a slightly modified Woodbury inversion formula, but I can't find the second one. $(P^{-1} + B^\top R^{-1} B)^{-1} = P-P B^\top (BPB^\top + R)^{-1} BP$ $(P^{-1} + B^\top ...
2
votes
1answer
31 views

Need a hint on how to solve this matrix equation

\begin{equation} AX+XA=B \end{equation} $A^{-1}$ and $B^{-1}$ is available. The furthest I've gotten is: \begin{equation} X+A^{-1}XA=A^{-1}B \end{equation} which doesn't help at all.
0
votes
1answer
45 views

Normalizer and Centralizer of Upper Triangular Matrix

Consider the group $H_3(\mathbb{Z})=\left\{\begin{pmatrix}1 & a & c\\0 & 1 & b\\0 & 0 & 1\end{pmatrix}:a, b, c\in\mathbb{Z}\right\}$. Find $Z\left(\begin{pmatrix}1 & 1 ...
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votes
0answers
12 views

Different forms of the plane

Given $V=span ((-5,4,4), (-5,7,5))$ how to show that equation of V is $-8x+5y-15z=2$ I was trying to do the rref form but still it does not give me any hints about that. How to show that?
1
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1answer
46 views

Linear Algebra - matrix equation

solve the following matrix equation for $X$: $$ A(X-B)^{-1}=B$$ where $$A = \left [\begin{array}{ccc} 1 & 2 \\ 3 & 4 \\ \end{array} \right ]$$ $$B = \left [\begin{array}{ccc} 1 & 1 \\ ...
1
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1answer
66 views

Linear operator form of a symmetric matrix

I want to determine what it means for a symmetric matrix to be written in terms of linear operators. Perhaps the key? it looks like the form I am looking for might be the self-adjoint operators? ...
1
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0answers
61 views

Proof that border rank and the rank of a matrix (order 2 tensors) are equivalent

Recall the definition of border rank for a matrix (order 2 tensor, which can be easily be extended to any order tensor): border-rank(T) is the minimum r such that $\forall \epsilon > 0$ there ...
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0answers
13 views

Augmenting a matrix with a highly-incorrelated column

Consider a binary matrix: $$\begin{pmatrix} 1 & -1 & 1 \\ -1 & 1 &1 \\ \vdots & \vdots &\vdots \\ 1 & 1 & -1\end{pmatrix}$$ with a random distribution of 1 and -1 ...
0
votes
1answer
26 views

Normal Block Upper Triangular Matrix

What is an economical proof that a block upper triangular matrix is normal iff its off-diagonal blocks is zero and each of its diagonal blocks is normal? By economical proof I mean a short proof, ...
1
vote
2answers
32 views

Pfaffian of unitary transformed matrix

Let $U$ be a unitary matrix, and $U^\dagger$ be its Hermitian conjugate. What is $\mathrm{Pf}(U^\dagger AU)$? Since $\mathrm{Pf}(U^\dagger AU)^2=\mathrm{det}(U^\dagger AU)=\mathrm{det} ...
1
vote
1answer
17 views

An infinite number of solutions available to the sparse representation problem

I would like to analyze the following problem (different cases leads to which solutions to the problem and such): $$||y-Dx||_2 \leq \epsilon$$ (an overcomplete dictionary matrix $D \in \mathbb{R}^{n ...
0
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0answers
19 views

Get the maximum permutation matrix from logical matrix

$A_{mn}$ is a $(0,1)$-Matrix (or logical matrix). How to get a sub matrix $B_{pp}$ from $A$, satisfying that $B$ is a permutation matrix and p is the maximum? For instance, PS: A permutation ...
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0answers
12 views

How can we construct a matrix with size $n\times 2^n$ , i.e., each column vector is the vertex of the hyperplane generated by $[L,U]^n$?

In matlab, if we have two vectors L and U (with size n ), how can we efficiently construct a matrix with size $n\times 2^n$ , each column vector is the vertex of the hyperplane generated by $[L,U]^n$ ...
2
votes
2answers
75 views

Matrix rank and determinant

Given two matrices $A,B \in M_{4 \times 4} (\mathbb{R})$ such that $\det (AB) \ne 0$ what is the rank of the matrix B? How to solve this I have no idea how to approach this problem.
1
vote
1answer
21 views

Positive semi-definiteness of a matrix comprising block matrices of special form

I am interested in characterizing the positive semi-definiteness of a matrix in the form (A B C; D E F; H I G), where each one the the blocks A..G is a 3X3 matrix whose components are the same, e.g., ...
0
votes
0answers
20 views

Need some clarification in how to calculate heat equilibrium over a surface

Okay. So I'm sort of following the technique describe in the paper here: http://people.csail.mit.edu/ibaran/papers/2007-SIGGRAPH-Pinocchio.pdf to calculate the heat equilibrium over a surface in ...
2
votes
1answer
78 views

Solving matrices with unknown constants?

I am struggling with solving systems of linear equations with unknown constants. I can solve simple ones, such as: In the following system, for which values of k would produce: Infinitely ...
0
votes
1answer
25 views

What do Small/Big eigenvalues indicate on?

If a matrix has: 1) Small eigenvalue 2) Big eigenvalue 3) Eigenvalue equal to one What do each of them mean? Is there any conclusion about the matrix properties based upon its eigenvalue?
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0answers
39 views

the spectral norm of the subtraction of two diagonal matrices

I saw this equation in some papers (e.g., http://cs-www.cs.yale.edu/homes/mmahoney/pubs/l2sample.pdf ), but I am wondering if it is wrong. The original equation is where $\|\|_2$ is the spectral ...
1
vote
1answer
109 views

Cholesky, Inverse, and Determinant when updating the diagonal of a symmetric positive definite matrix

Suppose that $A$ is a symmetric positive definite matrix and assume its dimension $n$ is large. Let $I$ be the $n \times n$ identity matrix and $m \neq 0$ be a scalar. I'm interested in computing as ...
0
votes
2answers
107 views

Finding determinant of 4x4 by using echelong form and multiplying across diagonal

I have a matrix and I'm supposed to find the determinant. I chose to use the method of row reduction into echelon form and then multiplication across the diagonal. I row reduce the matrix but the ...
0
votes
1answer
86 views

If $A^2=0$, then $I−A$ is invertible [duplicate]

If $A^2=0$, then show that $I−A$ is invertible. I am getting nowhere that leads me to the hint: $I+A$.