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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2answers
187 views

matrix representation of polynomial

Here is a polynomial $p(x,y) = (ax + by)^2$, it can be written like this $$p(x,y) = \left(\left[ \begin{array}{cc} a & b \\ \end{array} \right] \left[ \begin{array}{c} x\\ y\\ \end{array} ...
4
votes
2answers
151 views

Product of matrices; MAPLE giving a strange answer

Either my brain is seriously fried up right now or the computer is wrong. If I have a matrix $\begin{bmatrix} 4 & -2\\ 2 & -1 \\ 0 & 0 \end{bmatrix}$ multiply by its transpose ...
1
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1answer
120 views

FFT of a matrix and its square.

I am doing something computationally intensive that requires that I compute the fast fourier transform of a matrix, let's say $A$, and also compute the FFT of its square, $A^2$. I am wondering if ...
7
votes
1answer
443 views

Determinant game - winning strategy

I came across this problem while looking at Putnam problems a while ago: "Alan and Barbara play a game in which they take turns filling entries of an initially empty 2008 x 2008 array. Alan plays ...
5
votes
1answer
184 views

Prove by using diagonalization

Can anyone give me some hints on how to prove this question? Q: Use diagonalization to prove that if $A \subset B$ are lattices then $[B:A ]=\frac{\Delta(A)}{\Delta(B)}$. Added: Definition: A ...
2
votes
3answers
65 views

Multiples of determinant are elements in a matrix

Suppose we have an $n \times n$ square matrix, $A$. Let the determinant be $|A|.$ We also constrain the elements of $A$ such that each element of $A$ is an integer multiple of $|A|.$ Is there an ...
3
votes
1answer
51 views

Is there a matrix decomposition $P = AA^{+}$, given P?

Suppose one could experimentally obtain $P$, a $N\times N$ matrix. Is there a way to decompose this into two matrices $AA^{+}$, where $A$ is $N\times M$ and $A^{+}$ is the pseudo-inverse of $A$? ...
1
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2answers
257 views

Norms of idempotent matrices

So if I have a random idempotent matrix (a matrix that when squared equals itself) how do I go about calculating its 2-norm? I know that a idempotent matrix has eigenvalues of only 0 or 1 and I know ...
1
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2answers
74 views

How to write this equation in matrix form?

If I have a $n\times n$ matrix $A$ and a column vector $v$ of $n$ elements, I would like to define vector $x$ as: $$x_{i} = \sqrt{\sum_{j}^{n}(A_{ij}v_{j})^{2}}$$ How can I write this in matrix ...
0
votes
1answer
39 views

How to prove that to reduce $B$ to echelon form no row interchanges are needed?

Suppose that to reduce a matrix $A$ to row echelon form are necessary $n$ elementary operations $E_1,...,E_n$. Suppose that $E_{n_1},...,E_{n_k}$ are the permutation operations that are needed. How to ...
0
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1answer
50 views

Can this function with modulo and truncated division be simplifed?

Can this function with modulo and truncated division (DIV) be simplifed? f(x)=(x%c)*r+DIV(x,c)%r Basically, I use this ...
3
votes
1answer
91 views

Identity makes every matrix invertible?

I have found this in a proof and do not understand where this comes from: If A is singular, then there exists $\delta \in \mathbb{R}_{>0} \forall \epsilon\in (0,\delta): \epsilon ...
10
votes
4answers
5k views

Prove that $A+I$ is invertible if $A$ is nilpotent [duplicate]

Possible Duplicate: Units and Nilpotents Given $A^{2012}=0$ prove that $A+I$ is invertible and find an expression for $(A+I)^{-1}$ in terms of $A$. ($I$ is the identity matrix).
1
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0answers
56 views

Is there any solution for this over-determined system of equations?

Under what condition(s) a unique solution is available for this over-determined system of equations? $$ x^TA_1x = x^TA_2x = \cdots = x^TA_{N-1}x $$ where $$ A_d = [w^{d(p_i-p_j)}]_{K \times K} ...
0
votes
2answers
43 views

Proof that if $n<k$ and $A$ is an $n\times k$ matrix, then $A^{T}A$ is not invertible

Can I get a proof of the fact that if $n<k$ and $A$ is an $n\times k$ matrix, then $A^{T}A$ is not invertible?
4
votes
3answers
800 views

$4 \times 4$ matrix and its inverse. Is my method ok?

I have been struggling to derive inverse matrix of a $4 \times 4$ Lorenz matrix $\Lambda$. $$ \Lambda = \begin{bmatrix} \gamma&0&0&-\beta \gamma \\ 0 & 1 & 0 & 0\\ 0 & 0 ...
1
vote
1answer
66 views

Is there a good strategy for computing eigenspace corresponding to $1$ of a matrix with entries of trigonom

For example, say $A= \left ( \begin{matrix} \cos x & -\sin x & 0 \\ \cos y \sin x & \cos x \cos y & -\sin y \\ \sin x \sin y & \sin y \cos x & \cos y \end{matrix} \right)$. ...
2
votes
2answers
332 views

Need help calculating this determinant using induction

This is the determinant of a matrix of ($n \times n$) that needs to be calculated: \begin{pmatrix} 3 &2 &0 &0 &\cdots &0 &0 &0 &0\\ 1 &3 &2 &0 &\cdots ...
1
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2answers
46 views

Do we have $x^TDAx\ge \min(\lambda_D)\min(\lambda_A)x^Tx$ if $A$ is PD and D is both diagonal and PD?

Suppose matrix $A\in\mathbb{R}^{n\times n}$ is symmetric positive definite and $D\in\mathbb{R}^{n\times n}$ is both diagonal and positive definite, do we have the following result? $$x^TDAx\ge ...
-2
votes
2answers
81 views

Is there any good strategy for computing null space of a matrix with entries $\cos x$ and $\sin x$?

For example, say $A= \left ( \begin{matrix} \cos x & -\sin x & 0 \\ \cos y \sin x & \cos x \cos y & -\sin y \\ \sin x \sin y & \sin y \cos x & \cos y \end{matrix} \right)$. ...
1
vote
0answers
47 views

Instead of iteration method, can we solve this matrix computation?

I have a simple problem with matrix compuation. Matices what I have are $A_f$ matrix with $(M \times N)$, $B_f$ matrix(or vector) with $(N \times 1)$. I just want to calculate $|A_fB_f|^2$ for $F$ ...
1
vote
1answer
96 views

Endomorphisms and projection operators in vector spaces over finite fields

In the process of studying for a qual I've become hung up on the following simple problem: Let $T$ be an endomorphism of a finite-dimensional vector space over a finite field. Show that there is a ...
1
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1answer
89 views

Proof of Nonnegativity Inequality

Prove the Inequality: $$\sum_{i,j}\left ( (PAQ)_{i,j}\frac{B_{i,j}^2}{A_{i,j}}- (PBQ)_{i,j}B_{i,j}\right ) \geqslant 0$$ Given that: $P$ and $Q$ are $n$x$n$ and $m$x$m$ symmetric matrices, $A$ ...
1
vote
1answer
51 views

Invertibility of row and column operations

I have a problem and a proposed plan for a solution. Please tell me if I'm on the right track. Problem: What happens if instead of $1$ row operation and then $1$ column operation, the reverse order ...
1
vote
1answer
50 views

Maximum matrix simplification

What is the most that a matrix can be simplified if row or column operations are both allowed? Intuitively, I am guessing that everything is 0 except the diagonal entries, which are a mix of 0's and ...
1
vote
2answers
98 views

Inverting a special matrix

Consider matrices $A$ and $B$ of the forms below: $$A = \lambda \cdot I$$ $$B = \beta \cdot \pmatrix{ 1 & 1 & \cdots & 1\\ 1 & 1 & \cdots & 1\\ \vdots & \vdots & ...
3
votes
2answers
157 views

Let $Q$ be a special orthogonal matrix. Show that $Q(u\times v)=Q(u)\times Q(v)$ for any vectors $u, v\in\mathbb R^3$.

Let $Q$ be a $3\times3$ special orthogonal matrix. Show that $Q(u\times v)=Q(u)\times Q(v)$ for any vectors $u, v\in\mathbb R^3$. I have no idea how to start. I'm not sure if $Q(u)\cdot ...
0
votes
1answer
70 views

$4$ way heat distribution multiplier problem

I'm making a simple heat distribution program. It's a $2D$ matrix with cells holding heat value. Every iteration looks for cells near current which have lower heat value and gives them some of its ...
6
votes
2answers
94 views

$2$ question on matrix with some condition

we need to tell whether $a,b$ true or false, I know that there does not exist $n\times n$ $A,B$ such that $(AB-BA)=I$,as if we take trace of both side they are not equal, so $a$ is false? $b$ I ...
2
votes
1answer
96 views

Commutative matrices and symmetric property

Assume we have two commutating matrices, [A,B]=0. Can we say that A and B are symmetric? Regards
7
votes
2answers
138 views

Unsure about a maths symbol

Help, help, help! I've come across this maths symbol, $[n(i,j)]^{0.5}$ where $n$ is a square matrix. Does this mean that it is the $(i,j)$ element of $n^{0.5}$? or $n(i,j)^{0.5}$? source: ...
3
votes
3answers
88 views

The matrix notation of signum?

The following question on a notation might look trivial but I am really not sure how to deal with it. If I have a variable $x$, I could write out: $$x=|x|\;\text {sgn} (x)$$ a notation that helps ...
2
votes
1answer
1k views

what are the conditions for the product of 2 symmetric matrices being symmetric

In generally, the product of two symmetric matrices is not symmetric, so I am wondering under what conditions the product is symmetric. Likewise, over complex space, what are the conditions for the ...
0
votes
1answer
54 views

Matrix expansion does not decrease norms

Given a block matrix $A = \left[ {\begin{array}{*{20}{c}} {{A_{11}}}&{{A_{12}}}\\ {{A_{12}}}&{{A_{22}}} \end{array}} \right]$, where $A \in {R^{N \times N}}$, it is true that the euclidean ...
8
votes
1answer
222 views

Maurer-Cartan 1-form

Can anyone help me with the following? Let $\rho$ be the right-invariant Maurer-Cartan 1-form $$\rho = dg\ g^{-1}$$ I want to show that the MC equation $$d\rho - \rho \wedge\rho = 0$$ holds. So ...
-1
votes
1answer
39 views

What elements does this set have

What elements does the set $T_{2,3}$ in this paper exactly have? and how about $T_{3,3}$, and $T_{3,4}$?
2
votes
1answer
117 views

Basis of a $n \times n$ matrix which trace is zero

For a matrix $2 \times 2$ it's easy. But for$n \times n$, I don't understand. Someone can help me?
3
votes
2answers
233 views

How to find a positive defective matrix?

When can a positive $n\times n$ matrix (with strictly positive entries only) be defective (non-diagonalizable)? It is not hard to show that it's not possible for $n=2$. I was able to find an example ...
0
votes
1answer
24 views

convulotion associative between vectors

We learnt that convolution is commutative, meaning that: $$xh = hx.$$ However if I take: $$h=[-1,0,1] \mbox{ and } x=[1,1,1]^T $$ ($T$ is transpose) I get that $xh$ is not equal to $hx$. Could ...
0
votes
1answer
149 views

Multiplying a matrix by a plane

I have a question - make a sketch showing the effect of multiplication by the matrix $A=\begin{bmatrix} 2 & -1 \\ 2 & 3\end{bmatrix}$ on the plane $\Bbb R^2$. How do you multiply a matrix by ...
3
votes
2answers
2k views

Matrix determinant using Laplace method

I have the following matrix of order four for which I have calculated the determinant using Laplace's method. $$ \begin{bmatrix} 2 & 1 & 3 & 1 \\ 4 & 3 & 1 & 4 \\ -1 ...
2
votes
1answer
371 views

Rotating a Matrix by an angle

So I have a matrix like so \begin{pmatrix} x_0 & x_1 & x_2 & x_3 \\ y_0 & y_1 & y_2 & y_3 \end{pmatrix} And I need rotate the matrix by an angle - for say $45$ degrees. I ...
0
votes
1answer
158 views

Conditions for $v \otimes v$ to be positive semidefinite for complex $v$

I have a complex-symmetric matrix (in the sense $A=A^{T}$ not $A=A^{H}$), which is required to be positive semi-definite in the following sense (sometimes referred to as positive real): $ \Re(x^{*} A ...
3
votes
3answers
2k views

Determining whether a set of vectors forms a basis using Gaussian elimination

Given the following three vectors: $$ b_1 = \begin{pmatrix} 1 \\ 3 \\5 \end{pmatrix},\space b_2 = \begin{pmatrix} 2 \\ 1 \\ 7 \end{pmatrix},\space b_3 = \begin{pmatrix} 4 \\ 2 \\3 \end{pmatrix}, ...
0
votes
1answer
59 views

Complex matrix with a single Eigenvalue

Im not sure about a question, and need your help. Is a complex matrix with a single Eigenvalue necessarily diagonalizable? I'm thinking that the answer is true because the opposite case does happen ...
1
vote
1answer
152 views

Derive Rigid Transform Matrix from Axes and Origin

I'm trying to derive the matrix of a rigid transform to map between two coordinate spaces. I have the origin and the axis directions of the target coordinate space in terms of the known coordinate ...
0
votes
1answer
26 views

Symmetric $N\times N$ matrix, multiplicity $N-1$, for any $N$

The $N\times N$ matrix has $1-s$ along the diagonal and $s/(N-1)$ on the off diagonal. For $N=2,\dots,5$ the characteristic polynomial is $(X-1)(X+\frac{N}{N-1}s-1)^{N-1}$ where $X$ denotes ...
7
votes
1answer
118 views

Matrix + combinatorial or conditional probability: bit patterns

I'm trying to get my head around a problem, and it's not working. The problem: consider an NxN matrix that represents a binary number. For instance, a 4x4 matrix is a 16 bit number, a 6x6 matrix is ...
4
votes
2answers
997 views

Showing that if an equation has a unique solution for one variable, then it has unique solutions for all.

I have a problem and a proposed solution. Please tell me if I'm correct. Problem: Let $A$ be a square matrix. Show that if the system $AX=B$ has a unique solution for some particular column vector ...
0
votes
1answer
128 views

condition for upper triangular matrix

Consider the following condition from this other post Define $S_k = {\rm span} (e_1, \ldots, e_k)$, where $e_i$ the standard basis vectors. Clearly, the linear map $T$ is upper triangular if and ...