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
20 views

Find the $n^{th}$ power of a $2$x$2$ matrix.

Let $A=\begin{pmatrix}3&-2\\2&-2\end{pmatrix}$. Using Lagrange's interpolation compute $A^n$ for $n\in\mathbb{N} $ So far I've worked out the minimum polynomial of $A$ to be $(x-2)(x+1)$ but ...
0
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1answer
15 views

finding the inverse of a matrx

In order to decrypt a cipher text using hill cipher, we must first find the inverse matrix of a given matrix. From this link http://en.wikipedia.org/wiki/Hill_cipher, ...
1
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2answers
25 views

How to calculate row sums of a power of a matrix

Let $P $ be an $n\times n$ matrix whose row sums $=1$.Then how to calculate the row sums of $P^m$ where $m $ is a positive integer?
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1answer
15 views

Derivative involving inner product

How would I take the derivative of a function $$f(x) = < x,x >=x^{T}x?$$ The answer seems to be 2x but I don't know how to explicitly show this other than saying "there are 2 x's being operated ...
2
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1answer
33 views

Moving a point around a circle

we're currently working on a game which involves a character that rotates around a point. We are using a rotation matrix to rotate a given a point (x,y) around another point by first translating to ...
1
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1answer
15 views

How Do I Find The Permanent of a Double Stochastic Matrix n * n size

I am reading a book on Stochastic Models, and I don't understand this practice questions: A doubly stochastic n × n matrix S has all entries equal to 1/n. The permament of a n × n matrx A is ...
4
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4answers
91 views

Why is the volume of a parallelepiped equal to the square root of $\sqrt{det(AA^T)}$

Why is the $\sqrt{det(AA^T)}$ equal to the volume of a parallelepiped? Is is somehow related to the fact that $det(A) = det(A^T)$? EDIT: To clarify, the parallelepiped is spanned by the columns of ...
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0answers
15 views

Is there a single matrix norm such that for all stochastic $P$, $\| P \| = 1$?

By a stochastic matrix I mean a square real non-negative matrix with rows summing to one. Denote the set of all such matrices $\mathcal{S}$. By matrix norm I mean a norm in the vector space of ...
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2answers
16 views

Proving a theorem about trace of matrix which involving generalized inverse matrix

can you prove that theorem for me: Let A be mxn matrix of rank r then, $\ tr[I-A(A'A)^-A'] = m-r $  .   $\ A' $(transpose of A) ,$\ A^- $(generalized inverse of A)
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1answer
22 views

Norm of the sum of inverse matrices

Let $A,B$ be two invertible matrices. Is there a way to compute $\|A^{-1} -B^{-1}\|$ in terms of $\|A-B\|$?
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3answers
26 views

How do I prove matrix irreversibility without determinants?

I have to prove that if matrix has two identical rows or columns then it is not a reversible matrix. I know that in such scenario matrix determinant is equal zero, but I cannot use determinants in my ...
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1answer
17 views

Does there exist a Unit Matrix for a m x n matrix?

By definition, a Unit/Identity matrix (I) is a matrix such that, I A = A I = A If the matrix A is of dimension m x n, then unit matrix in IA must be of dimention m x m, while in A I should be of ...
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0answers
17 views

Solve the following system using Gauss-Jordan elimination

$4x - 8y = 12$ $3x - 6y = 9$ $-2x + 4y = -6$ So the augmented matrix will be: $$ \begin{bmatrix} 4 && -8 && 12\\ 3&& -6 && 9\\ -2 && 4 && -6 ...
3
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0answers
42 views

Proving that table totals can always be preserved with ceiling and floor

$\begin{array}{|c|c|c|c|} \hline 11.998& 9.083 & 2.919 & &24 \\ \hline 12.983&10.872&3.145&&27\\ \hline 1.019&2.045&0.936&&4\\ \hline & & ...
1
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1answer
28 views

Is $vv^{T} - v^{T}vI$ non-singular? [on hold]

Is $vv^{T} - v^{T}vI$ non-singular ? Why? $v$ is vector
0
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1answer
24 views

Derivation of Trace

I am curious about a formula in http://zh.wikipedia.org/wiki/%E8%B7%A1 $$\frac{\partial\text{tr}(A^{-1})}{\partial A}=-(A^{-2})^T$$ I have tried to prove this. We have $A^{-1}=\frac{A^*}{|A|}$, and ...
0
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1answer
22 views

An Inequality in Numerical Optimization

I am reading Jorge Nocedal and Sepher J. Wright's Numerical Optimization and stuck at an exercise 4.6 in chapter 4. The Canchy-Schwarz inequality states that for any vector $u$ and $v$, we have ...
0
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1answer
8 views

Number of Distinct Elements in Set of Products of 2 Matrices

Let $X=\begin{pmatrix}\cos\frac{2\pi}{5} & -\sin\frac{2\pi}{5}\\\sin\frac{2\pi}{5} & \cos\frac{2\pi}{5}\end{pmatrix}$ and $Y=\begin{pmatrix}1 & 0\\0 & -1\end{pmatrix}$. Find the ...
1
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0answers
26 views

Eigenvalues with constraints?

Note: This is a short version of About diagonalizing a matrix for a quadratic expression (with the goal of uncoupling mixed terms) For a $n$-dimensional symmetric matrix A, orthogonal matrix C exists ...
0
votes
0answers
16 views

System solving with Substitution and Matrices

My class was able to produce solutions using Substitution on the following System: $$ \left\{ \begin{array}{c} x+y+z=0 \\ 2x+3y+2z=-1\\ x-y+z=2 \end{array} \right. $$ The solution was: x = 1, y = ...
0
votes
1answer
21 views

Linear Algebra - Give an example for $3x3$ matrix for these eigenvalues

I'm having trouble with this problem : Give an example for matrix $A$ with these eigenvalues $\lambda_1-1,\lambda_2=1,\lambda_3=0$ while : $$v_1=(0,1,1)$$ $$v_2=(1,-1,1)$$ $$v_3=(0,1,-1)$$ ...
0
votes
2answers
32 views

Let $A$ be a single $p\times p$ Jordan block. Find general solution to $\dfrac{dx}{dt} = Ax$

Let $A$ be a single $p\times p$ Jordan block. Find the general solution to $\,\dfrac{dx}{dt} = Ax$. What should I approach first? Please help!
4
votes
4answers
177 views

Question about determinants

I am working on some practice problems and I just cant see to even begin to understand how to do this question. It starts off by giving some facts such as det= 1 for the following:$$ \begin{matrix} a ...
2
votes
2answers
14 views

Is the null space inside the collumn space of a matrix?

From what I've seen online, it seems that the null space isn't in the column space, but I don't understand why that is the case. If the null space is the set of all combinations that equal 0, and the ...
3
votes
2answers
22 views

Proof that theorems about trace of matrix :

Can somebody help me about proofs of this theorems A is an nxn matrix and $\ A^2$ = mA then, tr(A) = m rank(A) . A is an nxn matrix and k is a positive integer then, tr($\ A^k$) = $\sum_{i=1}^n ...
0
votes
1answer
19 views

Need to prove $(JC=0=CJ,\,JJ=nJ)\implies (C-aJ)^{-1}-(C-bJ)^{-1}=\frac{b-a}{ab n^2} J$

I can't prove that matrix $C$: $$\big(JC = 0 = CJ\text{ and } JJ = nJ\big) \implies \left((C-aJ)^{-1} - (C-bJ)^{-1} = \frac{b-a}{abn^2} J\right)$$ I know that $$(JC = 0 = CJ\text{ and }JJ = nJ) ...
2
votes
1answer
26 views

Get normalised eigenvectors

I am given the matrix: $\begin{pmatrix} a & b \\ b & -a \end{pmatrix}$ and I already calculated the eigenvalues $\lambda = \pm \sqrt{a^2+b^2}$. Now, I want to get the normalised ...
0
votes
1answer
25 views

Element matrix multiplication representation

Matrix element by element multiplication defined : $C=A*B$ $c_{ij}=a_{ij}b_{ij}$ Is this multiplication can be represented with stardant matrix multiplication or Kronecker product ?
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0answers
21 views

2D convolution: how to eyeball it?

I have a question of doing simple convolution in 2d by just "eye-balling" it without doing the actual computation. In 1D discrete time, when we have a simple input ...
2
votes
3answers
60 views

Roots of a cubic equation with coefficients based on unknown values $a$, $b$ and $c$.

I want to find the eigenvalues of the following matrix: $$ \left( \begin{array}{ccc} 0 & a & b \\ a & 0 & c \\ b & c & 0 \end{array} \right) $$ So, I found the characteristic ...
0
votes
0answers
20 views

Proof involving projections and column spaces

Let $A \in \mathbb{M}_{m×n}(\mathbb{R})$ with linearly independent columns. If $\overrightarrow{b} \in \mathbb{R}^m$, then prove $proj_{Col(A)}(\overrightarrow{b}) = ...
1
vote
3answers
46 views

Prove that a matrix with a given characteristic polynomial is diagonalizable

Matrix $A$ is defined over real number. Characteristic polynomial : $p(x)=(x+3)^2(x-1)(x-5)$ It also known that : $$\text{rank}(A+2I)+\text{rank}(A+3I)+\text{rank}(A-5I)=9$$ prove $A$ ...
0
votes
1answer
28 views

Domain/codmain + range/kernel for linear mappings

Consider the linear mapping: $$L(x_1,x_2)=(2x_1-3x_2,4x_1+5x_2,2x_1-x_2)$$ Solve for: (a) Domain and codomain of L (b) Standard matrix of L (c) Basis for the range of L (d) Basis for the kernel ...
0
votes
1answer
19 views

Is the derivative of the characteristic polynomial equal to the sum of characteristic polynomial of principle submatrices?

Let $A$ by an $n \times n$ matrix over the complex numbers and let $\phi(A,x) = \det(xI-A)$ be the characteristic polynomial of $A$. Let $B_i$ be the principal submatrix of $A$ formed by deleting the ...
1
vote
2answers
40 views

Show that a set of vectors is linearly dependent

Show that the set $S = \{(3, 2), (−1, 1), (4, 0)\}$ is linearly dependent by finding a nontrivial linear combination of vectors in the set whose sum is the zero vector. (Use $s_1$, $s_2$, and ...
-1
votes
1answer
29 views

Linear Algebra - Prove trival solution eigenvalue

A is an $2\times2$ matrix with $\operatorname{trace}=1$, and $\det A=-6$. Prove that $(2A+5I)x=0$ has only trival solution. I need to show that $(-A-\frac{5}{2}I)x=0$ Therefore I need to show that ...
0
votes
1answer
29 views

Can't understand matrix based derivation

$\beta(k,d)=(X'X+kI)^{-1}(X'y+kdB_L)$ $=[I+k(X'X)^{-1}]^{-1}(X'X)^{-1}(X'y+kdB_L)$ $=[I+k(X'X)^{-1}]^{-1}(B_L+kd(X'X)^{-1}B_L)$ $=[I+k(X'X)^{-1}]^{-1}(B_L- dB_L)+dB_L$ $B_L=(X'X)^{-1}Xy$ X=n*p ...
1
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0answers
12 views

median eigenvalue

When I look at the spectral density plots of my (usual) laplacian graphs, they spike at the median eigenvalue. But what significance for the graph/matrix (which originates from a network) does the ...
1
vote
2answers
79 views

Solving ODE containing matrices

We have an ODE $ \psi'(t)_{_{3 \times 3}}=\psi(t)_{3 \times 3}(A_{3 \times 3}+B_{3 \times 3}t)\tag 1$ Given Data in Question We have no quarentee that $\psi'(t),\psi(t)$ both have inverse A,B are ...
6
votes
1answer
50 views

What is the quickest way to find the characteristic polynomial of this matrix?

Let $e_k$ be the $k$-th vector of the canonical base of $\mathbb R^n$ and let $$B = [e_2 \mid e_3 \mid \dots \mid e_n \mid e_1]$$ What it the quickest way to show that the charachteristic polynomial ...
0
votes
0answers
17 views

Is any triangular matrix with positive diagonal elements a Cholesky factor?

I'm having a hard time finding information about Cholesky factors, and I'm sure it's a very simple question if it was asked to the right person. I need to create positive semi-definite matrix using ...
1
vote
1answer
34 views

Gram matrix to be cancelled

Let $V$ be a $n$ dimensional Euclidean space with inner product $<\cdot,\cdot>$, with basis $e_1,\cdots,e_n$. Then the Gram matrix is $A=(a_{ij})$ with $a_{ij}=<e_i,e_j>$. It is well-known ...
1
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2answers
28 views

Positive definite matrix to be cancalled

From $ax\geq 0$ for $a>0$, we have $x\geq 0$. So I suggest that if $Ax\geq 0$ for $A$ positive definite matrix, $x$ a column vector, $0$ is the column vector with $0$ as elements, then $x\geq 0$, ...
0
votes
0answers
15 views

Understanding the difference in (double) diagonally traversing trough a square matrix

I have been struggling with an algorithm to solve the PE Problem #149. I was able to find a solution (algorithm) for this problem on the internet, which can be found here. I do understand this ...
1
vote
0answers
18 views

Determinant of specific infinite matrix

What is the limit, as n approaches infinity, of the determinant of an n x n matrix where each cell has the value cos(n * row + column)? My friend and I believe the answer to be 0, but can't figure ...
4
votes
1answer
43 views

Finding unknown matrices in a set of simultaneous matrix equations

I've come across a thorny problem in my research, which is too complicated and specific to ask here. However, it bears some similarity to the following problem, and understanding how to solve this ...
2
votes
1answer
22 views

Is there a 3D equivalent of a 2D matrix?

Just thinking, is there a 3D 'equivalent' of a matrix. I know it's possible to get matrices that only have one row or column (i.e. vectors) thus making there a sort of 1D equivalent, but is there a 3D ...
3
votes
0answers
47 views
+50

Proof of the conjecture that the kernel is of dimension 2, extended

Pursuing my research, I am now looking for a proof of an extension of the problem proposed here and answered. It's an extension in the sense that I'm now considering two different $t_1$ and $t_2$. The ...
0
votes
1answer
38 views

Linear Algebra vs Matrix Algebra [on hold]

Hi I don't know if this would be a proper question to ask here or not Anyway, I am an undergraduate electrical engineering student and I am considering taking another math course. What is the ...
1
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0answers
39 views

Diagonalization of Hermitian matrix

I would like to perform diagonalization of a Hermitian matrix $A$ and I know the steps but at the end I am not getting diagonal matrix with eigenvalues on the main diagonal, can anyone help me why? ...