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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1
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0answers
38 views

Optimal VCV matrix solution of multivariate loglikelhood

I asked a related question yesterday and got a brilliant answer from Ross B. However I still have difficulties. I have the following analog of multivariate loglikehood function (minus 2*log-likehood ...
1
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2answers
70 views

Conjugates of the upper triangular matices

It's a shame...I want to give an explicit description of the set $\bigcap_{m \in GL_n(K)} mUm^{-1}$, $U$ being the upper triangular subgroup of $GL_n(K)$. It seems to be $K^\times I_n$ but I do not ...
0
votes
1answer
51 views

Real orthogonal matrix

Let $A\in \mathbb{R}^{n \times n}$ be an orthogonal matrix $(AA^t=I)$ and $a+ib$ an eigenvector $$A\cdot (a+ib)=\lambda \cdot (a+ib)$$ where $\lambda \in \mathbb{C} -\mathbb{R}$. How to prove that ...
3
votes
2answers
65 views

If $A$, $B$, $A-B$ and $I+A$ are invertible $n×n$ matrices then prove the following

i. $(A-B)^{-1} = A^{-1} + A^{-1}(B^{-1} - A^{-1})^{-1}$ ii. $(I+A)^{-1} = I-(A^{-1} + I)^{-1}$ iii. $tr((I+A)^{-1}) + tr((A^{-1} + I)^{-1}) = n$ I'm stuck on these. So far I only thought about ...
0
votes
2answers
47 views

$T: V \rightarrow V$ a linear transformation such that $T^2 = I$ and $H_1= \{v \in V | T(v) = v\}\ $ and $H_2= \{v \in V|T(v) = -v\}\ $

Let V a vector space and $T: V \rightarrow V$ a linear transformation such that $T^2 = I$ and $H_1= \{v \in V | T(v) = v\}\ $ and $H_2= \{v \in V|T(v) = -v\}\ $ then $V = H_1 \bigoplus H_2$ I stuck ...
-1
votes
1answer
16 views

How to switch the rows in a random matrix?

I have an M by N matrix $A$. Does anyone know an easy way to find a N by N matrix $B$ such that $A*B = C$, where $C$ is the result of M by N matrix when switching the first and last row in $A$. Note: ...
3
votes
1answer
77 views

Matrix multiplication of $A \in M_{3 \times 2}$ and $B \in M_{2 \times 3}$.

Given two matrices $A \in M_{3 \times 2}$, $B \in M_{2 \times 3}$, and $AB=$ \begin{bmatrix}8 &2&-2\\2&5&4\\-2&4&5 \end{bmatrix}, find $BA$. I noticed that $\det{AB}=0$, ...
0
votes
5answers
70 views

If $A =\begin{pmatrix} -1 & 0 & 1\\ 0 & 1 & 1\end{pmatrix}$ and $AB = I$ find the $3\times 2$ matrix $B$.

Alright so you multiply $A$ and $B$ and you get four equations. Then you do $\det[AB] = \det[I] = 1$ and you get a fifth. I'm stuck here now. What else can I do to find $B$? I'm trying to get this ...
0
votes
1answer
46 views

Is it true that every eigenvalue has at least one eigenvector?

As mentioned above: Is it true that every eigenvalue has at least one eigenvector? Or is it possible that while trying to find the basis of a specific eigenspace, i will get only the zero vector ...
0
votes
2answers
64 views

Showing $A^2 - 3A^{147} + 2I = \begin{pmatrix} 9 & -9\\ 3 & -3\end{pmatrix}$. [closed]

Let $ A = $$ \begin{pmatrix} 9 & -9\\ 3 & -3\end{pmatrix} $$ $. Show $$A^2 - 3A^{147} + 2I = \begin{pmatrix} 9 & -9\\ 3 & -3\end{pmatrix}$$
0
votes
2answers
50 views

What could be the rank of a matrix multiplied by its transpose ?

Let $A$ be a full rank $m×n$ matrix $(m<n)$, i.e. $\operatorname{rank}(A)=m$. Can the rank of $A'A$ be $n$? Under what condition would this hold? Thanks!
0
votes
1answer
62 views

Find Eigenvalues of Circulant Matrices

$$ A=\begin{pmatrix} \alpha^R&\beta^R &\gamma^R&-\alpha^I&-\beta^I &-\gamma^I\\ \gamma^R&\alpha^R&\beta^R &-\gamma^I&-\alpha^I&-\beta^I \\ \beta^R ...
13
votes
3answers
896 views

Why is an orthogonal matrix called orthogonal?

I know a square matrix is called orthogonal if its rows (and columns) are pairwise orthonormal But is there a deeper reason for this, or is it only an historical reason? I find it is very confusing ...
2
votes
1answer
44 views

Summing infinite numbers of matrices

Let $\mathbf{I}$ be a identity matrix, and $\mathbf{A}$ be a symmetric matrix in which every entry $a_{ij}$ follows $0 \le a_{ij} < 1$. I want to get $\mathbf{S} = \mathbf{I} + \mathbf{A} + ...
0
votes
2answers
147 views

Decompose $A$ to the product of elementary matrices. Use this to find $A^{-1}$

Decompose $A$ to the product of elementary matrices. Use this to find $A^{-1}$ $$ A = \begin{bmatrix} 3 & 1\\ 1 & -2 \end{bmatrix} $$ I understand how to reduce this into row echelon form ...
2
votes
1answer
32 views

Markov chains by hand

If I have a starting point: $A_T=[0,1]$ at $T=1$ and a one step transition matrix of: $B=\left[ \begin{align} &\frac34 & \frac14& \\& \frac1{20}& \frac {19}{20} &\end{align} ...
1
vote
0answers
110 views

Newton's method for multidimensional functions

Can Newton's method be used to find the root of a function f : $\mathbb{R}^n\to\mathbb{R}^m$. Can anyone provide a proof for this? (I have checked the method of solving system of equations with ...
0
votes
0answers
21 views

Interpreting & Analysing a Transitional Matrix

How do you interpret such a problem Are we expect to add the rows, and that would be the one with larger number of goats in the long term. Therefore A(row 1) and b(row 2)... therefore the answer is ...
5
votes
4answers
1k views

Is every self-inverse matrix diagonalizable?

If $A=A^{-1}$, is there always a matrix C such that $C^{-1}AC$ is a diagonal matrix (containing only -1 and 1 in the main diagonal) ? How can I check with PARI/GP, if a given matrix is ...
0
votes
1answer
55 views

real matrices $2\times 2$ and $3\times 3$ that are not similars to a diagonal matrix

Example of real matrices $2\times 2$ and $3\times 3$ that are not similars to a diagonal matrix. I find that $A =\begin{bmatrix} 2 & 1 \\ 0 & 1 \end{bmatrix} $ then i suppose that its ...
3
votes
2answers
138 views

Norm of Hilbert matrix is it equal to $\pi$?

Let $A$ be a Hilbert matrix, $$a_{ij}=\frac{1}{1+i+j}$$ We have the following result : $\Vert A\Vert\leq \pi$. I am using the subordinate norm of the euclidean norm i.e. $$ \Vert A\Vert=\sup\{\langle ...
0
votes
1answer
71 views

Type of this Conic section

I want to determine, to which type the following Conic sections belong to: $$ \begin{align} \textrm{(i)}&\quad-8x^2+12xy-6x+8y^2-18y+8=0\\ \textrm{(ii)}&\quad5x^2-8xy+2x+5y^2+2y+1=0 ...
1
vote
2answers
92 views

maximum and minimum dimension of the space generated by $\{v_1,v_2,v_3,v_4\}$

I'm confused about this problem. I have four vectors $v_1 = (1,1,1,a), v_2 = (1,2,3,a), v_3= (b,1,0,1), v_4 = (0,b,0,0)$ with $a,b$ real numbers. Determine the maximum and minimum dimension of the ...
0
votes
2answers
29 views
1
vote
7answers
435 views

positive definite quadratic form

Is $\sum_{i=1}^n x_i^2 + \sum_{1\leq i < j \leq n} x_{i}x_j$ positive definite? Approach: The matrix of this quadratic form can be derived to be the following $$M := \begin{pmatrix} 1 & ...
0
votes
1answer
115 views

How to find the determinant of a NxN matrix

Here is my matrix. How do I find the determinant of this one? I'm really trying to solve it but I can't think of anything. $$ \begin{pmatrix} 3 & 2& ...& 2\\ 2& 3& ...& 2\\ ...
0
votes
0answers
173 views

Finding minimum of a distance function using matlab

I have a function for that I want to find the minimum. The function calculates the distance between two sets where a set is defined as matix of row vectors $ D = [ d_1, d_2, ..., d_n]$, $d_n$ is a $m ...
1
vote
1answer
60 views

Derivative of loglikehood like matrix function

I have a function $$ f(C)=trace(XX^TC -log(C)), $$ Here $X$ is $nxk$ matrix , $C$ is strictly positive definite symmetric matrix parametrized as follows $$ C=(D+UFU^T)^{-1} $$ $D$ is positive ...
0
votes
1answer
59 views

Find the value of a so that the 2 x 2 matrix A is invertible

Find the value of a so that the $2 \times 2$ matrix $A = \begin{pmatrix} a-3 & 1\\ 2a+14 & a \\ \end{pmatrix}$ is invertible. do I just use the $\frac{1}{ad-bc}$ rule and solve for $a$, ...
3
votes
1answer
725 views

How do I find the initial state Matrix?

The question gives a $2\times2$ transition matrix: $$ \begin{bmatrix} 0.8 & 0.2\\ 0.3 & 0.7 \end{bmatrix}. $$ And then it gives me the initial state matrix but I'm wondering how do I find the ...
1
vote
2answers
130 views

Fastest Gaussian Elimination Method?

I have this matrix and I want to know is there a method that I can always rely on to get the inverse without much trial and error. The matrix is; $$ \begin{bmatrix} 1 & 1 & 1\\ 0 & 3 ...
0
votes
1answer
77 views

Homework: canonical form of quadratic form

X=(x,y,z) Q(X) = $x^2 + 4xy + 6xz + 3y^2 +8yz +5z^2 $ I got by using completing the square method: Q(X) = $(x+2y+3z)^2 - (y+2z)^2$ so as I learned now I do: $u = x+2y+3z$ $v = y+2z$ $w = 0 $ ...
1
vote
1answer
33 views

If the first r columns of U are linearly independent, then so are the first r columns of A?

Let $U$ be a row echelon form of a square matrix $A$. If the first $r$ columns of $U$ are linearly independent, then should the first $r$ columns of $A$ be linearly independent? In my opinion, "Yes" ...
5
votes
2answers
1k views

Cholesky decomposition in positive semi-definite matrix

While trying to apply the algorithm described in the article: Robust adaptative metropolis algorithm with coerced acceptance rate (2011), Matti Vihola I used the a Cholesky decomposition to find ...
2
votes
1answer
65 views

Find $e^{AT}$ where $A$ is a Matrix that is given

How to find the value of $e^{At}$ where $A$ is the matrix $A =\begin{bmatrix} 4 & 3 \\ 2 & -1 \end{bmatrix}$
8
votes
3answers
129 views

Find the expansion for $\det(I+\epsilon A)$ where $\epsilon$ is small without using eigenvalue.

I'm taking a linear algebra course and the professor included the problem that prove $$ \rm{det}(I+\epsilon A) = 1 + \epsilon\,\rm{tr}\,A + o(\epsilon) $$ Since the professor hasn't covered the ...
1
vote
0answers
85 views

Calculate Geodesic Path of $N\times N$ matrix on Riemannian manifold of fixed rank

If I have two matrices $A(0)$ at $t=0$ and $A(1)$ at $t=1$, they are $N\times N$ matrices, and they are on the Riemannian manifold of rank $K$. How to calculate the geodesic path $A(t)$? I haven't ...
3
votes
2answers
412 views

Orthogonal Projection of a matrix

Let $V$ be the real vector space of $3 \times 3$ matrices with the bilinear form $\langle A,B \rangle=$ trace $A^tB$, and let $W$ be the subspace of skew-symmetric matrices. Compute the orthogonal ...
1
vote
1answer
56 views

Left shift operator $L: l^2 \rightarrow l^2$ on the sequence space $l^2$

$$L: l^2 \rightarrow l^2$$ is defined by $$b = (b_1,b_2,...) \mapsto Lb = (b_2,b_3,...)$$. $(Lb)_n = b_{n+1}$ respectively. How can I determine the adjoint endomorphism $L^*$? Kind regards George
0
votes
1answer
76 views

Spectral radius and Dominant Eigenvalue

What is the difference between the spectral radius and dominant eigenvalue? If they are one and the same then why do both get mentioned, for instance here ...
0
votes
0answers
59 views

Why is smallest singular value of a singular matrix zero?

In my book, it is stated that the smallest singular value ($\sigma_n$ ) of a singular matrix is zero. I don't understand what it is so, please someone explain the reason to me.
0
votes
1answer
73 views

Properties of invertible matrices

Show that if $A$, $B$, and $A + B$ are invertible matrices with the same size, then $A(A^{-1}+B^{-1})B(A+B)^{-1}=I$ What does the result in the first part tell you about the matrix ...
1
vote
1answer
1k views

Prove that if A is an invertible matrix, then A*A is Hermitian and positive definite.

If I'm not mistaken, if a matrix $M$ has its conjugate $M^*=M$, then $M$ is Hermitian. In this case then, am I asked to show that $(A^*A)^*=A^*A$? What does it have to do with $A$ being invertible ...
2
votes
3answers
138 views

Prove that if $A$ is invertible then $AA^\top$ is positive definite [duplicate]

I need to prove that if $A$ is a square invertible matrix then $AA^\top$ ($A$ multiply $A$ transpose) is positive definite. I tried to prove that all the eigenvalues are positive. I know that ...
2
votes
0answers
100 views

Self-inverse matrices with integers with pairwise different absolut values.

Let A be a self-inverse matrix ($A=A^{-1}$) with integer values such that no two integers have the same absolut value. Let M be the maximum of the absolut values (maximum-norm) of A. Which M is the ...
1
vote
1answer
32 views

Shears using Matrix Methods

Determine the equation of the image of the graph: $$y=(x-1)^3 -2$$ after a shear of factor $1$ away from the $y$-axis, relative to the line $y=1$.
6
votes
2answers
633 views

What can be said about a matrix which is both symmetric and orthogonal?

I tried to find matrices A, which are both orthogonal and symmetric, this means $A=A^{-1}=A^T$. I only found very special examples like I, -I or the matrix $$\begin{pmatrix} 0 &0& -1\\ ...
5
votes
2answers
197 views

What linear transformations preserve these conditions?

Main Question Let's define $\Gamma(n)$ as the set of real antisymmetric matrices of size $n$ ($n$ is an even Integer), fulfilling: $$ \forall \gamma\in \Gamma(n) \Rightarrow \gamma^2=-\mathbb I_n$$ ...
0
votes
1answer
22 views

Forms - unitary group?

If $Id$ is the $n$ by $n$ identity matrix and $J$ is the $2n$ by $2n$ matrix with $Id$ in the upper right corner and $-Id$ in the lower left corner, then $Sp_{2n} = \{G\in Gl_{2n} : G^{tr}JG = J\}$ is ...
1
vote
1answer
135 views

Rewriting the simplified google algorithm in linear algebra form

I have the expression for the rank ($x_{i}$) of a page $i$ in an internet with $n$ sites, each site contains $n_{i}$ links to other sites and is linked to by the pages $L_{i}\subset\{1,\dots,n\}$. The ...