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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0
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0answers
171 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
710 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
129 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
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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
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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 ...
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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
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2answers
409 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
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1answer
75 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
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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
136 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
95 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
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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 $y=1$.
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votes
2answers
623 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
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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 ...
1
vote
1answer
33 views

How do I find matrix that satisfies following conditions?

How do I find matrix $A$ with integer entries given two $2\times 1 $ vectors $\vec{x}, \vec{a} $ such that $$\vec{x} = A \vec{a}$$
1
vote
1answer
87 views

How can I find $T^{-1}(x,y,z)$ (inverted matrix) of a linear operator $T:V_3 \to V_3$

How can I find $T^{-1}(x,y,z)$ (inverted matrix) of a linear operator $T:V_3 \to V_3$, which matrix relative to the basis: $A=\{ (1,0,0), (1,1,0), (1,1,1)\}$ is: $$T_A= \begin{bmatrix} 2 &0 ...
3
votes
1answer
59 views

Skew-symmetric matrix and exp function $e^A$

Let $A_{nXn}(\mathbb{R})$ Skew-symmetric matrix $A=-A^t$ prove that $e^A(e^A)^t=I$ while: $e^A=\sum_{i=0}^{\infty} \frac{A^n}{n!}$ I tried this: $A=-A^t \Rightarrow A$ is Diagonalizable with ...
1
vote
1answer
47 views

Why is this finding inverse of a matrix by row operation not working?

the correct answer is $\begin{pmatrix} -5&3&-6\\-6&3&-7\\-2&1&-2 \end{pmatrix}$ So I think the mistake might be in the first two row operations but I see nothing?
12
votes
1answer
338 views

A sharper bound for $\|\cos(kA)\|_{\infty}$ for symmetric stochastic matrices

Given $A \in \mathbb{R}^{n \times n}$ that is symmetric, stochastic and diagonalizable, and $k \in \mathbb{N}$, I am interested in bounding $\|\cos(kA)\|_{\infty}$ from above. $\| \|_{\infty}$ is ...
0
votes
1answer
98 views

Symmetric matrix over inner product space

I try really hard to prove this Question. let $A_{nXn}(\mathbb{R})$ Symmetric matrix $A=A^t$ let $\lambda$ be the greatest Eigenvalue of A. we will define over the field $\mathbb{R}$ with the ...
0
votes
1answer
47 views

Show: $\phi: \mathbb{R}^3 \rightarrow \mathcal{su}(2)$, $h \mapsto h \cdot \sigma$ is an isometric isomorphism

I found this problem and need some help. It is given: $$ \sigma_1 = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} $$ $$ \sigma_2 = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix} $$ ...
0
votes
1answer
20 views

Transformation of matrix with variables

I have this matrix: $$\eqalign{\pmatrix{1&-1&-1&|&-2\cr 3&1&-1&|&b\cr a&8&2&|&7\cr} &\sim\pmatrix{1&-1&-1&|&-2\cr ...
2
votes
2answers
101 views

Prove that $A^{t}A$ is positive definite

$A$ is an invertible matrix over $\mathbb{R}$ (nxn). Show that $A^{T}A$ is positive definite. I looked up for it and found this two relevent posts but still need help. positive definite and transpose ...
2
votes
1answer
28 views

Acquiring $Df(\mathbf{x})$

Sorry for the probably easy and silly question, but I try to teach myself linear algebra and I am stucked at "the derivative as a matrix" part. I know how to differentiate partially and I know how ...
0
votes
1answer
116 views

Finding loci of possible points satisfying vector simultaneous equations

I recently had an exam and a question came up which I was only partially able to answer. The question was the following: Let $\vec{a}$, $\vec{b}$ and $\vec{c}$ be constant vectors in ...
-1
votes
1answer
199 views

Reverse of matrix multiplication

If I have matrices A, B, and C so that C = A * B, how can I get A from B and C? This page ( http://mathworld.wolfram.com/MatrixInverse.html ) tells me that A sould be C*B^-1, but using python and ...
3
votes
2answers
62 views

Find all $3\times3$ square matrices which commute with any $3\times3$ upper triangular matrix.

I'm not sure how to proceed. Let us find all possible solutions for the matrix $A$ which commutes with any other matrix $X$. In other words: $$AX=XA$$ Stating the matrix multiplication explicitly ...
3
votes
6answers
1k views

How can I prove that a square matrix is invertible if it satisfies this polynomial equation?

For a 3x3 matrix $C$, it is given that $$C^3+I=3C^2-C$$ I am then required to prove that $C$ is invertible. I have attempted a proof, below, but I am not sure it is valid or if there is a better ...
0
votes
1answer
56 views

Operations with $\text{SL}_2(\mathbb{Z})$

We define $\Omega :=\left\lbrace z \in \mathbb{H}\colon -\frac{1}{2} \leq \operatorname{Re}z \leq \frac{1}{2} \wedge |z| \geq 1\right\rbrace$. I want to show that the following holds: $$ \forall \, ...
0
votes
1answer
24 views

Exponent of polynomials (of matrices)

$A$ is a matrix over $\mathbb R$ (reals). Prove that for every $f,g\in \mathbb R[x]$, $\displaystyle e^{f(A)}\times e^{g(A)} = e^{f(A)+g(A)}$ I tried using the sigma writing but got stuck (I ...
3
votes
1answer
40 views

Matrix $A$ with characteristic polynomial

Given: Matrix $A$ with characteristic polynomial $p(x) = (x+3)^2(x-1)(x-5)$ Also given: $\rho(A+2I) + \rho(A+3I) + \rho(A-5I) = 9$ (btw $\rho$ means rank of the matrix) Prove: $A$ is ...
0
votes
3answers
74 views

What do you mean by the subspace spanned by the matrices?

What do you mean by the subspace spanned by the matrices $\{1,A,A^2,\dots,A^n\}$ where A is and $n\times n$ real or complex matrix.
1
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0answers
36 views

element subset for adjacency matrix

I am trying to create an element of a matrix that is a subset of a larger matrix. However, I am told that my subscripts do not match. I wanted other people's opinion as to what I am doing wrong and ...
0
votes
0answers
12 views

explicit expression for the solution of a lower-triangular Toeplitz linear system

I need to find an explicit expression for $c_k$ ($k=0,1,2,\cdots$) in terms of $A_m$ ($m=0,1,2,\cdots$) and $b_n$ ($n=0,1,2,\cdots$)from the following lower-triangular Toeplitz linear system of ...
1
vote
2answers
64 views

Determinant of $U$, Determinant of $U^T$

Given an $n\times n$ matrix $U$ such that $U^TU = I_n$, the $n\times n$ identity matrix. Then what are the possible values of the determinant of $U$?
2
votes
3answers
66 views

How to check whether this matrix is diagonalizable or not.

Let $\rm A$ be a complex $3\times3$ matrix with $\rm A^3=-1$. Which of the following statements are correct: $\rm A$ has three distinct eigenvalues. $\rm A$ is diagonalizable over $\Bbb ...
2
votes
1answer
739 views

2x2 symmetric matrix is a subspace of vector space.

Can you kindly check my proof of the problem and correct if possible. The following $S=\{A\in M_{2,2} | AA^T=A^TA\}$ is a subspace of $V=M_{2,2}$ all real $2\times2$ matrices. My proof: S ...
1
vote
2answers
50 views

trace function ($2\times2$) with ordered bases as linear transformation

We got trace function as following: $$\operatorname{tr}\begin{pmatrix} a & b\\ c & d\\ \end{pmatrix}=a+d$$ So now have to write down $[\operatorname{tr}]_{S_1,S_2}$, ...
2
votes
1answer
145 views

Some questions on Nilpotent matrix [closed]

Q & A style. Just wanted to share the following question which came in a competitive exam and so college level maths students may find it useful: A non-zero matrix $A\in M_n(\mathbb{R})$ is said ...
2
votes
1answer
44 views

Inverse of partitioned matrix, checking result

$A$ is an $n\times n$ matrix, partitioned as $$A=\begin{bmatrix} A_{11} & A_{12}\\ A_{21} & A_{22} \end{bmatrix},$$ where $A_{11}$ has dimensions $k\times k$ and $A_{11}$ and $A_{22}$ are ...