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
13 views

derivative of gradient involving inverse of matrices

I need to take three partial derivatives of this squared mahanalobis distance with respect to these three matrices: $Q, A,$ and $S$ $$(x+Ab)^T(A^TQA+S)^{-1}(x + Ab)$$ $x$ and $b$ are vectors of ...
1
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1answer
30 views

Do positive-definite matrices always have real eigen values?

Do positive-definite matrices always have real eigenvalues? I tried looking for examples of matrices without real eigenvalues (they would have even dimensions). But the examples I tend to see all ...
0
votes
0answers
4 views

Determining pitch and roll angles from the coordinates of a vector

I want to know, given the measurement of an accelerometer at rest (so not really an acceleration but a force per unit of mass) the inclination of this accelerometer, along the X and Y axis. So, In ...
0
votes
0answers
15 views

A inquality in matrix norm

Let $A,I \in {M_n}$($I$ is identity matrix) and $\left| {\left\| . \right\|} \right|$ is matrix norm.Suppose $\left| {\left\| A \right\|} \right| < 1$ and $\left| {\left\| I \right\|} \right| \ge ...
0
votes
0answers
10 views

Subtraction of quadratic forms with positive-definite matrix?

In linear regression, the OLS vector of estimators minimizes the sum of squares of the residuals (e'e). This means that for any other vector j of estimators, it must follow that: (1) b'(X'X)b - ...
1
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1answer
8 views

Solving a matrix for color manipulation

I'm making an application that deals with color transforms. The idea is that if you give it an RGB color and apply a color matrix transform it outputs another color. In this case I'm giving the color ...
0
votes
2answers
25 views

Suppose $A$ is an invertible matrix. Is it true that there always exists a polynomial $p(x)$ such that $A^{-1}=P(A)$?

Suppose $A$ is an $ \times n$ invertible matrix. Is it true that there always exists a polynomial $p(x)$ such that $A^{-1}=P(A)$? The question is from Moscow Institute of Physics and Technology My ...
0
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1answer
22 views

Proving two matrices are cogredient over $\mathbb{Q}$

Two matrices $A,B$ are said to be cogredient if there exists an invertible matrix $P$ such that $B = P^{t}AP$. I know how to tell if two matrices are cogredient in algebraically closed fields, its as ...
3
votes
2answers
37 views

How to define a specific ring using a homomorphism

If we have a ring $R$ then I can form a ring of matrices isomorphic to $R$ by setting $r \overset{\phi}{\mapsto} \left( \begin{array}{ccc} r & 0 \\ 0 & 0 \end{array} \right) $ and defining ...
0
votes
0answers
14 views

Orientation of a link in a link-system in space

my question is that I have a system of three links, all connected by spherical joints. There's three joints. I have the coordinates of all three joints labeled $a$, $b$ and $c$, plus the end-effector ...
3
votes
1answer
39 views

An Extension to the Generalized Eigenvalue Problem

Given two square matrices $A_1,A_2 \in \mathbb{R}^{n\times n}$, the generalized eigenvalue problem is finding the scalar $\lambda \in \mathbb{C}$ and vector $x \in \mathbb{C}^{n}$ such that $$ ...
0
votes
1answer
32 views

Proof: $ |||\frac{1}{2} AB + \frac{1}{2} BA||| \leq |||AB|||$

I am looking for a proof of the following:$$ |||\frac{1}{2} AB + \frac{1}{2} BA||| \leq |||AB|||$$ For positive hermitian matrices A and B, and a unitarily invariant norm $ |||\cdot|||$.
0
votes
1answer
30 views

Let A and B be n*n matrices such that trace(A)<0<trace(B).

Let A and B $n\times n$ such that trace(A)$\lt0\lt$trace(B). Then, $f(t)=1-det(e^{tA+(1-t)B})$ has 1) infinitely many zeros in $0\lt t\lt1$ 2) at least one zero in $\Bbb R$ 3) no zeros 4) either ...
3
votes
3answers
198 views

How do I restrict k to ensure my matrix has exactly 3 distinct eigenvalues?

$$A=\begin{bmatrix}-1&-1&0\\-12&3&-1\\k&0&0\end{bmatrix}$$ How do I restrict $k$ to ensure that my matrix has 3 distinct real eigenvalues? I tried going about it the long way ...
1
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2answers
41 views

Can I use eigenvalues to find the inverse of a vector?

I have two 1D matrices (say dimension 1xn) called A and B. Multiplying these: A . B = M. Where M is a scalar. Knowing B and M, can I find A? One cannot take the inverse of a vector, but is it ...
-1
votes
0answers
11 views

math abstraction of out product with condition

I have two arrays to compare. Label True/False from a comparing b : a=c(2.9,3.7,3.8, 2.7,3.3, 3.9) and b=c(18,21, 30 ,21, 17, 27) And I use ...
7
votes
3answers
84 views

Is $[X,Y] \neq 0$ the sufficient condition of $e^{X+Y} \neq e^Xe^Y$?

We know that if X commutes with Y, where X and Y are $n\times n$ matrices, then we have $$e^{X+Y}=e^Xe^Y$$ However, can we conclude that $e^{X+Y} \neq e^Xe^Y$ if X doesn't commute with Y ? Is there ...
1
vote
2answers
46 views

Distance between points

Suppose I have two matrices each containing coordinates of $m$ and $n$ points in 2 D. Is there an easy way using linear algebra to calculate the euclidean distance between all points (i.e., the ...
0
votes
1answer
17 views

Determinant of Gram matrix is non-zero, but vectors are not linearly independent

From Wikipedia: a set of vectors is linearly independent if and only if the Gram determinant (the determinant of the Gram matrix) is non-zero. But consider the matrix M: ...
-2
votes
0answers
43 views

A question in matrix norm [on hold]

Let $I,A \in {M_n}$ and suppose $\left| {\left\| . \right\|} \right|$ be a matrix norm $\left| {\left\| I \right\|} \right| \ge 1$ and $\left| {\left\| A \right\|} \right| < 1$($I$ is identity ...
1
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0answers
5 views

An equality between maximums of two logdet expression

I have the following question. Let $K$ be a positive-definite $N\times N$ real-valued matrix (I'll denote this by $0\prec K$ and will subsequently assume all matrices are $N\times N$ and real-valued) ...
3
votes
1answer
49 views

Determinant proof using its properties

Prove without expanding: \begin{equation} \begin{vmatrix}bc&a^2&a^2\\b^2&ac&b^2\\c^2&c^2 & ab\end{vmatrix} = ...
1
vote
3answers
81 views

$\frac{1}{{1 + {\left\| A \right\|} }} \le {\left\| {{{(I - A)}^{ - 1}}} \right\|}$

Let a matrix norm $ {\left\| . \right\|}$ have the property that $ {\left\| I \right\|} = 1$ and $ {\left\| A \right\|} < 1$. Why does the following inequality hold? $$\frac{1}{{1 + \left\| A ...
0
votes
0answers
43 views

How to find one matrix, which is subject to $B^3 = A$. How much is such matrices? [duplicate]

Here I have a problem with row echelon form. $$A := \begin{bmatrix}-6 & 3 & 7 \\ 0 & -1 & 0 \\ -14 & 12 & 15\end{bmatrix}$$
4
votes
1answer
82 views

If $\det A=1$ and the matrices $A^{2015}$ and $A^{2017}$ are integer, is $A$ an integer matrix?

Assume $\det(A) = 1$ and all the numbers in the matrices $A^{2015}$ and $A^{2017}$ are integers. Can I say that all numbers in $A$ are integers too? How can I prove it?
3
votes
3answers
27 views

Determinant of symplectic matrix

A $2n \times 2n$ matrix $S$ is symplectic, if $SJ_{2n}S^T=J_{2n}$ where \begin{equation} J_{2n} = \begin{bmatrix} 0 & I_n \\ -I_n & 0 \end{bmatrix}. \end{equation} My question is, how to ...
1
vote
1answer
13 views

Invertibility of Product implies invertibility of factors

Say $C=AB$ where $A,B,C$ are all $n\times n$ matrices. It's easy to show that if $A$ and $B$ are invertible then $C$ is invertible --> $C^{-1}=B^{-1}A^{-1}$. Does the converse hold? That is, if $C$ ...
0
votes
1answer
41 views

$\left\| {\left| {BA - I} \right|} \right\| < 1$ $ \Rightarrow $ $A$ and $B$ are both nonsingular

Let $A,B \in {M_n}$ satisfy the inequality $\left\| {\left| {BA - I} \right|} \right\| < 1$ and $\left\| {\left| . \right|} \right\|$ be a matrix norm on ${M_n}$.Why do $A$ and $B$ are both ...
0
votes
1answer
19 views

reordering the indices of a matrix

Let $A$ be an $n \times n$ matrix of rank r. Then by reordering the indices if necessary we can bring the matrix in the form $(\frac{A_1}{A_2})$ where $A_1$ is an $r \times n$ matrix, $A_2$ is an $n-r ...
0
votes
0answers
32 views

Sum of two independent Continuous-Time Markov Chains [on hold]

This is the first time I have come across a question involving the sum of two independent continuous time Markov Chains.I know you can find the sum of two random variables Z = X + Y by finding the ...
1
vote
1answer
25 views

Is $\left\|A^TA(x-y)\right\| = \left\|A^TA\right\|\times \left\|x-y\right\|$ correct? $A \in \mathbb{R}^{n \times n}$

In the derivation of following, I meet a dumb problem: Note: 1. $\left\|\: \cdot \,\right\|$ is the $l_2$ norm. 2. $A \in \mathbb{R}^{n \times n} $ 3. $x,y \in \mathbb{R}^{n}$ ...
4
votes
1answer
45 views

Is $\det(U_1\Lambda_1 U_1^t +U_2\Lambda_2 U_2^t +I)\le \det(\Lambda_1 +\Lambda_2 +I)$ correct?

I want to simplify or find an upper bound for the determinant $|K_1+K_2+I|$ where $I$ is identity matrix, $K_1$ and $K_2$ are positive semi-definite matrices of size $n$ and thus can be written as ...
0
votes
0answers
46 views

eigenvalues of $A^TA$ and $AA$

I am a little bit confused about such fundamental problems: Suppose 1. $Ax=\lambda x$. 2. $A \in \mathbb{R^{n \times n}} $. Case I: $$A^TAx = \lambda A^Tx=\lambda \lambda x=\lambda^2x$$ ...
1
vote
2answers
33 views

Eigenvalues of a transition probability matrix

I have read that, for $$I - \alpha P$$ where $I$ is the $n\times n$ identity matrix, $\alpha \in (0,1]$, and $P$ is the transition probability matrix with dimensions $n \times n$, $I - \alpha P$ is an ...
0
votes
1answer
16 views

“Averaging” transformation matrices?

I have a question on how best to "average" transformation matrices. Say that I have n number of 4x4 transformation matrices, and I wanted to find a matrix that approximated each one of the n 4x4 ...
2
votes
1answer
36 views

Is my algorithm correct? (Polar decomposition)

I cant seem to find my mistake. Consider this matrix $T = $\begin{bmatrix} 2 & 1 & 1 \\[0.3em] -1 & 2 & 0 \\[0.3em] 0 & 1 & -1 \end{bmatrix} I need ...
1
vote
0answers
52 views

Can a matrix be similar to more than one matrix?

I have a little query about similar matrices I've been struggling with. Suppose I have a 5x5 diagonal matrix A with 5 distinct eigenvalues as entries in the main diagonal. The question is, to how ...
0
votes
0answers
29 views

Reflection matrix and algebraic multiplicity

Let $Q\in\mathbb{M}_4(\mathbb{R})$ a reflection matrix onto $R(A)$ subspace, where $A\in\mathbb{M}_{4\times 3}(\mathbb{R})$ is defined by ...
2
votes
4answers
33 views

Finding a matrix representation of the transpose transformation

Define $T : M_{n×n}(\mathbb{R}) → M_{n×n}(\mathbb{R})$ by $T(A) := A^t$. I know this transformation is linear and just takes a matrix and spits out it's transpose. I also know that the transpose is ...
3
votes
1answer
38 views

Inverse of a matrix and its transpose

I'm trying to figure out why the calculation below works. I do know that $(A^T)^{-1} = (A^{-1})^T$. The matrix A = $\begin{bmatrix} 1 & -1 & 0 \\ 1 & 1 & -1\\ 1 & 2 & -1 ...
1
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1answer
40 views
3
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1answer
110 views

Possible to solve $A + P^{-1}AP = B$?

Is it possible to solve for a matrix $A$ in an equation involving a matrix similar to $A$, of the form $$A + P^{-1}AP = B$$? The solution I'd be looking for would be for $A$ in terms of $P$ and $B$, ...
1
vote
2answers
41 views

Rotation matrix

I'm finding different results for the 3D rotation matrix in the XY plane from different sources and I was hoping for someone to help clarify. In my "applications of vector calculus" book, the matrix ...
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votes
0answers
70 views

Calculate the fifth root of the matrix [on hold]

I have got the following matrix that I want to calculate the fifth root of. What is a good approach? $$\begin{bmatrix}1&2&0\\-1&-2&0\\3&5&1\end{bmatrix}$$
0
votes
2answers
22 views

methods of constructing a matrix from its null space span

I have a matrix of size $4\times3$ and its null-space span is $\{(1,2,3), (2,5,7)\}$. How can I find the original matrix? It is not obvious from the span which vectors are free.
0
votes
2answers
34 views

I'm struggling to find this transformation matrix

$T:\Bbb{P}_3 \to \Bbb{P}_3$ is a linear transformation such that: $$\begin{align} T\left(-2 x^2\right) &= 3 x^2 + 3 x \\ T(0.5 x + 4) &= -2 x^2 - 2 x - 3 \\ T\left(2 x^2 - 1\right) ...
1
vote
1answer
25 views

How to find that two adjacency matrices are equal

What is the easiest way to tell if these two graphs are isomorphic and how do I know which nodes in both graphs are the same. I've made the adjacency matrices but they are pretty big. I think I need ...
2
votes
2answers
25 views

Algebric and geometric multiplicity and the way it affects the matrix

Given a matrix $A$. Suppose $A$ has $\lambda_1,\dots,\lambda_n$ eigenvalues each with $g_i$ geometric multiplicity and $r_1,\dots,r_n$ algebric multiplicity, $g_i\leq r_i$. Given this information ...
4
votes
3answers
67 views

Why does $\frac{1}{{\left\| {\left| {{A^{ - 1}}} \right|} \right\|}} \le \left\| {\left| B \right|} \right\|$?

Let $A,B \in {M_n}$ suppose that the following statements are true: $A$ is nonsingular, $A+B$ is singular, $\left\| {\left| . \right|} \right\|$ is matrix norm. Why is it true that: ...
1
vote
1answer
76 views

For which $\beta \in \mathbb{C}$ is the matrix $A=\bigl(\begin{smallmatrix} 0&1\\1&\beta \end{smallmatrix}\bigr)$ diagonalisable?

I have got a question refering to the following problem. Let $K=\mathbb{C}$. For which $\beta \in \mathbb{C}$ is this matrix diagonalisable? $$A=\pmatrix{0&1\\1&\beta}$$ I think that it is ...