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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0answers
20 views

Closed form of a matrix product

Is there any closed form or a bound for a matrix product of this kind $$ P=\prod_{i=1}^n \begin{pmatrix} 1-a & a \\ b_i & 1-b_i \end{pmatrix}, \quad a,b_i \in [0,1] $$ for an arbitrary ...
0
votes
0answers
12 views

Comprehensive easy to understand resource for learning matrix decompositions?

I am working on my thesis which is widely depended on knowledge about matrix decompositions. I have studied linear algebra with the help of YouTube videos, MITOpenCourseWare videos and Prof.Gilbert ...
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votes
0answers
27 views

augmented matrix question [on hold]

Please show me the augmented matrix solution with steps for the system $$ \begin{cases} 3x + y+z=18 \\ 4x + 2y+3z=12 \\ 7x + 8y+5z=9 \end{cases} $$
0
votes
1answer
12 views

Finding an ordered basis to diagonalize Transpose matrix.

We define $T : M_{n \times n}R \to M_{n\times n}R$ by $T(A) = A^t$. We can write the matrix representation of this transformation as: $[T]_\beta^\beta = \begin{pmatrix} ...
1
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3answers
29 views

For which values of $a$ does the matrix can be diagnolized?

Given $$A=\begin{pmatrix} 2 & 0 & 0\\ a & 2& 0\\ a+3 & a &-1 \end{pmatrix}$$ For which values of $a$ can $A$ be diagonal? I found that $p_A(x)=(x-2)^2(x+1)$ and tried to ...
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0answers
10 views
0
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1answer
21 views

What is $\max(\operatorname{Re} \{ \frac{x^* Ax}{x^* x}:0 \ne x \in C^n\} )$?

Let $A = \left( \begin{array}{*{20}{c}} 1&2\\ 0&1 \end{array} \right)$. What is $\max\left(\operatorname{Re} \left\{ \dfrac{x^* Ax}{x^* x}:0 \ne x \in C^n\right\} \right)$?
5
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1answer
38 views

Understanding a part of a proof involving Hilbert-Schmidt norm

I came across a proof I do not seem to understand fully, a screenshot is provided below. my concerns are the following: Why does the fact that $||T||_2 = ||UT||_2$ for every unitary U, allow us ...
1
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1answer
30 views

What are the facts used in each step of this proof?

What are the facts used in each step of this proof ? Suppose that $A\in F^{nm}$ and $B\in F^{ml}$ $$\begin{align}rank A + rank B &= rank\begin{bmatrix}0 & A\\B & 0\\ \end{bmatrix}\\ ...
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5answers
30 views

Eigenvalues of different symmetric $(2n+1)\times(2n+1)$ matrix

I ve looked at other similar post but I could not find help with them
0
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0answers
11 views

find Jordan form

Determine the jordan form of $A = \begin{pmatrix} 1 & 2 & 3\\ 0 & 4 & 5\\ 0 & 0 & 4 \end{pmatrix} $ First, I find the characteristic polynomial. $C_A(x)=(x-1)(x-4)^2$. ...
1
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4answers
26 views

Does an arbitrary matrix $X \in M_{n \times p}$ have a SVD?

I have proven, as below, that if $X \in M_{n \times n}$ is symmetric, then it has a SVD. $D(\lambda_i) = \text{Diag}(\lambda_i)$ is a diagonal matrix with entries $\lambda_1, \lambda_2, \dots$. ...
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0answers
20 views

Is this proof correct? Matrix ring. Center. [duplicate]

I found this https://crazyproject.wordpress.com/2010/08/23/the-center-of-a-matrix-ring-over-a-commutative-ring-is-precisely-the-scalar-matrices/ and I have very bad day, so I ask you to confirm.
0
votes
1answer
37 views

Prove that if $A,B\in M_n(\mathbb{F})$ are $(n-1)$-nilpotent then they are similar.

If $A,B\in M_n(\mathbb{F})$ are $n-1$ nilpotent, prove they are similar. Can I say that, since their minimal polynomial is $X^{n-1}$ they are similar? I know that If $A,B$ are similar, they have ...
1
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0answers
23 views

Solving System of Linear Equations

These are the two known equations: $$(I_2+I_3)-\frac{I_1+I_4}{I_1+I_2+I_3+I_4} = \frac{2x}{L}$$ $$(I_2+I_4)-\frac{I_1+I_3}{I_1+I_2+I_3+I_4} = \frac{2y}{L}$$ where I know the values of $(x,y,L)$. How ...
7
votes
2answers
87 views

$A^2=A^*A$.Why $A$ is Hermitian matrix?

Let $A$ be $n \times n$ matrix and $A^2=A^*A$. Why is $A$ a Hermitian matrix?
1
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1answer
37 views

Spectrum of the matrix $A=(a_{ij})$ where $a_{ij}=i+j$

What is the spectrum of the matrix $A=(a_{ij})_{n\times n}$ where $a_{ij}=i+j$ for any $n$. Also, what are the eigenvectors corresponding to their eigenvalues? Progress. This matrix is definitely ...
2
votes
1answer
34 views

Why does ${\lambda _{\max }}(A) = \max \{ \frac{1}{{{x^*}x}}:{x^*}Ax = 1\} $?

Let $A \in {M_n}$ be hermitian and suppose that at least one eigenvalue of $A$ is positive ($\lambda $ is eigenvalue of $A$). Why does ${\lambda _{\max }}(A) = \max \{ \frac{1}{{{x^*}x}}:{x^*}Ax = 1\} ...
2
votes
0answers
31 views

Second order derivation of Quadratic form

I would like to find the second order derivative of a Quadratic form. Assume we have a random complex column vector $x$ and a real constant value $C$. I am interested in computing the following: $$ ...
2
votes
1answer
64 views

Matrix and field extension

It is given that $F\subset K$ are fields. $A$ is a matrix of size $n\times n$ over $K$. I need to prove that there exist $c_1,\ldots,c_k\in K$, linearly independent over $F$, and matrices ...
1
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1answer
15 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
44 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
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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
28 views

A inquality in matrix norm [duplicate]

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
12 views

Subtraction of quadratic forms with positive-definite matrix? [on hold]

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
vote
1answer
10 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
27 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
votes
1answer
23 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
43 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
15 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
49 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
39 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
31 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
203 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
44 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 ...
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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
86 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
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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
18 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
48 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
vote
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) ...
4
votes
1answer
51 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
89 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
44 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
84 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
30 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
20 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
49 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 ...