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
7 views

$A,B$ be Hermitian.Is this true that $tr(AB)^2\le tr(A^2B^2)$?

Suppose $A,B \in {M_n}$ be Hermitian.Is this true that $tr(AB)^2\le tr(A^2B^2)$?
0
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0answers
9 views

Equivalence class of matrices on linear form

We well know that if $M$ is a matrix on a field $k$ then the equivalence class of $M$ is uniquely determined by its rank (where $A \sim B$ if $\exists P,Q $ invertibles such that $PAQ^{-1}=B$). ...
1
vote
0answers
24 views

About an inequality

Say $\{a_1,a_2,..,a_n\}$ and $\{b_1,b_2,...,b_n\}$ be two sets of n real numbers. Say that there is at least one element different between the two sets. Can one estimate for how large a positive ...
0
votes
0answers
28 views

Why does $rank(A) \ge \dfrac{{{{(trA)}^2}}}{{(tr{A^2})}}$? [duplicate]

Let $A \in {M_n}$ and Hermitian.Why does $rank(A) \ge \dfrac{{{{(trA)}^2}}}{{(tr{A^2})}}$?
0
votes
3answers
61 views

Let $A = \left( {\begin{array}{*{20}{c}} 0&1\\ 0&0 \end{array}} \right)$.What is numerical range $A$ [on hold]

Let $A = \left( {\begin{array}{*{20}{c}} 0&1\\ 0&0 \end{array}} \right)$. How to find its numerical range $W(A) = \{ {x^*}Ax:x \in {S^1}\}$?
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0answers
22 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 ...
1
vote
0answers
18 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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0answers
24 views

How to get real irreducible matrix representations from the complex irreducible matrix representations?

I'm trying to get real symmetry adapted orbitals for molecules with icosahedric symmetry (point groups $I$ and $I_h$) using the complete projector operator (truly projector if i=j): \begin{equation} ...
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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 $$ \begin{cases} 3x + y+z=18 \\ 4x + 2y+3z=12 \\ 7x + 8y+5z=9 \end{cases} $$
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votes
1answer
13 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} ...
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vote
3answers
31 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
votes
1answer
23 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 $\max\left(\operatorname{Re} \left\{ \dfrac{x^* Ax}{x^* x}:0 \ne x \in C^n\right\} \right)$?
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votes
1answer
39 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
vote
1answer
31 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
31 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
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0answers
12 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
27 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 $\lambda_1, \lambda_2, \dots$. ...
-2
votes
0answers
21 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.
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votes
1answer
38 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
91 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
39 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
37 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
33 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: $$ ...
3
votes
1answer
68 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
vote
1answer
16 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
vote
1answer
45 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
50 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
32 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
206 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
vote
2answers
45 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
87 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
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
50 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
11 views

An equality between maximums of two logdet expressions

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
52 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. $$A := \begin{bmatrix}-6 & 3 & 7 \\ 0 & -1 & 0 \\ -14 & 12 & 15\end{bmatrix}$$