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
12 views

Proof of a trace property

$Tr(XY) = 1$ and $Tr(Y) = 1$ implies that $Tr(X) = 1$. I tried to prove by contradiction and switch the dummy variable of $X$ and $Y$. But I don't think my approach is right and if there is any much ...
0
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1answer
15 views

If $rankA=k$ then $A=A_1+…+A_k$ such that $rankA_i=1$

Let $A\in M_n$ and $rankA=k$. It is true that: «There are $A_i\in M_n$ ($i=1,...,k$), such that $rankA_i=1$ and $A=A_1+....+A_k$ »
2
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2answers
14 views

Find the Jordan normal form of a nilpotent matrix $N$ given the dimensions of the kernels of $N, N^2, N^3$

Let $N\in \text{Mat}(10 \times 10,\mathbb{C})$ be nilpotent. Furthermore let $\text{dim} \ker N =3 $, $\text{dim} \ker N^2=6$ and $\text{dim} \ker N^3=7$. What is the Jordan Normal Form? The only ...
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0answers
17 views

Is $U(1)$ a normal subgroup of $U(2)$ and does the question even make sense?

I have been wondering whether $U(1)$, defined as the group of complex phases (edit for clarity: I mean complex numbers of unit absolute value, such as $e^{i\alpha}$ with $\alpha \in \mathbb{R}$) with ...
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0answers
14 views

$SU(n)$ generators

What is the generalization of the Pauli matrices and Dirac matrices in higher dimensions? I am actually looking for $\sqrt{\mathbb{I}}$ but I can't use the principal root which is just $\mathbb{I}$. ...
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0answers
19 views

Gaussian elimination algorithm performance

I am developing the quadratic sieve algorithm and I reached a new bottle neck: The matrix processing. I been reading quit a lot about this topic and I found many solutions Gaussian elimination: ...
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0answers
22 views

Getting The Inverse Of A Positive Definite Matrix By Mutiplying It On A Diagonal One

Is the following true ? The inverse of a positive definite matrix is also positive definite and since symmetric then we could write the following: $A=PP^T, \space B = A^{-1} = P\Lambda P^T$ since ...
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0answers
6 views

how to write amatlab code for document representation using second order tensor [on hold]

My project is on document representation using tensor.How i will represent a document using second order tensor in matlab?
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1answer
23 views

System of differential equation (Matrix form)

I'm trying to solve this system $$ M\ddot{X}(t) = KX(t) $$ where M is a known diagonal matrix and K is a symmetrical known matrix. I'm asked to do the ansatz $Y(t) = M^{1/2}X(t)$ where $M^{1/2} = ...
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1answer
24 views

How can a non-zero matrix $A$ be found such that Adj$(A) = 0$? [on hold]

Is it possible to find a $3 \times 3$ matrix $A$ such that it's adjoint is $0$?
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1answer
20 views

Is $X'X$ positive definite a necessary condition for $X'X$ to have full rank?

Let $X$ be a $T \times K$ matrix. $X'X$ positive definite means that for all $c \not = 0$ $c'X'X c >0$, so then $(Xc)' Xc > 0$ which implies $(X\cdot c)\cdot (X\cdot c)$ (I'm not sure what ...
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1answer
18 views

Find upper triangular matrix C such that Cx=y

In the image above, how does one know that $c=e$ and $c$ is not equal to $f$? and $e$ is not equal to $f$? How does one know that $b=d$?
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0answers
14 views

Spectral norm of lower triangular perturbation

Suppose $A\in R^{n×n}$ is a matrix equal to sum of the Identity matrix and a lower triangular matrix $L$. \begin{equation} A=I+L \end{equation} All diagonal entries of $L$ are equal to $0$, so that, ...
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0answers
17 views

Bounding lower triangular perturbation

Suppose $A\in R^{n\times n}$ is a matrix equal to sum of the Identity matrix and a lower triangular matrix $L$. Diagonal entries of $L$ are $0$. \begin{equation} A=I+L \end{equation} Define spectral ...
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0answers
23 views

$M_n$ is the subspace of all square matrices with trace $0$, what is the dimension of $M_n$?

There is an older post with many explanations of a more specific and less general case of a $4$ by $4$ Find the dimension of the space of $4\times 4$ real matrices with zero trace I didn't quite ...
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3answers
22 views

Proof: dimension of the vector space of solutions to the system Bx=0

I've run into a matrix dimension proof I'm having some trouble with: Let $A,B$ be $n\times n$ matrices, and let $P(A),P(B),P(AB)$ be the vector spaces of solutions to the systems ...
1
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1answer
28 views

Comparing matrix norm with the norm of the inverse matrix

I need help understanding and solving this problem. Prove or give a counterexample: If $A$ is a nonsingular matrix, then $\|A^{-1}\| = \|A\|^{-1}$ Is this just asking me to get the magnitude of ...
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2answers
48 views

Prove that elementary matrices perform row operations

How to prove that elementary matrices actually perform their intended row operations: multiplying by a constant, adding a multiple of one row to another, and switching two rows? I've seen examples ...
3
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1answer
24 views

$A \in SO(3,\mathbb R)\setminus\{I\}$ , then there are exactly two points in $S^2:=\{(x,y,z)\in \mathbb R^3:x^2+y^2+z^2=1\}$ which are fixed by $A$?

Let $A \in SO(3,\mathbb R)\setminus\{I\}$ , then is it true that there exist exactly two points in $$S^2:=\{(x,y,z)\in \mathbb R^3:x^2+y^2+z^2=1\}$$ which are fixed by $A$? Or equivalently we ...
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1answer
22 views

If $A$ is unitary and $f_A(x)=f_B(x)$ and $m_A(x)=m_B(x)$ then $A$ is similar to $B$

Given $A_{n\times n},B_{n\times n} \in \mathbb C$ then: if $A$ is unitary and the characteristic polynomial $f_A(x)=f_B(x)$ then $B$ is also unitary. if $A$ is normal and $f_A(x)=f_B(x)$ ...
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3answers
53 views

The formula $\DeclareMathOperator{tr}{tr}\mathrm{adj}(A)=\tfrac{1}{2}[(\tr A)^2-\tr(A^2)]I_3-[\tr A]A+A^2$ for the adjoint of a $3\times 3$ matrix

Let $A$ be a square matrix of order $3$. Prove that $$ \operatorname{adj}(A) = \tfrac{1}{2} \bigl[ (\operatorname{tr} A)^2 - \operatorname{tr}(A^2) \bigr] I_3 - [\operatorname{tr} A] A + A^2 ...
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2answers
33 views

The matrix $A = (a_{i,j})$ in which $a_{i,j} = 0$ if $i ≥ j$ is nilpotent.

A square matrix $A$ is called nilpotent if $A^m = 0$ for some positive integer $m$. Show that A $n \times n$ matrix $A = (a_{i,j})$ in which $a_{i,j} = 0$ if $i ≥ j$ is nilpotent.
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2answers
37 views

how to determine a matrix has a single eigenvalue

Find the jordan form of the matrix $$A = \begin{pmatrix} 1 & 1 & 2 & 2\\ 1 & -2 & -1 & -1\\ -2 & 1 & -1 & -1\\ 1 & 1 & 2 & 2 \\ \end{pmatrix}$$ ...
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1answer
24 views

Symmetric Matrix with Positive Eigenvalues

Not all matrix with positive eigenvalues is positive definite, i.e. $\mathbf{x}^\mathsf{T}A\mathbf{x}>0$ for all non zero vector $\mathbf{x}$. For example consider matrix $$A = \begin{bmatrix} 1 ...
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2answers
28 views

Physical meaning of cofactor and adjugate matrix

I like the way there a physical meaning tied to the determinant as being related to the geometric volume. Since the determinant can be calculated through Laplace's formula where the cofactor matrix is ...
1
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1answer
24 views

Finding an orthonormal basis for $\mathbb{C}^2$ with respect to the Hermitian form $\bar{x}^TAy$

Problem Statement: Let $A=\begin{bmatrix} 2 && 1 \\ 1 && 2 \end{bmatrix}$. Find an orthonormal basis for $\mathbb{C}^2$ with respect to the Hermitian form $\bar{x}^TAy$. I am ...
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0answers
16 views

satellites attitude determination TRIAD - how are orbital reference frame vectors constructed?

I posted this same question on space.stackexchange but never received any answer. So I am posting here hoping to get an answer as this is a quite mathematical topic. I am trying to fully understand ...
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2answers
27 views

How can we know if the minimal polynomial of a matrix has no multiple products?

If the characteristic polynomial $f_A(x)$ has multiples of the same product, for example $f_A(x)= (x+2)^2(x-1)$ so $(x+2)$ has a multiple of $2$, then is there a condition on $A$ such that we know ...
0
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1answer
34 views

If the characteristic polynomial of $A,B$ is the same does it mean that $A,B$ are similar?

If the characteristic polynomial of $A,B$ is the same does it mean that $A,B$ are similar? So I read that it's true only if $A,B$ are diagonalizable, but why? if the characteristic polynomial is ...
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0answers
25 views

What is the maximum value of coefficient $f_v$ with the constraInt that the matrix is positive semi-definite?

I am trying to solve this equation my self with my knowledge about characteristic polynomials, etc but I have placed it here earlier because I'm not a mathematician and maybe you give me ideas to ...
2
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1answer
26 views

If $x^{T}By = 1$, should $\operatorname{Tr}(Byx^{T}) = 1$?

would appreciate any hints with this question: Assume $x$, $y$ are both $n \times 1$ vectors, and that $B$ is $n\times n$. Given that $x^{T}By = 1$, should $\operatorname{Tr}(Byx^{T}) = 1$ ? Thank ...
1
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1answer
23 views

If $A^4=4A^2$ then $m_A(x)=x^2-4$ and if it isn't diagonalaziable over $\mathbb R$ then $0$ is an eigenvalue

Given $A_{n\times n} \in \mathbb R$ such that $A^4=4A^2$ then if $A$ is invertible and isn't of the form $cI, c\in \mathbb R$ then $m_A(x)=x^2-4$. if $A$ isn't diagonalaziable over ...
1
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1answer
37 views

Calculate A^8 using Cayley Hamilton Therorem

Find $A^8$ using Cayley Hamilton Therorem, when $$A = \begin{pmatrix} 0 & 0 & 0 & -1 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 2 \\ 0 & 0 & 1 & 0 ...
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3answers
38 views

Representing a linear operator on $V$ with an element of $V \otimes V^*$

I got interested by the first sentence of this wikipedia subsection. It claims that any linear operator $f:V\to V$ can be represented by an element of $V\otimes V^*$ in a very concrete way: the ...
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0answers
34 views

how to compare trace($B^4$) and trace[$(B-D)^4$] with $D$ the diagonals? [on hold]

If $B=(b_{ij})_{n\times n}$ is a real symmetric $n$ by $n$ matrix, $D = (d_{ij})_{n\times n}$, defined as $d_{ij}=b_{ij}$ if $i=j$ and $0$ otherwise. then how to compare $\text{Trace}(B^4)$ and ...
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2answers
39 views

Given $A$, $A^{-1}$ can be expressed with: $A^{-1}=bA+dI$

Given the matrix $A=\begin{pmatrix} -1 &3 &3 \\ 3& -1 & 3\\ 3& 3 & -1 \end{pmatrix}$ then $A$ is invertible and $A^{-1}$ can be expressed with: $A^{-1}=bA+dI, ...
2
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2answers
30 views

Let $A$ be a real symmetric matrix with rank $1$ , then can all the diagonal entries of $A$ be $0$ ?

Let $A$ be a square real symmetric matrix with rank $1$ , then can all the diagonal entries of $A$ be $0$ ? I know that real symmetric matrices are diagonalizable . Also if all the diagonal entries be ...
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0answers
39 views

How to find a onto homomorphism between two groups?

Consider the following subgroups of $\text{SL}(2,\mathbb{Z})$ : $A$ the subgroup of matrices with determinant $1$ : ...
3
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1answer
19 views

Frobenius norm and submultiplicativity

I read (page 8 here) that if $A$ and $B$ are rectangular matrices so that the product $AB$ is defined, then $$(1)\quad||AB||_F^2\leq ||A||_F^2||B||_F^2$$ Does that mean that the inequality above ...
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0answers
37 views

How to find out generators of the following free group?

The following is the quotient group of SL($2,\mathbb{Z}$). Consider $(H/\{-1,1\} \cap H)$ where $H=\begin{bmatrix}2\mathbb{Z}+1&4\mathbb{Z}\\2\mathbb{Z}&2\mathbb{Z}+1\end{bmatrix}$ How do ...
0
votes
2answers
29 views

how to find the index of following subgroup?

if I denotes the principal congurence group of level 2 i.e. $I=\{ M \in SL(2,Z) ; \:M \:\:\text{congruent to I} \mod(2)\}$. or I= ...
0
votes
1answer
37 views

Given a normal $A_{n\times n}$ matrix, then $\lVert A^*v \rVert = \lVert Av\rVert$ and $\langle Av,v\rangle = \langle A^*v,v\rangle$

Let a normal $A_{n\times n}\in \mathbb C^n $ matrix, then: $\forall v \in \mathbb C^n:\lVert A^*v \rVert = \lVert Av\rVert $ $\forall v \in \mathbb C^n : \langle Av,v\rangle = \langle ...
1
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1answer
50 views

Finding the inverse of A where A is of the form $A = D (I − N)$, where $D$ is diagonal with nonzero entries and $N$ is nilpotent

If a matrix can be written as $A = D (I − N)$, where $D$ is diagonal with nonzero entries and $N$ is nilpotent, then $A^{−1} = (I − N)^{−1}D^{−1}$. Use this to find inverse of: $\begin{bmatrix} 2 ...
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0answers
28 views

Why does $\bar A = \left\{ {{P_\Delta }(\lambda ):\left\| {{\Delta _j}} \right\| \le \varepsilon ,j = 0,1,2…m} \right\}$? [on hold]

Suppose ${P_\Delta }(\lambda ) = ({A_m} + {\Delta _m}){\lambda ^m} + ....... + ({A_1} + {\Delta _1}){\lambda ^1} + ({A_0} + {\Delta _0})$ is a matrix polynomial, and $\lambda $ is a complex ...
1
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0answers
40 views

Show $SO_2(\mathbb{R}) \cong\{A \in GL_2(\mathbb{R}) : A^tA = I, \det A = 1\}$.

Show $SO_2(\mathbb{R}) \cong\{ A \in GL_2(\mathbb{R}) : A^tA = I, \det A = 1\}$, where $SO_2(\mathbb{R})$ is the group of rotations of the circle under the operation of composition. Attempt: ...
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0answers
22 views

Finding transform matrix from resulting multiplypoint function

Two matrix transformation functions exist within the Unity3D API: 1) MultiplyPoint 2)MultiplyPoint3X4 3X4 matrix (2) preforms a standard transform against a vector (And ofc is easily replicated ...
0
votes
1answer
33 views

Matrix multiplication to make all numbers in a 3x3 matrix negative

Let's say I have the matrix called Delta, $$ \begin{matrix} a & b & c \\ d & e & f \\ g & h & i \\ \end{matrix} $$ What would I have ...
0
votes
2answers
28 views

Operations on 3x3 matrix through matrix products

What would I have to multiply the following matrix ... $$ \begin{matrix} a & b & c \\ d & e & f \\ g & h & i \\ \end{matrix} $$ by so ...
0
votes
1answer
20 views

Determinant of block matrix with null row vector

I'm a bit confused on a problem. I've been given an $(n+1)\times(n+1)$ square matrix, which is written in the form of a block matrix with the following dimensions $ \begin{bmatrix} (1x1) ...
0
votes
0answers
25 views

Let $X : S_3 → GL_2(\mathbb{R})$ . Compute the six matrices {$X(\pi) : \pi \in S_3$} and show they faithfully represent $S_3$.

Consider an equilateral triangle $V_1V_2V_3$ with center at the origin, and vertex $V_1 = (0,1)$ and vertices $V_1, V_2, V_3$ in counterclockwise order. Consider the action of the symmetric group ...