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), ...

learn more… | top users | synonyms (2)

0
votes
1answer
17 views

If $A$ and $B$ are $n\times n$ matrices, then $((AB)^{-1})^T=(A^{-1})^T (B^{-1})^T$

Please help me to solve this. Prove that if $A$ and $B$ are $n\times n$ matrices, then $((AB)^{-1})^T=(A^{-1})^T (B^{-1})^T$. a problem involve transpose and inverse of matrices. check the ...
-2
votes
2answers
10 views

Finding the values of the rank and nullity

$Q$ is a $3 \times 3$ matrix that is not invertible. What are all the possible values of the rank and nullity of $Q$ ?
0
votes
0answers
16 views

centralizer of A is equal to < A >

A=(0 a; 1 b) How to prove or disprove that centralizer of A is equal to < A > (matrices generated by A)? For a matrix to be in the centralizer I think by its definition it must commute with A. I ...
0
votes
0answers
10 views

Translate and Rotate mesh

I have a mesh constituted of some vertices in 3d space, let's call them $(x_1,y_1,z_1),(x_2,y_2,z_2),\cdots,(x_n,y_n,z_n)$. The mesh's central point is $(0,0,0)$. How to find out the new coordinates ...
2
votes
2answers
24 views

Scaling a matrix to make its eigenvalues fall within a certain interval

Suppose I have a diagonalizable matrix $M$ which has all its eigenvalues between $a$ and $b$. Is it possible to scale $M$ to $M_S$ such that all the eigenvalues of $M_s$ lie in the interval $[-1,1]$? ...
3
votes
1answer
31 views

Prove a vector in $\ell^2(\mathbb{Z})$ is zero

Suupose we take a vector $\vec{c}\in\ell^2(\mathbb{Z})$ where $$c(i)=\sum_{k=1}^\infty\frac{c(-k+i)+c(k+i)}{k+1}$$ That is, every elements of the vector is a series with the other terms in $\vec{c}$. ...
0
votes
1answer
16 views

Let $A$ be a $2\times2$ matrix with real entries such that $A$ is invertible. If $Det(A)=k$,and $Det(A+kadj(A))=0$

Let $A$ be a $2\times2$ matrix with real entries such that $A$ is invertible. If $Det(A)=k$,and $Det(A+kadj(A))=0$, then find the value of $Det(A-kadj(A))$ My attempt: $Det(A+kadj(A))=0$ ...
0
votes
1answer
14 views

Simplification of a product of three matrices

Define $$\mathbf{c}_t = \begin{bmatrix} x_{1t} \\ x_{2t} \\ \vdots \\ x_{Nt} \end{bmatrix}\in \mathbb{R}^N$$ where all entries are in $\mathbb{R}$, $t = 1, 2, \dots, p+1$. I am trying to simplify ...
1
vote
0answers
27 views

Matrices representing a map between free modules of infinite rank and Fitting's Lemma (Eisenbud)

p.497 of Commutative Algebra with a View Toward Algebraic Geometry, Eisenbud: If $\phi: F \rightarrow G$ is a map of free modules, then $I_j\phi$ is the image of the map $$\Lambda^j F ...
0
votes
1answer
26 views

Find the dimension and a basis of a subspace

Let $U$ is the set of all commuting matrices with matrix $A= \begin{bmatrix} 2 & 0 & 1 \\ 0 & 1 & 1 \\ 3 & 0 & 4 \\ \end{bmatrix}$. Prove ...
0
votes
2answers
22 views

System of linear equations - Resolution

$$ \left( \begin{matrix} \pi_1 & \pi_2 & \pi_3 \end{matrix} \right) = \left( \begin{matrix} \pi_1 & \pi_2 & \pi_3 \end{matrix} \right) \begin{bmatrix} 0.6 & 0.3 & ...
0
votes
0answers
23 views

If $ 0<A <B$, is it true that $||A||_p < ||B||_p$ for all $p$ positive integer?

I have got some questions regarding matrix norms and inequalities. We only consider square, nonsingular matrices in the following. If $ 0<A <B$, is it true that $||A||_p < ||B||_p$ for all ...
0
votes
2answers
14 views

Finding the general solution of a system of linear equations

so I've come across this question in preparation for an exam: Let $A$ be a $4\times 4$ matrix where $rank(A)=3$. The vectors $(1,2,0,-1),(0,2,1,1)$ are solutions to the system ...
0
votes
1answer
22 views

Multiplication of block matrices

Let $J_{m \times n}$ be an $m \times n$ matrix of $1$'s (and to abbreviate we write $J_m=J_{m \times m}$) and let $M=\begin{pmatrix} 0_n & J_{n \times m} \\ J_{m \times n} & 0_m ...
1
vote
2answers
23 views

Linear maps of polynomials, the bases of the space and their corresponding matrix.

Suppose $T \in \mathrm{Hom}(\mathscr{P}_3(\mathbb{R}),\mathscr{P}_4(\mathbb{R}))$ is defined by: $$Tp(x)=(x^2p(x))',$$ for all $x \in \mathbb{R}$ and $S \in\mathrm{Hom} ...
0
votes
1answer
40 views

Theorems restricting the eigenvalue of a matrix

I have a square matrix $C$, whose entries I will denote by $c_{ij}$, and I would like to bound the magnitude of its eigenvalues. Each $c_{ij}$ is defined in terms of $s_{ij}$ and $S_j$ as follows: ...
0
votes
1answer
20 views

How to calculate the variation of a matrix?

Suppose we have two diagonal matrices $$ A_{\mu \nu}=\left(\begin{array}{cccc} \rho(t) & 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0 ...
2
votes
1answer
37 views

Group isomorphism between two groups .

Let $SO_2(\mathbb{R})$ be the group of rotations of the circle under the operation of composition. And consider $\mathbb{R}$ as an additive group. Prove that $SO_2(\mathbb{R}) \cong \begin{Bmatrix} ...
2
votes
1answer
25 views

Check if two square matrices are similar.

Check if matrices $A= \begin{bmatrix} 1 & 1 & 5 \\ 0 & 2 & 0 \\ 0 & 0 & 2 \\ \end{bmatrix}$ and $B=\begin{bmatrix} 1 & 7 ...
0
votes
0answers
6 views

Is $X_{k+1}=\frac{1}{N}\sum_{i=1}^N \Pi_{X_{k}^{1/2}v_i}$ globally convergent?

Let $X_0=X_0^\top\in\mathbb{R}^{n\times n}$ be a symmetric positive definite matrix, let $v_i\in\mathbb{R}^n$, $i=1,\dots, n$, be a set of $n$-dimensional real vectors and pick an integer $N>0$. I ...
0
votes
0answers
8 views

Entry Expansionof Power Matrix

Suppose $A:=\{a_{i,j}\}, 1\le i,j, \le n$ is a $n\times n$ matrix with real positive entries. Now replace the constant $a_{1,1}$ with a real variable $x$. Denote by $A_x$ the resulting variable-Matrix ...
0
votes
1answer
24 views

How to show that $M_{2\times 2}(\mathbb{R})=W_1\oplus W_2$ based on the following assumption?

Let the subspaces $W_1=\{\begin{pmatrix}a&b\\-b&a \end{pmatrix}|a, b\in \mathbb{R}\}$ and $W_2=\{\begin{pmatrix}c&d\\d&-c \end{pmatrix}|c, d\in \mathbb{R}\}$ of $M_{2\times ...
1
vote
2answers
20 views

How to show if the following subset $W$ is a subspace of a vector space $V$?

$1.$ $V=P_n(\mathbb{R}), $and $ W=\{p(x)\in P_n(\mathbb{R})\mid p(1)+p(2)+p(3)=0 \}$ $2.$ $V=M_{n\times n}(\mathbb{R}), $and $ W=\{A\in M_{n\times n}(\mathbb{R}) \mid A \text{ is not symmetric}\}$ ...
0
votes
0answers
25 views

An inequality on the rank of a block matrix

Let $\mathbb F$ be a field, and let $r_1, r_2, s_1, s_2$ be positive integers. Consider the matrix $$X:=\left[\begin{array}{cc} A & B \\ C & D \end{array} \right],$$ where $A \in \mathbb ...
0
votes
3answers
28 views

How can you enlarge a shape about a point other than (0,0), using matrices?

If I want to enlarge a shape, $A$, by scale factor $k$ about $\left(0,0\right) $ I multiply each point (in the form $\begin{bmatrix}x\\y\end{bmatrix}$) by $kI$. However, I can't work out a general ...
0
votes
0answers
25 views

In matrix algebra, what's the name for the inverse operation of pre- or post- multiplication?

For example, in this typical equation: $$\mathbf{Mv}-\lambda \mathbf{v}=\mathbf{0}$$ (where $\mathbf{M}$ is a symmetric matrix, $\mathbf{v}$ is a vector, $\lambda$ is a scalar, and $\mathbf{0}$ is a ...
0
votes
1answer
35 views

Compare analytic model with numerical, mass spring system.

So I'm trying to solve a problem here and I have been working on it all day, clearly i'm in need of some guidance. I have a rod of length $L$ and cross section area $A$, Young's modulus $E$ and ...
3
votes
2answers
36 views

Intuition behind: Integral operator as generalization of matrix multiplication

So I am teaching myself more in-depth about integral operators and every once and awhile I see this little 'factoid', that integral operators are generalizations of matrix multiplications. In ...
2
votes
1answer
25 views

For every integer $n>1$ , does there exist a diagonal matrix $D \in M(n,\mathbb R)$ such that $AD=DA $ holds only if $A$ is diagonal?

Is it true that for every integer $n>1$ , there exist a diagonal matrix $D \in M(n,\mathbb R)$ such that $A \in M(n,\mathbb R)$ and $AD=DA \implies A$ is also a diagonal matrix ?
0
votes
1answer
39 views

If $\operatorname{rank}A=k$ then $A=A_1+…+A_k$ such that $\operatorname{rank}A_i=1$ [on hold]

Let $A\in M_n$ and $\operatorname{rank}A=k$. Is the following true? There are $A_i\in M_n$ ($i=1,...,k$), such that $\operatorname{rank}A_i=1$ and $A=A_1+....+A_k$.
3
votes
3answers
31 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 ...
1
vote
0answers
31 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 ...
1
vote
0answers
17 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 $\mathbb{I}$. ...
2
votes
1answer
45 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: ...
1
vote
0answers
36 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 ...
0
votes
0answers
7 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?
0
votes
1answer
28 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} = ...
-1
votes
1answer
25 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 $0$?
0
votes
1answer
23 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 ...
0
votes
1answer
20 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 $b=d$?
1
vote
0answers
22 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, ...
0
votes
0answers
22 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 ...
0
votes
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 ...
2
votes
3answers
27 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
vote
1answer
30 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 ...
0
votes
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
votes
1answer
25 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 ...
1
vote
1answer
23 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 $f_A(x)=f_B(x)$ ...
1
vote
2answers
62 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 ...
1
vote
2answers
35 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.