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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4
votes
4answers
87 views

is it true every left inverse of a matrix is also right inverse of it?

I am wondering that, consider there are $m$ linear equations with $n$ unknowns. We can represent it as $AX=B$. Let $L$ is the left inverse of $A$ therefore $LA=I$. Again from $AX=B$, we get $LAX=LB$ ...
5
votes
1answer
70 views

Is $SO(n)$ a topological space?

I am reading some articles about covering space in Wikipedia. It says that $\operatorname{Spin}(n)$ is the universal cover of $SO(n)$ for $n>2$. I cannot understand how people view groups as ...
1
vote
3answers
20 views

Checking whether the result is positive definite or positive semi-definite with two methods

Given, $$A = \begin{bmatrix} 1 &1 & 1\\ 1&1 & 1\\ 1& 1& 1 \end{bmatrix}.$$ I want to see if the matrix $A$ positive (negative) (semi-) definite. Using Method 1: ...
2
votes
2answers
44 views

What is the good way to remember the signs of the rotational matrix?

Recall rotational matrix in (x,y) is given by: $R = \begin{bmatrix} \cos(\theta) & -\sin(\theta) \\ \sin(\theta) & \cos(\theta) \end{bmatrix}$ For the life of me I cannot remember if the ...
2
votes
4answers
31 views

Is the matrix $A$ positive (negative) (semi-) definite?

Given, $$A = \begin{bmatrix} 2 &-1 & -1\\ -1&2 & -1\\ -1& -1& 2 \end{bmatrix}.$$ I want to see if the matrix $A$ positive (negative) (semi-) definite. Define the ...
-5
votes
0answers
22 views

Matrices and Linear Algebra- Determine if the list is linearly independent in the real vector space. [on hold]

1.Determine if the list $((3,2,0,1),\,(2,1,4,0),\,(0,-1,12,-2))$ is linearly independent in the real vector space $\mathbb R^4$. 2.In the real vector space $C(\mathbb R,\mathbb R)$ of all continuous ...
0
votes
1answer
23 views

Nullspace of Original Matrix multiplied by Transpose

I need help in a question. I need to show that, given an $ m \times n$ matrix $A$, $\bar x \in Null(A)$ if and only if $\bar x \in Null(A^tA)$. I found this answer: "Let $X=Null(A)$; then $\forall x ...
0
votes
1answer
21 views

If I have $m \times n$ matrix A and a vector $x \in \mathbb{R}^m$, Can I make Ax working? Will it be possible for Ax to do row operations?

If I have $m \times n$ matrix A and a vector $x \in \mathbb{R}^m$,where m>n, Can I make Ax working? Will it be possible for Ax to do row operations? If yes, then how do they operate?
-1
votes
1answer
17 views

Making a basis from the Column Space of a Matrix in MatLab [on hold]

Starting with matrix A whose entries are all zeros or ones, I want to make a new matrix B whose columns form a basis for the column space of A. I know that rref puts A in Gauss Jordan form and the ...
1
vote
0answers
19 views

Counting solutions to matrix equations

Given these equations: $$a_{1,1} x_1 + a_{1,2} x_2 + \cdots + a_{1,n} x_n = b_1 \bmod p $$ $$a_{2,1} x_1 + a_{2,2} x_2 + \cdots + a_{2,n} x_n = b_2 \bmod p $$ $$\vdots$$ $$a_{m,1} x_1 + a_{m,2} x_2 + ...
0
votes
0answers
45 views

How many $2\times3$ real matrices are needed to guarantee that at least one of them is a linear combinations of the others?

The only thing I know is that $$\left(\begin{array}{ccc}1&0&1\\0&1&1\end{array}\right)$$ Seems to have a column to be linear combinations of the others.
0
votes
1answer
19 views

Deducing bounds on operator norms of matrix differences

Let $A$ and $B$ be two $n\times n$ matrices with entries $0\leq A_{ij}, B_{ij}\leq 1,$ for all $i,j.$ Suppose we are given a bound on the operator norm as follows: $$\|A-B\|\leq \delta.$$ What is a ...
0
votes
0answers
20 views

Finding the maximum Eigen vector in MATLAB

I am trying to apply something I learned in paper regarding finding the maximum eigenvector of a matrix. Lets assume the matrix is $T$ and the max eigen vector of matrix ${\bf T}$ is ${\bf w}$ $${\bf ...
0
votes
0answers
14 views

How to calculate the differential of vector/matrix?

Suppose we have $L=AXB$, where $A\in R^{m\times n}$, $X\in R^{n\times p}$ and $B\in R^{p\times q}$. Then, how can we obtain the following differential: $$\frac{\partial L}{\partial X}.$$ If possible, ...
0
votes
0answers
19 views

Finding a variable in the determinant of sum of matrices [on hold]

I Don't Know how to earn P that is scalar from Below Formula : R = log2(abs(det(I + P * H*H'))) Everything is known except P. P is scalar and positive. I is an Identity NxN , H is complex NxN ...
2
votes
1answer
27 views

If I had a matrix A, what is the meaning of $A^T Ax$, given $A^T$ is transpose of A and x are vectors of variable?

If I had a matrix $A$, what is the meaning of $A^TAx$, given $A^T$ is the transpose of $A$ and $x$ is a vector? Is it operation on $x$ by the result of the multiplication of two matrices, or is it ...
4
votes
5answers
73 views

Matrix exponential: $\begin{pmatrix} 0 & 1 \\ -4 & 0 \end{pmatrix}$

It is asked to calculate $e^A$, where $$A=\begin{pmatrix} 0 & 1 \\ -4 & 0 \end{pmatrix}$$ I begin evaluating some powers of A: $A^0= \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}\; ; ...
0
votes
1answer
24 views

What should I do to tackle the following matrices calculation?

Through chapter 3 of Group Theory by Morton Hamermesh in part 3-6 (Equivalent representations; characters.) I stopped in some point. It's told "If we change the basis in the n-dimensional space $L$, ...
0
votes
1answer
21 views

Confusion between eigen value decomposition and singular value decomposition

The Singular Value Decomposition of matrix $H$ gives $$H = U \Sigma V^H$$ The Eigen value decomposition of $$HH^H= U \Sigma \Sigma^t U^H$$ I took an example in matlab and performed EID and SVD ...
1
vote
0answers
17 views

Calculating central elements of Universal Enveloping Algebras?

Simply put, how do I calculate (in general) the central elements of the UEA of some Lie algebra given some desired degree in the algebra generators? I know the so-called 'quadratic Casimir', of ...
1
vote
1answer
28 views

For any linear operator $\phi$ on $V$, prove such an integer $m$ exists.

Suppose $V$ is an $n$-dimensional vector space over some infinite number field $K$, $\phi\in\mathcal L(V)$, prove there exists such a (positive) integer $m$ that $$\text{Im} \phi^m=\text{Im} ...
3
votes
1answer
52 views

Show that $Ax=0, Bx=0$ share the same solution space iff there is some invertible $P$ s.t. $B=PA$.

The question is said in the title, suppose $A,B\in M_{m\times n}(K)$, where $K$ is some infinite number field. If we regard $A,B$ as linear maps from $K^n$ to $K^m$, then they share the same ...
1
vote
0answers
43 views

A question about product of three positive definite matrices

Assume that $A,B$ and $C$ are symmetric positive definite matrices. I guess that the eigenvalues of the matrix $D=ABC$ can be any complex numbers. Is that true?
0
votes
1answer
29 views

Matrix polynomials/eigenvalues

$\begin{pmatrix} 7 & -2\\2 & 2 \end{pmatrix}$ The eigenvalues for this matrix are $\lambda=6$ and $\lambda=3$ It also happens that $(A-6I)(A-3I)=0$ I've checked for various $2$ x $2$ ...
-2
votes
0answers
20 views

A question on matrix norm

Definitions: ${A_j},{\Delta _j} \in {C^{n \times n}},(j = 0,1,2....m)$ ${\rm{P(}}\lambda {\rm{) = }}{{\rm{A}}_m}{\lambda ^m} + .....{A_1}\lambda + {A_0}$ is a matrix polynomial, and $\lambda $ is ...
-1
votes
1answer
43 views

Three lines that intersect in a plane.

Find a condition for three lines (𝑖 = 1,2,3) in a plane given by $π‘Ž_𝑖 π‘₯ + 𝑏_𝑖 𝑦 = 𝑐_𝑖$ to intersect in one point. I decided to form a matrix and to find the identity matrix since it will ...
1
vote
0answers
16 views

Property of hermitian matrices (eigen values)

In a paper I have read the following ${\bf G}$ is a Hermitian matrix, that 1) ${\bf G}$ is diagonalizable 2) the singluar values are same as the eigen values Is number 2 correct? I cant seem to ...
0
votes
1answer
31 views

Is there any inner product on $M_{n \times n}$ inducing this norm?

The set $M_{n \times n}$ is the collection of all $n \times n$ matrices over $\mathbb{R}$. Definition: $\|A\|_2=Sup_{\|u\|_2=1} \|Au\|_2$. Is there any inner product on $M_{n \times n}$ inducing ...
2
votes
1answer
38 views

A question in matrix norm.

Suppose $A \in {\mathbb C^{n \times n}}$ and $\left\| A \right\| \le \varepsilon $ $v \in {\mathbb C^n}$ and ${v^*}v = 1$ Is this true that $\left\| {{v^*}Av} \right\| \le \varepsilon $?
0
votes
0answers
9 views

Response Matrix with finit actuator

I have a system of penalties ($P$) and actuators($A$). Whereby: d$P_i/$d$A_j$ = close to constant $\quad\forall i,j$ In order to minimize $P$, I create a response Matrix ($M$). With its ...
0
votes
1answer
27 views

A matrix with one non-zero singular value

I have a question regarding matrices and eigen values. If SVD decomposition was performed on matrix, and the inner matrix of singular values has only one non zero value. Should the left and right ...
0
votes
1answer
37 views

A question on spectrum [duplicate]

Let $A,B \in {C^{n \times n}}$ and ${\sigma (A + B)}$ is spectrom of $(A+B)$. Suppose $M = \left\{ {\lambda \in C:\lambda \in \sigma (A + B),\left\| B \right\| \le \varepsilon } \right\}$ $F(A) = ...
0
votes
1answer
19 views

In a matrix does every set of r row vectors need to be linearly independent for rank to be r?

Rank of a matrix is the maximum number of linearly independent row vectors , does every set of r row vectors need to be linearly independent or finding only one set of r row vectors which are linearly ...
4
votes
4answers
923 views

Why is the matrix product of 2 orthogonal matrices also an orthogonal matrix?

I've seen the statement "The matrix product of two orthogonal matrices is another orthogonal matrix. " on Wolfram's website but haven't seen any proof online as to why this is true. By orthogonal ...
0
votes
1answer
29 views

Show that if the leading principal minors of a nonsingular $n\times n$ matrix $A$ are all nonzero then the matrix $A$ has $LU$ factorization

I am stucked at this problem: Prove by induction that if the leading principal minors of an $n\times n$ nonsingular matrix $A$ are all nonzero then the matrix $A$ has $LU$ factorization. (The ...
0
votes
0answers
33 views

Show that for every $n\times n$ matrix $A$, there exist an $n\times n$ permutation matrix $P$ such that $PA$ has $LU$ factorization

I am stucked at this problem: Prove by induction on $n$ that for every $n\times n$ matrix $A$, there exist an $n\times n$ permutation matrix $P$ (a matrix obtained by rearranging the rows (or ...
0
votes
0answers
18 views

finding all $m\times k$ matrices with prescribed row and column sums and zero elements

I'm looking for an algorithm constructing non-negative integer matrices with prescribed row and column sums and some predefined zero entries. For example, if column sums are [1 1 2 1 1] and row sums ...
0
votes
0answers
11 views

Question regarding Eigen Value Decomposition and Singular Value Decomposition

I have a product of matrices that have the following form $$ {\bf A} ^H {\bf A}$$ where subscript $H$ means hermitian transpose. I am trying to find the eigen value decomposition (EVD) of ${\bf ...
0
votes
0answers
22 views

Parametric vector form of cartesian equation

Cartestian equation: $$-2x-y+z=6$$ I know to find the parametric vector form we can find any 3 points P, Q and R which satisfy the cartesian equation. $$ \begin{pmatrix} x_1\\ y_1\\ z_1 ...
1
vote
2answers
35 views

How to solve for the matrix $X$ in the following equation $AXB + X = CD$

How to solve for the matrix $X$ in the following equation $AXB + X = CD$? $A$ and $B$ are full rank symmetric matrices, and there is no structure to $CD$. $CD$ just could be $C$.
0
votes
1answer
76 views

Is it true a matrix $A$ has determinant $0$ if and only if $A^N=0$?

I know that the determinant doesn't stay the same for a matrix $A$ for which the determinant $\neq 0$. I just calculated some determinants of a $3\times 3$ matrix to find that out. But I also ...
-1
votes
1answer
49 views

A question on numerical range

Let $A,B \in {C^{n \times n}}$ and ${\sigma (A + B)}$ is spectrum of $(A+B)$. Suppose $M = \left\{ {\lambda \in C:\lambda \in \sigma (A + B),\left\| B \right\| \le \varepsilon } \right\}$ $F(A) = ...
4
votes
0answers
51 views

Upper bound on infinity norm of inverse of a positive definite matrix [on hold]

Consider a positive definite matrix, $A$, and the following quantity: \begin{align} \|A^{-1}\|_\infty \end{align} Are there any upper bounds on the above normed term?
1
vote
0answers
23 views

How do you expand a matrix to a power?

Suppose I have an nxn matrix A, where t is a natural number >0. Is A^t=A^(t-1)A or A^t=AA^(t-1) I would think that the operation of splitting them up into these two should work. However, A and ...
0
votes
1answer
30 views

Does this matrix operation hold?

Suppose A is an nxn matrix and b is a constant scalar. t is some natural number >0 Can i apply binomial expansion on (A-Ib)^t?
1
vote
0answers
35 views

Matrix pencils of quadratic forms

Consider a matrix pencil of quadratic form $F-Ξ»B$ with $B$ positive definite. For which $Ξ»$ the pencil $F-Ξ»B$ less or equal to $0$ (negative definite)?
0
votes
0answers
15 views

Matrices of Ordered Bases

Let $V$ be a real finite-dimensional vector space and $T : V β†’ V$ be a linear map. Let $E$ be a basis of V . What does it mean to say that $A$ is the matrix of $T$ with respect to $E$. Let $S : V β†’ V$ ...
2
votes
1answer
61 views

Sign of $tr(A)$ given $I_n+A+A^2+A^3=0$

Let $A$ be a real matrix such that $I_n+A+A^2+A^3=0$, what is the sign of $tr(A)$ ($tr$ being the trace) ? What I have done : One can easily figure our the inverse of $A$ since ...
1
vote
2answers
51 views

Relation between norms of two matrices

Is there a relation between the norm $\|A\|$ of a nonsingular symmetric positive definite matrix $A$ and the norm of its inverse matrix $A^{-1}$?
0
votes
0answers
14 views

Augmented Matrix and Row echelon form

For which real numbers s and t does the following linear system have (a) no solution, (b) exactly one solution, or (c) infinitely many solutions? Justify your answers. (sβˆ’1)x +(s+3)y + z = 1 s x ...