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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2
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1answer
31 views

Matrices derivation and identities

Good day, I am having difficulty understanding the derivation below. This is adopted from Simon Prince's computer vision book, pg 543 for the derivation, pg 626 for the inversion relation. I can not ...
4
votes
1answer
169 views

Show that $A=0 \iff \mathrm{tr}(A)=0$ where $A= M_1+ \cdots +M_{\ell}$.

Let $G=\{M_1, M_2, \ldots ,M_{\ell}\} \subset \mathcal{M}_n(\mathbb{R})$, such that G form a group for the usual matrix multiplication. Denote $A= M_1+ \cdots +M_{\ell}$. Show that $$A=0 \iff ...
2
votes
1answer
47 views

Condition for convergence

Let $A \in \mathbb{R^{m\times{n}}}$ with full row rank. Let $B=I-\lambda A^T(AA^T)^{-1}A$ with $\lambda \in \mathbb{R}$. Determine the set of values of $\lambda$ for which $\exists \lim_{k \to ...
0
votes
1answer
17 views

Symmetric Parallelograms Under Linear Transfer Marticies

I am trying to show that a parallelogram which is symmetric about the origin stays symmetric about the origin under the action of a linear transfer matrix. It is a fairly trivial case to draw a ...
1
vote
0answers
167 views

How to compute time ordered Exponential?

So say you have a matrix dependent on a variable t: $$A(t)$$ how do you compute $$e^{A(t)}$$ It seems Sylvester's formula, my standard method of computing matrix exponentials can't be applied ...
1
vote
1answer
58 views

Proving $\displaystyle rang(AB) \le \inf(rang(A),rang(B))$

Supposing $\displaystyle A\in \mathbb{M}_{np}(\mathbb{R})$ and $B\in\mathbb{M}_{pq}(\mathbb{R})$: How can prove that: $\displaystyle rang(AB) \le \inf(rang(A),rang(B))$
5
votes
1answer
135 views

A challenge question in determinant of real matrices!

Suppose that $n\in \mathbb N -\{1\}$ and $a_{11},a_{12},\ldots,a_{nn}$ are $n^2$ distinct real numbers, prove that there is some enumeration of $a_{ij}$'s like $b_{ij}\ (i,j=1,2,\ldots,n)$ such ...
7
votes
2answers
320 views

Circulant determinants

Suppose that $a_1,a_2,\ldots,a_n$ are $n$ distinct real numbers; is the following statement true? There is a permutation of $a_1,a_2,\ldots,a_n$, namely $b_1,b_2,\ldots,b_n$, such that the ...
1
vote
0answers
100 views

Relation between determinant and L1 norm

Recently, I have coped with a problem about the relation between determinant of positive definite matrices and their L1 norm. More specifically, assume that $\Sigma_{1}$ and $\Sigma_{2}$ are two ...
3
votes
1answer
150 views

A Beautiful Determinant!

Find the determinant of the following matrix in the terms of $a_1,a_2,\cdots,a_n$ explicitly, $$ \begin{bmatrix} a_1 & a_2 & a_3 & \cdots & a_n\\ a_2 & a_3 & a_4 & \cdots ...
1
vote
1answer
67 views

Find the eigen values and vectors of the matrix

Find the eigenvalues and eigenvectors of the matrix $A$: $$A = \begin{bmatrix}-2 & 2 & -3\\2 & 1 & -6\\-1 & -2 & 0\\\end{bmatrix}.$$ $$A - \lambda I = ...
2
votes
1answer
117 views

Classification of parabolic elements of a subgroup of $PSL_2(\mathbb R)$

Let $G\subset PSL_2(\mathbb R)$ be the group generated by the matrices $$a_n=\begin{pmatrix} 1 & 2\cot\frac{\pi}{n}\\0 & 1\end{pmatrix},\; c_n = \begin{pmatrix} ...
1
vote
1answer
56 views

Do zeros present along the diagonal yield complex eigenvalues?

I was told today by a friend that having a zero along there main diagonal of a matrix will promote complex eigenvalues. I do not believe this is true because the below matrix Z has a zero present ...
2
votes
1answer
27 views

relationships of symmetric matrices

I came across the following relationships, but I have no idea how to prove them. I would love to know they can be proved. Suppose $X$ and $Y$ are both symmetric matrices, relationship: $$(X + ...
1
vote
3answers
60 views

Random matrices in coordinate independent way

How to generate a random matrix in a basis independent way (so that the random distribution does not change if the coordinates are rotated)? I am especially interested in generating random rotation ...
3
votes
1answer
173 views

Proof for real Jordan canonical form

Let A $\epsilon$ Mat(nxn, $\mathbb R$) be a matrix that is diagonalizable in $\mathbb C$ with k real eigenvalues of algebraic multiplicity 1 and (n-k)/2 pairs of complex-conjugated eigenvalues of ...
0
votes
1answer
18 views

Lower triangular matrices and identity question

If I have $(L^{−1}M)(L^{−1}M)^T=I$ and both L and M and are lower triangular matrices. Does that mean L = M ?? If yes please state the property .
1
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0answers
103 views

Finding an orthogonal $P$ and a diagonal matrix $D$ so that $D=P^tAP$

Given some matrix $A$, how do I find a diagonal matrix $D$ and an orthogonal $P$ so that $D = P^t A P$? I know how it's done with a regular $P$ which is invertible, how is it different with $P$ is ...
3
votes
3answers
45 views

Rank of $I_n + M$ where M is skew-symmetric?

Apparently the answer is $n$. I know that the rank of $M$ is always even, but how does this help? Many thanks.
0
votes
1answer
44 views

Concavity of quadratic form

I know that the quadratic form $x'Ax$ is a concave in vector $x$ if matrix $A$ is negative semi definite. What happens if $A$ depends on $x$ (so that I have $x'A(x)x$), but I still know that $A(x)$ is ...
1
vote
0answers
23 views

A certain product of C*-algebras

So, I am looking for some kind of 'product' $\bullet$ on the category of (unital?) $C^*$-algebras satisfying that $M_n(\mathbb{C})\bullet M_m(\mathbb{C}) = M_{m+n}(\mathbb{C})$ where ...
0
votes
1answer
51 views

Representation of a linear transformation

Let $U=(u_1,u_2,u_3)$ be a basis for $\mathbb{R}^3$, and let $A$ be a $3\times 3$ matrix such that $$Au_1=u_2, Au_2=u_3, Au_3=u_1.$$ ...
3
votes
3answers
403 views

$A^3 = I$ ($A$ is real Symmetric matrix). Does it imply that $A = I$?

Question of our assignment $A$ is a $3×3$ real symmetric matrix such that $A^3 = I$ (Identity matrix). Does it imply that $A = I$? If so, why? If not, give an example. Any help will be appreciated. ...
0
votes
1answer
47 views

Does eigenvalues bounded below imply matrix norms bounded below?

If we have a sequence of matrices $A_n$ such that all of the eigenvalues are positive and bounded away from $0$, is it true that $|A_nx| \geq \lambda|x|$ for some $\lambda>0$? Thank you
1
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1answer
45 views

Eigen Values and Nature of Matrix [closed]

Let J be a 3x3 matrix all of whose entries are 1.Then (i)0 and 3 are the only eigen value of J (ii)J is positive semi definite. (iii)J is diagonalizable (iv)J is positive definite. Here, J is ...
2
votes
3answers
480 views

matrix transpose * itself = identity

if $A^TA = I$ does that mean A has to be an identity matrix and nothing else? or is there other possibilities. please state the property if any.. Sorry forgot to mention that $A$ is a lower ...
1
vote
2answers
41 views

Matrix norm question.

When do we know that $|Ax| \geq \lambda|x|$ for all $x$ where $A$ is some matrix and $\lambda$ is some constant > 0. Is it enough if all of the eigenvalues are positive? If so, can you please prove ...
1
vote
3answers
35 views

Transponse and inverse

If I have two lower diagonal matrices L and M If I have $(L^TL)^{-1} (M^TM) = I$ where $L^T$ is the transpose of L can I say $(LM^{-1})^T (LM^{-1}) = I$ why? and using what lemma or theorem.. ...
0
votes
2answers
69 views

Matrices problem?

Does anyone know how to do this? 1.) Write two matrices A and B such that AB = BA, but neither A nor B is the identity matrix.
0
votes
1answer
130 views

Matrices word problem?

Julie and Bill are waiters at the Ogling Ogre Convention Center, which is well-known for serving the most deliciously disgusting meals to its guests. One of their tasks was to count the 2-eyed and ...
0
votes
1answer
43 views

Does relative positive definiteness have a good name?

Often, when there are two matrices $A$ and $B$ such that $A-B$ is positive definite, this relationship is written $A \gt B$, $A \succ B$ or similar. It has the advantage of being consistent with the ...
2
votes
1answer
119 views

Abstract Algebra Matrix Group Theory

The matrix group G = SL(n, $\mathbb{R}) = \{A \in M(n, \mathbb{R})\} \text{ acts on } X= R^n$ by left matrix multiplication: $\tau _A(x) = A\cdot x (\text{matrix product }(n \times n ) \cdot (n \times ...
2
votes
1answer
95 views

number of eigenvalues = dimension of eigenspace

Is this true in general? What about: number of negative eigenvalues = dimension of span(eigenectors for the negative eigenvalues)? Or even more generally: number of eigenvalues greater than 4.3 = ...
1
vote
1answer
85 views

Show that $[A,\exp(B)]=\exp(B)[A,B]$

Denote $\exp(A)=\sum_{k=0}^{+\infty} \frac{A^n}{n!}$ where $A\in M_n(\mathbb{R})$ and $[A,B]=AB-BA$ Assume that $A,B$ commute with $[A,B]=AB-BA$ Show that $$[A,\exp(B)]=\exp(B)[A,B]$$ ...
0
votes
1answer
27 views

Simplify quadratic polynomial with matrix

I am reading a paper and have trouble following equation (3): $$ (\mathbf{x}-\mathbf{d})^T \mathbf{A}_1 (\mathbf{x}-\mathbf{d}) + \mathbf{b}^T_1 (\mathbf{x}-\mathbf{d}) + c_1 = \\ \mathbf{x}^T ...
7
votes
3answers
344 views

Maximum number of linearly independent anti commuting matrices

Given n x n matrices, my book says the maximum size of a set of linearly independent mutually anti commuting matrices is $n^2-1$. I don't understand why this is true. Would appreciate any tips to ...
-1
votes
1answer
59 views

chosing between matrix theory and combinatroics

I have to take one more math course to finish my math minor , i am a computer science major and i want to know which course will benefit me more matrix theory or combinatorics and which takes more ...
0
votes
1answer
209 views

Inverse of a lower triangular Toeplitz matrix vs. the matrix size

I am recently trying to find the inverse of the lower triangular Toeplitz matrix ($\mathbf{A}$), with some special elements: $$ \mathbf{A}_{M\times M}=\left[\begin{array}{ccccc} 1\\ -a_{1} & 1\\ ...
0
votes
2answers
123 views

Eigen values and Eigen vectors

Let A be a 4x4 matrix with real entries such that $ \ -1,1,2,-2 \ $ are its eigen values.If $B=A^4-5A^2+5I$ ,where $I$ denotes the 4x4 identity matrix ,then which of the following statements are ...
2
votes
2answers
109 views

Finding matrix with respect to given bases

Given that A: \begin{matrix} a & b & c \\ d & e & f \\ \end{matrix} is a matrix of T : V -> W with bases G = {g1, g2, g3} and Q = {q1, q2}, respectively. Find the matrix of T ...
2
votes
1answer
524 views

If a set of 2x2 matrices are independent, do they also span M22?

I am looking for a basis in M22. So I need to get a linearly independent set that also spans the vector space. I have worked out how to tell independence, but I am stuck on the spanning requirement. ...
-1
votes
1answer
43 views

Matrices in the plane,polygon assignment, help. please?

a.) Given polygon P with vertices (1,5), (4,8), (8,5), (6,2) and (2,1), find the following: Find the area of P. Tip: Make sure to move COUNTERCLOCKWISE from point to point to ensure you get a ...
-1
votes
1answer
46 views

Singular matrices over a commutative ring $R$, with a given adjoint matrix

First, I apologize if this is a duplicate question. I also must apologize if this has a trivial solution. This question has two parts: Let $R$ be a commutative ring with $1$, and let $F = R^n$ be a ...
1
vote
2answers
60 views

Diagonalization problems (eigenvalues and vectors)

I am trying to diagonalize the following matrices: $$A = \begin{pmatrix}0 & 1\\-1 & 2\end{pmatrix}\qquad B = \begin{pmatrix}1 & 2\\-1&-1\end{pmatrix}$$ For matrix $A$, I find an ...
2
votes
3answers
77 views

Showing Orthogonality

How would I do this question..... I'm familiar with Gram-Schmidt and the basics but I have no idea how to do $a$ and $b$ in this question. Suppose $\{\vec x_1, \vec x_2, \vec x_3\}$ is an ...
0
votes
1answer
28 views

Given 2 matrices generate a reducible algebra, show they have a common eigenvector

Two matrices A, B generate an algebra.... the span of all words made with A and B... example of an element of the algebra: $A^kB^nA^m + I + B^sA^q$ etc... (exponents are all nonnegative). This algebra ...
1
vote
1answer
37 views

Find $P$ such $P^{-1}AP = kR$.

Let $ A=\begin{bmatrix} 3 & -5 \\ 1 & -1 \\ \end{bmatrix}$ Find P such that $P^{-1}AP = k R$ where $k \neq 0$ is a scalar and R an element of $SO(2)$ = {R element of ...
1
vote
1answer
1k views

Median of Medians in 2D Array/Matrix

This is a bit of a mathematics problem, and a MATLAB problem. In MATLAB, if I call median(M), where M is an ...
3
votes
4answers
160 views

Find the axis of rotation of a rotation matrix by $INSPECTION$ (NOT by solving $Kv=v$)

$$K=\ \begin{pmatrix} 0 & 0 & 1\\ -1 & 0 & 0\\ 0 & -1 & 0 \end{pmatrix}$$ Find the axis of rotation for the rotation matrix $K$ by INSPECTION. This is from my other thread ...
1
vote
1answer
61 views

Similar Matrices Conditions:

Sorry this question was already asked but my english is not good. For matrix to be similar, does it have to have all of these properties or SOME of them? Same determinant Same Trace Same ...