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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2answers
93 views

Constructing Matrix (Rotation, Reflection)

Construct the matrix corresponding to a rotation of 90 degrees about the y-axis together with a reflection about the (x,z) plane. Reviewing Linear Algebra and seem to have forgotten some stuff. ...
2
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1answer
92 views

Proof that two square, diagonal matrices A and B fulfill the first binomial formula

In the current exercise for linear algebra, we had to find conditions so that two arbitrary quadratic matrices A and B with the same dimension satisfy the first binomial formula: (A+B)^2 = A^2 + 2AB ...
1
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2answers
66 views

Linear Algebra - inverse of a complex matrix

This is from the UPenn prelim questions. http://hans.math.upenn.edu/amcs/AMCS/prelims/prelim_review.pdf (question 12 in Linear Algebra session) Let A be a real symmetric matrix and form the matrix ...
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1answer
43 views

Question about Hermitian Symmetric matrix

Assume that $A$ is a Hermitian symmetric n x n matrix with complex entries having all of its eigenvalues lying inside the interval, (-1, 1). Is $A^3 + Id$ always a positive definite matrix? My hunch ...
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0answers
40 views

Inverse of a non-singular linear transformation

The question is about showing that if A is a non-singular linear transformation of an n-dimensional linear space to itself, then there must be some polynomial $c_0 + c_1 z + ... + c_k z^k$ such that ...
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0answers
52 views

“Other” one-sided inverses

If a matrix has linearly independent columns, then if it's a square matrix it has linearly independent rows. Forget square matrices; think of this: $$ A=\begin{bmatrix} \bullet & \bullet \\ ...
1
vote
1answer
91 views

Why do lattice cubes in odd dimensions have integer edge lengths?

This is a spinoff from Characterization of Volumes of Lattice Cubes. That question claims a number of facts as being proven, but doesn't include the full proofs. That's fine for the question as it ...
2
votes
1answer
64 views

Can't show that these matrices are diagonalizable.

Consider that for each $n \times n$ (possibly complex) matrix, $A_{k}$, $0 \leq k \leq m$, we have that \begin{align} A_{0}A_{k} &= kA_{k}, \qquad 1 \leq k \leq m \end{align} and suppose that ...
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votes
1answer
63 views

Properties of a matrix and eigenvalues

A, B, C are three real-square matrices. A is an upper triangular matrix with all of its diagonal entries equal to zero. B is a matrix such that $b_{ij}=-b_{ji}$, and C is a matrix such that $\sum_j ...
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votes
1answer
84 views

Solve an matrix trace equation: find x in Tr[(Y-xWY-FB)'(WY)]=0

I have the following linear equation $$ \mathrm{Tr}[(Y-xWY-FB)'(WY)]=0$$ where $Y$ is an $n$-by-$b$ matrix, $W$ is an $n$-by-$n$ matrix, $F$ is an $n$-by-$m$ matrix, $B$ is an $m$-by-$b$ matrix, ...
1
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1answer
102 views

On Adjacency Matrix of a Graph with a Cut Vertex and a Bridge

Let $G$ be a graph. If $v_i$ (resp. $v_iv_j$) is a cut vertex (resp. a bridge) of $G$, what can you say about its adjacency matrix $A(G)$?
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0answers
30 views

Online tools for plotting and deriving formulas from set of data?

I have a matrix of 20x3 decimal values. I would like to have them in a graph, with one plot for each of the three y values. I would also like approximated formulas for each of the three. Are there any ...
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1answer
48 views

Commutative matrices of order 2

I am trying to give a simple proof of why $GL_n(K)$ spaces are not isomorphic for different $n$, by finding $2^n$ (I presume this is the maximal possible number) of pairwise commutative matrices who ...
3
votes
1answer
124 views

How find this matrix the inverse $A^{-1}$

Let $a,b>0$,and the matrix $A_{n\times n}$ and such $$A=\begin{bmatrix} a&b&0&\cdots&0&0\\ b&a&b&\cdots&0&0\\ 0&b&a&\cdots&0&0\\ ...
2
votes
3answers
107 views

Proving that $\lambda$ being an eigenvalue for $A$ implies $\lambda^{-1}$ is an eigenvalue for $A^{-1}$

Let $A$ be an invertible matrix, and let $\lambda$ be an eigenvalue for $A$. We have that $Ax = \lambda x$ for some eigenvector $x$. Note that $A^{-1}Ax = A^{-1}\lambda x$, which gives $x = ...
2
votes
1answer
65 views

How to find $A=UDU^H$ in this case

I am given a matrix $A$. I find out it is normal. And I compute $\det(A-\lambda)=0$ and find that not all $\lambda_i$ are different, i.e., the eigenvalues are not distinct. Thus, I am not sure if the ...
0
votes
1answer
44 views

Simple Matrix Caluculation

I am new to Math, and my knowledge is not good. Can anyone help me with this problem? Show that $(M^*)^* = M$, $(M^T)^T = M$, and $(M^t)^t = M$. I think I got some idea on the last 2, but the I am ...
0
votes
1answer
59 views

Find two matrices such that the sum of their ranks is the rank of the sum

Find $2$ non-zero, $2 \times 2$ matrices, such that $\mathrm{rank}(A+B)=\mathrm{rank}(A)+\mathrm{rank}(B)$ I want to start from the identity matrix and work backwards, but I cant seem to cook up two ...
4
votes
1answer
408 views

Cramer's Rule Question

Use Cramer's rule to solve this system for z: $$2x+y+z=1$$ $$3x+z=4$$ $$x-y-z=2$$ so my work is: $$\frac{\left|\begin{matrix} 2 & 1 & 1\\ 3 & 0 & 4\\ 1 & -1 & 2 ...
1
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1answer
87 views

Determinant by applying Gaussian Elimination

I understand when using Gaussian Elimination you have to get it in ref form (upper triangle) and calculate the product of the diagonal. Additionally you have to keep track of the number of swaps to ...
0
votes
2answers
107 views

Rank and Invertibility Problem - Non Square Matrix

Let $A \in \mathcal M_{m×n}(F)$. Prove that if $\text{rank}(A) = m$, then there exists $B \in \mathcal M_{n×m}(F)$ such that $AB = I_m$. I think I need to prove that $A^{-1}$ exists, such that ...
1
vote
1answer
103 views

Upper triangular matrix and nilpotent

How to show that an upper triangular matrix is nilpotent if and only if all its diagonal elements are equal to zero by using induction?
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1answer
56 views

Largest eigenvalue of a special m-matrix

How to estimate the largest eigenvalue of followed characteristics? Let $A={a_{ij}}$. Symmetric positive definite. Real. Very sparse. Diagonal elements are all positive, and off-diagonal elements ...
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votes
1answer
19 views

Order: multiplying matrix by a scalar

Let's say we have some matrix A and a constant c. Does it generally hold that cA=Ac? Thanks
1
vote
1answer
91 views

Proving properties of linear maps on one-dimensional vectors

An exercise from the book "Linear Algebra Done Right" asks to prove the following: 'Show that every linear map from a one-dimensional vector space to itself is multiplication by some scalar. More ...
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0answers
39 views

Linear transformations and bases

Let $V=R^3$ and let $T: V \to V$ be the linear transformation defined by $T(x,y,z)=(3x-z,2y+5z,-7x+4z)$ (a) Using standard basis $\mathscr{E} = {(1,0,0),(0,1,0),(0,0,1)}$ for $V$ find the ...
0
votes
1answer
71 views

Is it true that 2 matrices are similar **if and only if ** they have the same Jordan form?

Is it true that 2 matrices are similar if and only if they have the same Jordan form? I know that one direction is correct: if have the same Jordan form -> similar. Is the other direction correct - ...
0
votes
1answer
109 views

Fitting 2nd Order multivariate quadratic with matrices

Hopefully you at least entertain this question as it took forever to construct the below matrix using TeX. Any ways, so I have a list of data points ($X_1$,$X_2$,Y), with the X's being independent ...
1
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0answers
66 views

When is a matrix diagonalizable?

The matrix $A$ is diagonalizable with $A=SDS^{-1}$ where $D$ is a diagonal matrix and $S$ is the matrix with eigenvectors as columns iff $A$ has linearly independent eigenvectors. Correct?
-2
votes
2answers
60 views

Why can this product of rectangular matrices not be the identity matrix?

Let $m > n$ be positive integers. Show that there do not exist matrices $A ∈ R_{m×n}$ and $B ∈ R_{n×m}$ such that $AB = I_m$, where $I_m$ is the $m × m$ identity matrix If someone could explain it ...
2
votes
1answer
78 views

Let $k$ be a division ring, then the ring of upper triangular matrixes over $k$ is hereditary

I'm reading Ring Theory by Louis H. Rowen, and he claimed that The ring of upper triangular matrices over a division ring is hereditary (it's on page 196, Example 2.8.13 of the book). I think it ...
5
votes
3answers
74 views

A question on eigenvalues

Let $A,B\in M_{2}(\mathbb{R})$ so that $A^2 = B^2 = I$. Which are eigenvalues of $AB$? 1) $1\pm \sqrt 3$ 2)$3 \pm 2\sqrt2$ 3)$\dfrac {1}{2},2$ 4)$2 \pm 2\sqrt 3$
3
votes
1answer
287 views

Gershgorin Circle Theorem: counterexample to a statement in the proof?

I have been struggling to comprehend the proof of Gershgorin Circle Theorem for a long time now, but I think I have come upon a counterexample. I'm probably wrong, but please tell me where I'm ...
2
votes
1answer
64 views

Why does matrix-matrix product come close to the peak performance of a system?

In this paper, I read " The most important operation is GEMM (GEneral Matrix Multiply), which typically defines the practical peak performance of a computer system." But why? Why does matrix-matrix ...
1
vote
1answer
250 views

Obtaining rotation matrix from Euler angles if all three rotations happen at once. Does order of multiplication matter?

I'm having a problem getting my head around Euler Angles. Specifically if I wish to obtain a rotation matrix for a system where pitch, roll and yaw have all changed at once by various values... how ...
3
votes
2answers
120 views

$A,B \in \mathcal{M}_{n}(\mathbb{R})$ $ABA-BAB=I$ and $A^2B+B^2A=O$ $A,B$ are invertible

Let $A,B$ be two matrices with $A,B \in \mathcal{M}_{n}(\mathbb{R})$ and $ABA-BAB=I$ and $A^2B+B^2A=0$. Prove that $A,B$ are invertible matrices.
2
votes
1answer
218 views

Number of binary n x m matrices, with at most k consecutive number of 1 in each column

I am trying to compute the number of $n x m$ binary matrices with at most $k$ consecutive values of $1$ in each column. I've figured out that I it will be enough to find the vectors with $1$ column ...
3
votes
1answer
222 views

Linear Algebra: Identity map

I was asked to prove that the identity map $id : \Bbb R^n \to \Bbb R^n $ can be represented by the the identity matrix regardless of the basis My Attempt: Let $\mathcal B = \lbrace v_1 , ...,v_n ...
1
vote
1answer
25 views

Is it true that $|(Mv)\cdot(Nw) \leq C|v||w|$ (matrix-vector)?

If $M$ and $N$ are matrices such that each element of the matrices depends on $t$, so we have $M_{ij}(t)$, $N_{ij}(t)$, and we have the result $M_{ij}$, $N_{ij} \in L^\infty(0,T)$, is it true that ...
0
votes
1answer
67 views

Complexity of sparse back substitution

What is the complexity of sparse backsubstitution $Rx = b$, given $n$, the dimensions of dense $x$ and $b$ as well as of the sparse $R$ and $nnz$, the number of nonzero entries in $R$?
2
votes
2answers
134 views

diagonalize a non-normal matrix , without distinct eigenvalues

I wonder how to diagonalize a matrix that is non-normal, and does not have distinct eigenvalues. Let $\lambda_i$ be the eigenvalue, and $v_i$ be the eigenvector with that eigenvalue. I think the ...
1
vote
1answer
260 views

Cholesky decomposition of the inverse of a matrix

I have the cholesky decomposition of a matrix M. However, I need the cholesky decomposition of the inverse of the matrix, invM. Is there a fast way of getting this, without first inverting the matrix ...
3
votes
1answer
38 views

The exponention map of the matrix

Let $sl(n)$ denote the set of all $n\times n$ real matrices with trace equal to zero and let $SL(n)$ be the set of all $n\times n$ matrices with determinant equal to one. Let $\varphi(z)$ be a real ...
0
votes
1answer
37 views

$T$ is a linear operator on a IPS $V$ which has a basis $\beta$. Prove that $A_{ij} = \langle T(v_j),v_i \rangle$

I have trouble understanding a proof on textbook and I would appreciate your help! Corollary. Let $V$ be a finite-dimensional inner product space with an orthonormal basis $\beta = \{v_1, v_2, ...
2
votes
2answers
623 views

Simplfy a complex matrix into a real one

I encounter systems of linear complex equations (At most 3 equations) in my circuit analysis course. The calculator I am using is Casio fx-991ES and it only accepts real elements when in matrix or ...
4
votes
3answers
218 views

Show non-symmetric matrix has non-orthogonal eigenvectors

I'm struggling with a problem from Boas's Mathematical Methods in the Physical Sciences. The question is, for a 2x2 matrix M s.t. M is real, not symmetric, with eigenvalues real and not equal, show ...
1
vote
1answer
65 views

Rank-one modification of graph Laplacian

Suppose I have a Laplacain matrix for a 3-node-path graph as follows $L=\left[\begin{array}{ccc} 1 & -1 & 0\\ -1 & 2 & -1\\ 0 & -1 & 1 \end{array}\right]$ Now, I want to ...
1
vote
1answer
20 views

Function that transforms a Matrix to different dimensions

What is the name of a function that transforms a matrix into different dimensions? Say I have a matrix M of dimensions $(x,y)$ and I want to transform it to dimensions $(w,v)$. I can accomplish this ...
0
votes
1answer
66 views

Constructing a similarity matrix between points

I have two images with two sets of corresponding points. In order to align the images I'm trying to compute the similarity matrix that describes the relationship between the corresponding points. I ...
0
votes
1answer
109 views

Simple algorithm Hermite Normal Form for 3x3

In the scope of the implementation of a model, I need to reduce a 3x3 real matrix into its Hermite Normal Form. I am very new to this kind of reduction and only find algorithm using complex notions ...