Questions tagged [matrices]

For any topic related to matrices. This includes: systems of linear equations, eigenvalues and eigenvectors (diagonalization, triangularization), determinant, trace, characteristic polynomial, adjugate and adjoint, transpose, Jordan normal form, matrix algorithms (e.g. LU, Gauss elimination, SVD, QR), invariant factors, quadratic forms, etc. For questions specifically concerning matrix equations, use the (matrix-equations) tag.

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4
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3answers
86 views

Is there a matrix that can be used to find the transpose of a matrix?

Let $A$ be a general $n\times n$ invertible matrix. Let $T^A$ be the "transposer" matrix i.e. $T^A A = A'$. (Does that $T^A$ multiplied by $A$ equal the transpose of $A$?) Then does $T^A$ depend on ...
1
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0answers
23 views

Exponential stabilization of system of equation

Let $X$ be an infinite dimensional Banach space and let $A$ be a square matrix of size $n$. Suppose that the sequence of solutions $y_n$ of some partial differential equation problem is given by $$ ...
0
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0answers
22 views

How to express the orthogonal complement of a linearly transformed null space?

Let ${\mathcal A} = \mathrm{Null}(A)$ be the null space (kernel) of $A \in {\mathbb R}^{m\times n}$. Consider a linear map $T: {\mathbb R}^n \to {\mathbb R}^n$ (i.e., $T \in {\mathbb R}^{n\times n}$), ...
0
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0answers
16 views

Monotonically of symmetric positive definite matrix divided by its smallest eigenvalue

Let $P_1$ and $P_2$ be two symmetric positive definite matrices satisfying $P_1 \succ P_2$. Denote $\lambda_{min}(P_1)$ and $\lambda_{min}(P_2)$ as their smallest eigenvalues. Can we prove the ...
3
votes
3answers
87 views

Higher powers of a matrix's relation with its trace

Let $A=[a_{ij}]$, where $a_{ij}=u_{i}v_{j}, 1 \leq i \leq n$ and $1\leq j \leq n$ and $u_i,v_j$ belong to $R$ satisfies $A^5=16A$. Find trace(A). I denoted U as a column matrix having values u1,u2,......
0
votes
1answer
50 views

Show that $(A \oplus B)^n = A^n \oplus B^n$

So the question is to prove that $(A \oplus B)^n = A^n \oplus B^n$, for all $A \in M_{p,p}$ and $B \in M_{q,q}$. I could sort of see how this would work out, but I'm struggling to prove rigorously. I ...
3
votes
1answer
52 views

Diagonalizability of a block-diagonal matrix

I will very appreciate if someone would check my proof of the following problem: Consider $$B=\begin{pmatrix}L&M\\O&N\end{pmatrix}\in\mathbb{C}^{2n\times 2n}$$ for $L,M,N,O\in\mathbb{C}^...
0
votes
1answer
26 views

Find vector such that rotation matrix has required form

Let $v_1 = \frac{1}{3}(1,2,-2)^T, v_2 = \frac{1}{3}(2,1,2)^T$. Let $\Phi \colon \mathbb R^3 \to \mathbb R^3$ be the rotation by $\pi/2$ within the plane $E: v_1 \cdot x = 0$. Determine $v_3 \in \...
2
votes
2answers
137 views

Two matrices that are not similar have (almost) same eigenvalues

I have two matrices $$ A=\begin{pmatrix} a & 0 & 0 \\ 0 & b & 0 \\ 0 & 0 & c \end{pmatrix} \quad \text{ and } \quad B=\begin{pmatrix} d & e & f \\ d & e &...
0
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1answer
25 views

How to find the dimension of the intersection of hyperplanes?

Let P ⊂ R4 be the subspace with equation 2x + 3y − z + w = 0, let Q ⊂ R4 be the hyper-plane with equation x + y −z = 0, let R ⊂ R4 be the hyper-plane with equation x−y + w = 0, and let l = P∩Q∩R. I ...
1
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1answer
46 views

Problem on a set of all $3 \times 3$ real upper triangular matrices with all diagonal entries $=1$,

Let $W$ be the set of all $3 \times 3$ real upper triangular matrices with diagonal entries $1$ and let $B = (b_{ij})$ be a $3 \times 3$ real matrix trhat satisfies $AB = BA$ for all $A \in W$ then ...
2
votes
1answer
76 views

How much shift Eigenvalues?

Let $A$ be an $n × n$ matrix. Then, we can create a family of matrices $A(t) = tB + D$ where $D$ is the same as $A$ with all the off-diagonal entries reduced to zero and $B$ is the same as $A$ with ...
0
votes
0answers
17 views

Union of graphs is a sort of matrix concatenation

Just an intuition for discussion: If we consider a set $G=\lbrace G_1, G_2\rbrace$ of two connected simple graphs, where $G_1$ and $G_2$ have no vertex in common then the the graph union $G_1\cup G_2$...
2
votes
1answer
31 views

Projection Matrix Formulae Comparison and Intuition

I wanted some intuition behind the formulae of projection of point to a subspace. Particularly I wanted to compare it to the situation where the subspace is just a 1D line. Let $b$ be the point to ...
0
votes
2answers
49 views

Matrix theory, use Kronecker delta

I been stuck on this problem for a while. Use Kronecker delta to show when $AE_{i,j}$ is equal to $E_{i,j}A$, where $A$ is a square matrix and $E_{i,j}$ is the elementary matrix, that is $1$ on $(i,j)...
0
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2answers
51 views

2004 Georgia Tech Varsity Multiple Choice Version B Question 10

I was reviewing this test and came over this question: How many distinct solutions $x$ are there for the equation det($A-xI$) $=0$ where det stands for determinant, $I$ is the identity matrix, and $A$...
-1
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1answer
36 views

The inverse in the sense of MOORE-PENROSAE $AA^{-1}B=B$ [closed]

On which conditions we have $$AA^{-1}B=B$$ A, B are nonnegative definite symmetric $n\times n-$matrices, and $A^{-1}$ is the Moore-Penrose inverse of A.
0
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1answer
28 views

Matrix equation, simplified from $4\times 3$ to $3\times 3$

I am confused as to how this is simplified, The image shows the question and what the mark scheme says but I do not understand it.
2
votes
0answers
40 views

Maximum order of element of finite general linear group

The question may have been appeared, but I didn't find here. In matrix group ${\rm GL}_n(\mathbb{F}_q)$, the largest order of any element is equal to $q^n-1$. It is easy to show existence of ...
2
votes
1answer
79 views

Taking the $n$-th root of a diagonal matrix

I have the following problem: let $C$ be a $3\times 3$ real matrix such that $$ C^3= \begin{bmatrix} -1&0&0\\ 0&1&0\\ 0&0&2 \end{bmatrix} $$ What is $C$? The issue is that I ...
2
votes
0answers
44 views

Complex symmetric matrices are normal

In the taxonomy at https://en.wikipedia.org/wiki/List_of_matrices, it says that complex symmetric matrices are normal. For real symmetric matrices, you can prove that they are hermitian hence normal. ...
1
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2answers
24 views

Entry-wise square of orthogonal matrix

Let $O(n)$ be the space of all $n\times n$ orthogonal matrix. Consider the following set: $$S_n=\{(U_{ij}^2)_{1\leq i,j\leq n}: U\in O(n)\}$$ Then it is obvious that $S_n$ is a subset of the space of ...
-1
votes
1answer
13 views

How can I sort variables into groups of three such that the average similarity is the highest possible?

I have a spreadsheet in which the leftmost column and topmost row contain 12 names, and the values in the cells inside represent the similarity (on a scale of -1 to 1) between the names in the column ...
0
votes
0answers
23 views

Upper bound with matrices

Let $A(x)$ be a $n\times n $matrix with smooth coefficients and bounded and let $f$ be a vector in $(L^p(0,1))^n$. Is passage is correct? $${\left\| {A(.)f} \right\|_{{{\left( {{L^p}} \right)}^n}}} ...
3
votes
0answers
71 views

Why did we study “defect” in the late 1800s?

I just began reading Weibel’s An introduction to Homological Algebra and stuck at the first page, where the author attempts to give a historical motivation for the homology construction: ...
-2
votes
1answer
67 views

True or False elementary matrices [closed]

True or False: If E1 and E2 are two n x n elementary matrices then E1 x E2 is an elementary matrix.
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1answer
60 views

I tought a matrix that can be row reduced to the identity matrix is also invertible.

This matrix i can convert the identity matrix. $\begin{bmatrix} 1 &2 &-1 \\ -1 &0 &3 \\ 0 &1 &1 \end{bmatrix}.$ see here: --> still, the determinant is 0 of ...
0
votes
0answers
42 views

Matrix Derivative of the trace proof [duplicate]

Let $f: \mathbb{R}^{m\times n} \mapsto \mathbb{R}$ be a mapping from an $m$-by-$n$ matrix to the real numbers. Defining the derivative of $f$ to be: $$\nabla_A f(A) = \begin{pmatrix} \frac{\partial f}{...
0
votes
3answers
47 views

Matrix derivative with respect to matrix

Say I have a $ 2 \times 2$ matrix as $A$ and a $2 \times 1$ vector as $X$. I want the derivative of the matrix product with represent to $A$: Let $y= \begin{bmatrix} a & b\\ c & d \end{...
0
votes
2answers
40 views

Proving that a Matrix has an Inverse using a Polynomial

Suppose an $(n × n)$ matrix $A$ satisfies that an equation such as, $A^{2} − 3A + I = 0,$ where $I$ is the $(n × n)$ identity-matrix, prove that A has an inverse. Any tips on approaching a proof?
1
vote
2answers
37 views

Linear Functions View as Surfaces or Vectors

I am familiar(on a basic level) with concepts of inner product, dual space and linear functionals. Reviewing my understanding of simple linear functions of the type $Y=ax+by+cz$ I see that this can be ...
0
votes
2answers
53 views

Is there an easy way to prove the matrix $(XYZ)^{-1} = Z^{-1}Y^{-1}X^{-1}$? [closed]

I've been using pro numerals to show this relationship, and it is taking ages to complete because of all the steps necessary! Any other easier way to prove this statement?
1
vote
0answers
22 views

Complexity of calculating $(A^n)_{i,j}$

I'm looking for the most efficient way to calculate a single entry of a matrix that was $n$ times multiplied with itself . Pet example and motivation: Let us denote the Stirling numbers of the second ...
3
votes
0answers
40 views

Minimal set of $n\times n$ matrices whose products generate all $n\times n$ matrices with a single $1$

Let $n \in \mathbb{N}^*$ be the dimension of the matrices. Let $M_{i,j}$ be the $n\times n$ matrix with $1$ at position $(i,j)$ and 0 elsewhere. Let $S_n$ be the set containing all the $n^2$ ...
1
vote
2answers
14 views

How many numbers in the matrix?

We consider a matrix $\begin{bmatrix}a_{i,j}\end{bmatrix}$ with $r$ rows and $c$ columns. We fill this matrix only with zeros and ones. How many ones (maximally!) we can write into the matrix $r\...
2
votes
2answers
68 views

Inverse of $A^T Y A$

Is there a formula for the inverse of a matrix $$ X = A^\top Y A $$ in terms of $A$ and $Y$, given that $A \in F^{m \times n}$ is full rank with $m >n$, and $Y$ is positive definite?
4
votes
1answer
170 views

Showing $\det\big[ (B+K)^{-1} (A+K) \big] = O(1) $ when $A,B$ are rank 1 updates of $I_n$ and $K$ is symmetric PD with positive entries

In general, given $n$ define $m_A, m_B \in\{1,...,n-1\}$ by $$ m_A = floor(a \times n) $$ $$ m_B = floor(b \times n ) $$ where the constants $a,b \in (0,1)$ are independent of $n$ with $a \ne b$ . ...
1
vote
1answer
49 views

Can linear transformations which reduces the dimensionality of vectors preserve the Euclidean distances between them?

$y^{(i)}$ = M$x^{(i)}$, where $x^{(i)}$ is in 3D, $y^{(i)}$ is in 2D and Euclidean distance between $y^{(i)}$ and $y^{(j)}$ is same as $x^{(i)}$ and $x^{(j)}$. Does M exist?
0
votes
2answers
39 views

How to invert a block tridiagonal matrix?

I'm dealing with the inversion of a pretty large block-matrix, whereby I am only interested in a particular block of the inverse. I would like to avoid the inversion of the whole matrix and I wonder ...
0
votes
2answers
67 views

How to prove the determinant is $|A|^m$? [closed]

$A=(a_{ij})$, and $I_m$ is the identity matrix of order $m$. How to prove the following? $$\left|\begin{array}{cccc}a_{11}I_m&a_{12}I_m&\cdots&a_{1n}I_m\\a_{21}I_m&a_{22}I_m&\...
1
vote
2answers
108 views

Why is the sqrt of a matrix different to ^0.5

If H = [4 4; 4 4] Why is sqrt(H)=[2 2; 2 2] And, (H)^0.5=[1.4142 1.4142; 1.4142 1.4142]
2
votes
1answer
38 views

Let $\text{trace}(AM)\ge0$ where $M$ is psd, is $\text{trace}(A)\ge0$?

Let $\text{trace}(AM)\ge0$ where $M$ is positive semidefinite, so we know that $\text{trace}(M)\ge0$ since the trace of a matrix is the sum of its eigenvalues. Is $\text{trace}(A)\ge0$?
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votes
0answers
14 views

I am having this confusion over different methods of least squares and linear regression

So I am getting into least squares and linear regression. I am having trouble differentiating the multivariate calculus method vs the matrix method. For example Minimization method with ...
1
vote
1answer
51 views

Meaning of euclidean distance matrix's rank

It is well known that in dimension $m$, an Euclidean distance matrix has rank less than or equal to m+2. Wiki also knows about it Also another known fact is added there: If the points $x_{1},x_{2},...
0
votes
1answer
34 views

If $R$ is an integral domain and $M=\langle m_1, \dots, m_k \rangle$ is an $R$-module, then the rank of $M$ is less than or equal to $k$.

I'm supposed to use the fact that if $A \in M_{m,n}(R)$ with $m<n$, then the columns of $A$ are linearly dependent. This is true since $A$ represents a map $f: R^n \to R^m$ and the columns of $A$ ...
0
votes
1answer
58 views

Trace of $(K+ vv^T)^{-1} (K+uu^T)$

Let $K$ be a $n \times n$ symmetric real positive definite matrix. Let $u$ and $v$ be non-zero $n\times 1$ vectors taking values of $0$'s and $1$'s. Assume that the values in $u$ and $v$ are sorted so ...
1
vote
1answer
123 views

Given small symmetric matrix A, calculate cofactor for large matrix B made using A.

$$A = \begin{bmatrix} a_1 & a_2 & a_3\\ a_2 & a_4 & a_5\\ a_3 & a_5 & a_6 \end{bmatrix}$$ $$B = \begin{bmatrix} -1 & -1 & -1\\ -1 & -1 & -1\\ -1 & -1 & ...
4
votes
0answers
122 views

Show that a homomorphism from $SL_2(\mathbb{Z})$ to $ SL_2(\mathbb{F}_p)$ is surjective

The homomorphism sends the entries of matrices in $SL_2(\mathbb{Z})$ to their congruence classes mod $p$. After a lot of work I could prove that a matrix $$\begin{pmatrix}a & b\\\ c & d\end{...
2
votes
1answer
40 views

Decompose projection matrix into a matrix and its pseudoinverse

While researching regression, I encountered the situation where I am trying to reconstruct a data matrix $X \in \mathbb{R}^{n\times p}$ from the $n \times n$ hat matrix $H = X(X'X)^{-1}X' = XX^{+}$ (...
0
votes
1answer
51 views

Showing $\prod_{i=1}^n \lambda_i = O(1)$ for certain $n \times n $ matrix when 1 as an eigenvalue has multiplicity $n-2$

This is a follow-up question to this post: link . In general, given $n$, two matrices are defined as follows: \begin{equation} A = \begin{pmatrix} I_{n-m_A} & 0 \\ 0 & I_{m_A} + J_{m_A} \\ \...