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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Sum of all elements in a matrix with specified conditions
Let M be a matrix with 20 rows and 21 columns, containing the following elements
$M[i][j]=i*i$ if $i=j$
$M[i][j]=min(i,j)$ if $i≠j$
for $1\leqslant i\leqslant 20, 1\leqslant j\leqslant 21$.
What is ...
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0
answers
21
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Do coordinate changes only affect antisymmetric matrices linearly?
Let there be an antisymmetric tensor field $\Omega_{ab}(q)$ where $q^i$ are coordinates on a 2N dimensional manifold. For context, this is a general symplectic form on phase space. I want to find a ...
- 2,212
14
votes
5
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Find all matrices that commute with $\left(\begin{smallmatrix}2&3\\1&4\end{smallmatrix}\right)$
Find all $2\times 2$ matrices that commute with
$$\left( \begin{array}{cc}
2 & 3 \\
1 & 4 \end{array} \right)$$
My progress:
I know that a square matrix commutes with itself, the identity ...
- 6,873
2
votes
3
answers
216
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Find all matrices that commute with $\left(\begin{smallmatrix}12&9\\0&-9\end{smallmatrix}\right)$
Find all $2\times 2$ matrices that commute with
$$\left( \begin{array}{cc}
12 & 9 \\
0 & -9 \end{array} \right)$$
My progress:
I know that we need to find all possible matrices A that AB = BA.
...
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0
votes
1
answer
241
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Set of matrices that commute with $\left(\begin{smallmatrix}1&1\\1&1\end{smallmatrix}\right)$
Which are matrices $2\times 2$ that commute with the matrix
$$\left[\begin{array}{cc}1&1\\1&1\end{array}\right]?$$
- 6,873
0
votes
1
answer
27
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What even is a lower and upper triangular matrix in the context of Jacobi method?
I cannot understand how the hell L and U works in terms of the Jacobi Method.
\begin{equation}
x_{n+1} = D^{-1}(b-(L+U)x_n)
\end{equation}
Take this matrix system as an example:
\begin{equation}
...
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2
votes
2
answers
372
views
Trace of product of matrices with nonzero trace
I have the four matrices
$$\begin{pmatrix}1&0&0&0\\1&0&0&0\\0&1&1&1\\0&1&1&1\end{pmatrix},\quad
\begin{pmatrix}0&1&0&0\\0&1&0&0\...
- 609
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votes
0
answers
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Piling up a vector and multiplying by a matrix
Let $V$ be a complex inner product space. Suppose $V$ is the orthogonal direct sum $V = U\oplus W$ of two of vector subspaces $U$ and $W$. Suppose further that $\text{dim}U = \text{dim}V = n$, $\{e_{1}...
- 1,891
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votes
1
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Inverse $T$ matrix of a 3*3 matrix. [closed]
I have the matrix
$$ A=
\begin{bmatrix}
1 & 4 & 1\\
0 & 2 & 5\\
0 & 0 & 5
\end{bmatrix}.
$$
I have found the
$$ T=
\begin{bmatrix}
1 & 4 & 1\\
0 & 0 & 1\\
0 &...
- 13.9k
3
votes
1
answer
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If $A$ is normal with $\sigma(A)\subseteq \mathbb{R}\cup\mathbb{T}$, does $\text{dim ker}(AB-BA)=\text{dim ker}(A^*B-BA^*)$?
This clearly holds if $A$ is self-adjoint, and also if $A$ is unitary, because then $A(\text{ker}(AB-BA))=\text{ker}(A^*B-BA^*)$. To prove this, if $w\in\text{ker}(AB-BA)$, then $A^*B(Aw)=A^*ABw=Bw=BA^...
- 139k
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0
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Bounded (finite-rank) Inverse Operators
While studying for my functional analysis course I encountered bounded inverse theorem which states that given a bijective bounded linear operator $T:X\rightarrow Y$, then $T^{-1}:Y\rightarrow X$ is ...
- 16k
1
vote
1
answer
38
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Making a symmetric matrix positive (semi-)definite by adding a diagonal matrix
Say I have a matrix $A \in R^{p \times p}$ which is symmetric and with non-negative diagonal entries (i.e. $a_{ii} \geq 0 \forall i\in \{1, \ldots, p\}$). However, $A$ is not positive (semi-)definite.
...
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1
answer
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Does symmetric matrix adjoint of $A$ have all its entries equal?
Reading about graph theory they said that, for the Laplacian matrix of a graph, let's call it $A$, its adjoint $Adj(A)$ since $A$ is symmetric, ,
Here is an example of Laplacian matrix $A$ and its ...
- 139k
0
votes
1
answer
28
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Orthogonal block-matrix
Let $$M=\begin{bmatrix}A & C \\ 0 & B\end{bmatrix}\in \mathbb R^{m\times n}$$ be a block matrix.
(a) If $M$ is orthogonal and $C=0$, are $A$ and $B$ orthogonal?
(b) Suppose $A$ is orthogonal ...
- 1,031
0
votes
0
answers
32
views
direction of poles of 2 cascading systems
Given the following system: $$G(s)=G_2(s)G_1(s)=\begin{bmatrix}
> \begin{array}{c|c} A_2&B_2\\ \hline C_2&D_2 \end{array}
> \end{bmatrix} \begin{bmatrix} \begin{array}{c|c} A_1&...
- 3,635
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votes
5
answers
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views
Show that $\operatorname{rank}(A+B) \leq \operatorname{rank}(A) + \operatorname{rank}(B)$
I know about the fact that $\operatorname{rank}(A+B) \leq \operatorname{rank}(A) + \operatorname{rank}(B)$, where $A$ and $B$ are $m \times n$ matrices.
But somehow, I don't find this as intuitive as ...
- 5,155
0
votes
0
answers
16
views
Closure of equivalence relations problem at the end
The question is "Use Warshall’s Algorithm to find the transitive closure of the relation {(a, c), (b, d), (c, a), (d, b), (e, d)} on the set {a, b, c, d, e}."
My matrix at the end is:
( 1 0 ...
- 23
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votes
3
answers
141
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System of equations in terms of an unknown and its conjugate
I have a system of complex equations:
$$ A z + B \overline{z} = c$$
Where $A, B \in \mathbb{C}^{N \times N}$ and $z, c \in \mathbb{C}^{N \times 1}$.
I want to solve for $z$. If I could express it ...
- 11
11
votes
1
answer
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What does the degree of a matrix minimal polynomial encode?
Let $\mathsf{F}$ be any field. Let $A$ be an $n \times n$ matrix over $\mathsf{F},$ whose rank is $r \le n.$ Let $\mu \in \mathsf{F}[x]$ be the minimal polynomial of $A.$
What does $\deg(\mu)$ tell ...
CommunityBot
- 1
28
votes
5
answers
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How to prove $(AB)^T=B^T A^T$
Given an $m\times n$-matrix $A$ and an $n\times p$-matrix $B$. Prove that $(AB)^T = B^TA^T$.
Here is my attempt:
Write the matrices $A$ and $B$ as $A = [a_{ij}]$ and $B = [b_{ij}]$, meaning that ...
- 1
0
votes
2
answers
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views
Finding a change of Basis Matrix Of Polynomial Degree Less than 1
I am doing some practice questions and I'm not too sure where to start on this one:
Consider the bases $B = \{p_1,p_2\}$ and $B' = \{q_1,q_2\}$ of P$_1$, the polynomials of degree $\le$ 1, where $p_1 ...
CommunityBot
- 1
1
vote
1
answer
53
views
Square roots of Jordan blocks
It is known that for the Jordan block
$$ J_p = \left(\overbrace{\begin{array}{cccc}
\lambda & 1& \ldots & 0\\
0 & \lambda& & 0\\
0 & 0 & \ddots & 1 \\
0 &0 &...
- 450k
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votes
1
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32
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$\det(\bar B)=0$ for $\bar B$ columned matrix with only $2$ non-null entries equal to $1$ and $-1$?
Reading about graph theory, I am presented with the case in which a square incidence submatrix of a
directed graph has, in each column, only 2 non-zero entries (from the definition of the incidence ...
- 4,436
6
votes
1
answer
67
views
Proof of first step in SVD
I have to proof the following statement:
Prove that for a given $A\in \mathcal M _{n \times m}(\mathbb R)$ there exist two orthogonal matrices $U \in \mathcal O(n)$, $V \in \mathcal O(m)$ such that:
$...
- 52.2k
1
vote
0
answers
32
views
Search for matrix spectrum
One task asks to find the spectrum of the matrix
$$
\left( \begin{array}{*{20}{c}}
-21&6&2&-20&-6\\
-21&12&-7&-14&-7\\
-2 &1&-4&-1&-1\\
19&-6&-2&...
- 402
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votes
1
answer
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If the determinant of Hessian is negative, what can we say about the matrix?
This question is based on this answer for the question How do I prove that this objective function is not convex?
The objective expression becomes $2y^2x^2$. The Hessian of this expression is $$\...
CommunityBot
- 1
6
votes
3
answers
2k
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Is there a simpler, more abstract proof of the Cayley-Hamilton theorem for matrices?
The Cayley-Hamilton theorem is equivalent to: Let $R$ be a ring and let $M_n(R)$ be $n\times n$ matrices over $R$. Then the minimal polynomial of $A \in M_n(R)$ over $R$ divides the characteristic ...
- 2,124
1
vote
1
answer
95
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Monotonicity of Frobenius norm
$A,B\in \mathbb{R}^{n\times n}$,$A,B\succeq 0$,$B-A\succeq 0$, prove $\Vert A \Vert_F \leq \Vert B \Vert_F $
(explain of sign: If $X \succeq 0$, then there exists
another matrix $Y \succeq 0$ such ...
43
votes
1
answer
31k
views
Generalized rotation matrix in N dimensional space around N-2 unit vector
There is a 2d rotation matrix around point $(0, 0)$ with angle $\theta$.
$$
\left[ \begin{array}{ccc}
\cos(\theta) & -\sin(\theta) \\
\sin(\theta) & \cos(\theta) \end{array} \right]
$$
Next, ...
- 1,257
2
votes
1
answer
83
views
On the logarithm of a matrix
While teaching a course on ODE, I needed to introduce the notion of matrix logarithms. I intend to define it as follows.
Definition (Matrix Logarithm)
Let $A\in GL_n(\mathbb{C})$. We define
A) ...
- 10.9k
3
votes
1
answer
83
views
Solve $\| X A - B \|$ subject to $X C = C X$
Given $A, B \in \mathbb{R}^{n \times k}$ and S.P.D. $C \in \mathbb{R}^{n \times n}$, I would like to find an analytical solution for the matrix $X \in \mathbb{R}^{n \times n}$ that minimizes
\begin{...
- 361
1
vote
1
answer
32
views
Relation between pairwise outer products and Kronecker tensor product
Let $A,B$ be real $n\times n$ matrices (if we want complex entries, replace transpose with hermitian conjugate). Write them as an array of columns so that $A=\begin{pmatrix} A_1~|&\cdots& |~...
- 14.3k
1
vote
1
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How can I use cramers rule to solve this problem?
I have deduced and found three equations in which I need to solve this problem (Stated Below)
How can I format this to fit into matrices to be solved with Cramers rule.
The formulas I have are:
R=...
CommunityBot
- 1
31
votes
8
answers
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views
Is there such a thing as a matrix of functions?
Do we ever put functions as entries of a matrix? If so, are these matrices used in linear algebra or do they have some other special use?
There have been minor not neccessarily conflicts per se, but ...
- 11
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votes
3
answers
4k
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Cholesky decompostion: upper triangular or lower triangular?
Can I find and use $U$ such that
$$A = U U^{T}$$
where $U$ is an upper triangular matrix, to find a solution instead of finding $L$ such that $ A = L L^{T}$ (where $L$ is a lower triangular matrix) ...
- 185
18
votes
4
answers
29k
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$\det(I+A) = 1 + tr(A) + \det(A)$ for $n=2$ and for $n>2$?
Let $I$ the identity matrix and $A$ another general square matrix. In the case $n=2$ one can easily verifies that
\begin{equation}
\det(I+A) = 1 + tr(A) + \det(A)
\end{equation}
or
\begin{equation}
\...
- 11.1k
3
votes
1
answer
59
views
What is the exact form of the matrix algebra generated by a Jordan matrix?
In a paper it is claimed that the matrix algebra generated by a matrix $$J=\oplus_i J_{\lambda_i,m_i}$$ in Jordan normal form "is the algebra of block diagonal matrices with the same block ...
- 8,862
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votes
2
answers
2k
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How does the classification using the 0-1 loss matrix method work? [closed]
In this machine learning lecture the professor says:
Suppose $\mathbf{X}\in\Bbb R^p$ and $g\in G$ where $G$ is a discrete
space. We have a joint probability distribution $\Pr(\mathbf{X},g)$.
...
CommunityBot
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3
votes
2
answers
83
views
Understanding subspace-restricted isomorphisms and global isomorphisms
(It may be noted that the author was not aware of Endomorphisms & Automorphisms at the time of writing this question)
I'm trying to better understand the concept of a linear transformation acting ...
- 48.6k
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0
answers
25
views
matrix multiplication of the form (UP) X = B
I have sudden confusion in matrix multiplication:
If I have
(UP)X = B, where U and P are matrix and B is a vector,
can I write UX= P^-1 B?
Sorry for asking such a basic question. I got this confusion ...
- 109
2
votes
0
answers
9
views
How to describe the partition of $GL_3(\mathbb{C})$ by Bruhat decomposition accurately?
We know that $GL_n(\mathbb{C})$ can be decomposed as $GL_n(\mathbb{C}) = \bigsqcup_{w \in W}BwB$ where $B$ is the subgroup of upper triangular invertible matrices and $W$ is the Weyl group isomorphic ...
- 45
2
votes
2
answers
2k
views
Eigenvalues of a Real Symmetric Bilinear Form Independent of Basis
Question is from Artin's Algebra, p. 263.
If $A$ is the matrix of a symmetric bilinear form, prove or disprove: The eigenvalues of $A$ are independent of the choice of basis.
I suspect the result ...
- 449
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votes
2
answers
89
views
For $T\in L(V,W)$, prove that there are bases such that $M(T)$ is 1 on the first dim range $T$ elements of the diagonal and zero everywhere else.
I'd like to solve the following problem from Chapter 3, "Linear Maps", section 3.C "Matrices", from Axler's Linear Algebra Done Right.
Suppose $V$ and $W$ are finite-dimensional ...
- 4,558
3
votes
4
answers
2k
views
Show that orthogonal matrices have eigenvalues with magnitude $1$ without a sesquilinear inner product
Is it possible to consider complex eigenvalues without a Hermitian (i.e. sesquilinear) inner product over a complex vector space?
For instance: let $A$ be a real orthogonal matrix (so $A^TA = I$). ...
- 4,558
2
votes
2
answers
88
views
Confusion about matrix polynomials
In this answer the point is made that since a unitary and symmetric matrix $U$ commutes with all $U^j$ (I assume those are the matrix powers $U^2$, $U^3$ etc?) it must also commute with $S=\sqrt{U}$, ...
- 7,140
0
votes
3
answers
46
views
Any assured method for finding the derivative of p Euclidian norms?
Assuming both $y$ and $\beta$ are $p \times 1$ vectors, and $W$ is a $p \times p - 2$ matrix, how would one take the first derivative of this: $L(\beta) = || y - \beta||^2_2 + || W\beta||^2_2$.
I'm ...
- 36k
-3
votes
1
answer
45
views
Finding characteristic polynomial of a square matrix and how to proving that matrix is diagonalizable [closed]
Let $A$ be a square matrix of order $n$ such that $|A + I| = |A − 3I| = 0$ and also $\operatorname{rank}(A)= 2$. I need to find characteristic polynomial of $A$ and have prove that $A$ is ...
- 25.5k
1
vote
0
answers
20
views
Generalization of Sylvester's law of inertia to the case of rectangle matrix
Sylvester's law of inertia states given a symmetric matrix $A$ and a squared invertible matrix $S$ of the same size, then $A$ and $SAS^\top$ have the same number of positive, negative, and zero ...
- 71
0
votes
1
answer
22
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Square is not a monotone function on the set of PSD matrices
Let $X,Y$ be real symmetric matrices. We say $X \geq Y$ if $X-Y$ is PSD.
Question: What is a basic ($2\times 2$) example of matrices $X,Y$ s.t.:
$$
X \leq Y
$$
but:
$$
X^2 > Y^2
$$
e.g.$X^2 - Y^2$ ...
- 225k
0
votes
1
answer
38
views
A condition to guarantee invertibility of a matrix $A=BCD$ for every invertible matrix $C$
I have an $m$-by-$m$ matrix $A = BCD$ where $B$ is an $m$-by-$n$ matrix, $C$ is an $n$-by-$n$ matrix, and $D$ is an $n$-by-$m$ matrix. My question is: what are the requirements on matrices $B$ and $C$ ...
- 139k