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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7
votes
2answers
615 views

Square Root of a Matrix

From a problem set I'm working on: (Edit 04/11 - I fudged a sign in my matrix...) Let $A(t) \in M_3(\mathbb{R})$ be defined: $$ A(t) = \left( \begin{array}{crc} 1 & 2 & 0 \\ 0 & -1 ...
4
votes
3answers
92 views

Finding the solution for $Ax=0$

Find the solution for $Ax=0$ for the following $3 \times 3$ matrix: $$\begin{pmatrix}3 & 2& -3\\ 2& -1&1 \\ 1& 1& 1\end{pmatrix}$$ I found the row reduced form of that ...
3
votes
3answers
256 views

Is it possible for a matrix to not be onto or 1-1?

Is it possible for a matrix to be neither onto nor 1-1?
1
vote
2answers
974 views

Confusion about whether or not this matrix is onto/1-1?

I am a beginner at matrices and I am trying to find out whether or not the linear transformation defined by the matrix $A$ is onto, and also whether it is 1-1. Here is the matrix $A$: ...
1
vote
2answers
437 views

Linear Algebra: solution of homogeneous system of equation

Could someone please explain to me how they found $x_1$? By my calculations I got $x_2$ & $x_3=0$ and $0=0$.
5
votes
4answers
468 views

Linear Transformations $ \mathbb R^2 \rightarrow \mathbb R^3 $

If $ T : \mathbb R^2 \rightarrow \mathbb R^3 $ is a linear transformation such that $ T \begin{bmatrix} 1 \\ 2 \\ \end{bmatrix} = \begin{bmatrix} 0 \\ 12 \\ -2 \end{bmatrix} $ and $ ...
1
vote
1answer
173 views

A quesion in Fulton & Harris book “representation theory a first course”

In Section 11.2 A little plethysm, it discusses the tensor product of two different representations of $sl_2\mathbb{C}$. It says "If $V=\bigoplus V_{\alpha}$ and $W=\bigoplus W_{\beta}$ then ...
3
votes
1answer
66 views

Kernel of a Matrix

I have been asked to find the kernel of the plane: $x+y+z = 0$. The answer I got was $[-1,1,0] , [-1,0,1]$. The correct answer is $[1,-1,0] , [1,0,-1]$. I'm confused why this is. This makes it ...
1
vote
1answer
84 views

What is the name and notation for this matrix norm?

For a given matrix $\mathbf{A}\in\mathbb{C}^{m \times n}$, let $\|\mathbf{A}\| = \sum_{i=1}^m\sum_{j=1}^n|A_{ij}|$. Clearly, $\|\cdot\|$ is a matrix norm. Is there a special name and notation for ...
2
votes
1answer
76 views

Question on linear algebra-matrices

Let $A$ be an $m\times n$ matrix and $B$ an $n\times k$ matrix. Show that the columns of $C=AB$ are linear combinations of the columns of $A$. If $\alpha_1,\dotsc,\alpha_n$ are the columns of $A$ and ...
2
votes
1answer
3k views

What's the relationship between singular, nontrivial and linear dependent? Basic linear algebra question.

I understand that if a matrix is singular, it has no inverse. If it has nontrivial solutions, it means at least one solutions exists. If it is linearly dependent, it means that for $a_1 ...
5
votes
2answers
216 views

Is the set $ {\rm SL }(n, \mathbb {R}) $ connected in ${\rm M }(n, \mathbb {R}) $

Is the set $ {\rm SL }(n, \mathbb {R}) $ connected in ${\rm M }(n, \mathbb {R}) $? Can you give me hints? I have only concept on following tags.
0
votes
0answers
274 views

Determinant of a general circulant matrix

I'm dealing with a problem that is comparable to "How do I calculate the circulant determinant $C(1, a, a^2, a^3,\dots , a^{n-1})$?", yet slightly more difficult: I was asked to determine the ...
6
votes
2answers
104 views

Can you determine from the minors if the presented module is free?

Motivation (you can ignore this part): A problem in Hartshorne (II.5.8c) asks to show that if we have a coherent sheaf $\mathscr{F}$ on a reduced noetherian scheme $X$, and the function ...
3
votes
4answers
514 views

Normalizing a matrix

I came across a step in an numerical algebra algorithm that says "Normalize the rows of matrix A such that they are unit-norm. Call U the normalized matrix." I do something like this: ...
1
vote
3answers
415 views

A constrained linear least Frobenius norm problem:$\min_{X} \|A-XB\|_F$ subject to $Xv=0$?

Assume we are given two matrices $A, B \in \mathbb R^{n \times m}$ and a vector $v \in \mathbb R^n$. $\|\cdot\|_F$ is the Frobenius norm of a matrix. How can we solve $$\min_{X \in \mathbb R^{n ...
3
votes
1answer
139 views

Formula relating traces of a linear map, its restriction to an invariant subspace and the induced quotient map?

Let $V$ be a finite-dimensional vector space, $T\colon V \to V$ a linear map and $W \subset V$ a $T$-invariant subspace (i.e., $T(W) \subset W$). Then there is a well-defined induced quotient map ...
0
votes
1answer
57 views

Matrix vector addition

Given A a 3x3 matrix and B a 3x1 matrix (or column vector), I am asked to calculate A + B. Both are initially filled with one's. To my knowledge the two matrices would have to be of the same n×m ...
0
votes
1answer
121 views

There are exactly three $2\times 2$ row reduced matrices $\begin{bmatrix}a&b\\c&d\end{bmatrix}$ such that $a+b+c+d=0$

Let $A$ be $2\times 2$ matrix with complex entries, $$A=\begin{bmatrix} a & b\\ c& d\end{bmatrix}$$ Suppose that $A$ is row reduced and also that $a+b+c+d=0$. Prove that there are exactly ...
0
votes
1answer
142 views

$Z$-matrix properties

[Ciarlet 1.3-2] Let $A=(a_{ij})$ be a square matrix whose elements satisfy $a_{ii} \geq 0$, $a_{ij}\leq 0$ if $i\neq j$. Show that the following two properties are equivalent. The matrix ...
1
vote
1answer
103 views

Kronecker's $\delta$ and diagonal matrices

Let $A$ be an $n\times n$ matrix. Prove that $A$ is a diagonal matrix if and only if $A_{ij}=\delta_{ij} A_{ij}$ for all $i$ and $j$. $\delta_{ij}$ is kronecker delta. I think that statement is ...
1
vote
1answer
43 views

Are the eigenvalues limited?

While studying elliptic operators, I encountered the following problem, which I'm having problems to prove or give a counter-example: Let $\Omega$ be an open subset of $\mathbb{R}^m$, and suppose ...
1
vote
1answer
218 views

Exponential map and the special orthogonal group

I need to show that the map exp$: \mathfrak{so}(2) \rightarrow SO(2)$ is surjective. I already have that $\mathfrak{so}(2) = \{A \in M(2, \mathbb{R}) \ | \ A^T + A = 0\}$ and the map is given by ...
2
votes
1answer
131 views

Relation between positive definite Hermitian matrices with their inverses

Let $A$ and $B$ be two positive definite Hermitian matrices. Show that the Hermitian matrix $$C\ =\ A^{-1} + B^{-1} - 4(A + B)^{-1}$$ is also positive definite. Thanks in advance.
0
votes
1answer
256 views

Column space of a $QR$-factorization

Let A be an $m$ x $n$ matrix with linearly independent columns and let $A=QR$ be a $QR$-factorization of $A$. Show that $A$ and $Q$ have the same column space. I honestly don't have a clue where to ...
1
vote
1answer
184 views

Differential Equation $y'' + 8y' - 9y = 0$ and Wronskian

given that $$y(1) = 1, y'(1) = 0$$ I have solved the general solution for this homogenous equation as follow: which is $$r^2 + 8r -9 = 0$$ $$ r = -9, 1$$ $$y = C_1e^t + C_2e^{-9t}$$ $$y' = C_1e^t ...
2
votes
5answers
294 views

Proving a system of n linear equations has only one solution

I have been given the system: $-x_1 + 2x_2 + ... + 2x_{n-1} + 2x_n = 1$ $2x_1 - x_2 + ... 2x_{n-1} + 2x_n = 2$ $.$ $.$ $.$ $2x_1 + 2x_2 + ... + 2x_{n-1} - x_n = n$ And the assignment to prove ...
0
votes
1answer
420 views

Column space of a matrix?

Recently I started learning about matrices and know for example that the pivot columns of a matrix form a basis for the column space of this matrix. I just can't seem to find out when two matrices ...
1
vote
1answer
69 views

Embedding of $PGL_n\mathbb{C}$ and friends

I would like the find an embedding/faithful representation from the projective linear group $PGL_n\mathbb{C}\to GL_m\mathbb{C}$ for some $m$, and likewise for the other projective groups ...
2
votes
1answer
315 views

Block diagonalizing a real matrix

I need to prove that for every real linear operator $T:\mathbb{R}^n\longrightarrow \mathbb{R}^n$, there exists an orthonormal basis of $\mathbb{R}^n$ such that the corresponding matrix is block ...
1
vote
2answers
40 views

Why the first row of the product $MJM^{-1}$ is just the first row of $M$ multiplied by $JM^{-1}$?

Let's define $A_{1,\cdot}$ as the first row of a matrix $A$. The Jordan form of $A$ is $J = M^{-1}AM$. Thus we can write $A = MJM^{-1}$. Question is, what's $(MJM^{-1})_{1,\cdot}$? My textbook ...
0
votes
1answer
48 views

inequalities for optimization over psd matrices with constraints

Consider two p.s.d. matrices $A$ and $B$ both in $\mathbb{R}^{d \times d}$. Define $$a = argmax_{x \in \mathbb{R}^d} x^\top A x $$ and $$b = argmax_{x \in \mathbb{R}^d} x^\top B x $$ both subjected to ...
0
votes
1answer
57 views

Basis of the orthogonal component

Let $\{v_1,\dots,v_n\}$ be an orthogonal basis for $R^n$ and let $W=\operatorname{span}\{v_1,\dots,v_k\}$. Is it necessarily true that $W^\perp=\operatorname{span}\{v_{k+1},\dots,v_n\}$? Either prove ...
2
votes
1answer
2k views

Prove that if the sum of each row of $A$ equals $s$, then $s$ is an eigenvalue of $A$.

Let $A$ be an $n$ x $n$ matrix. $i)$Prove that if the sum of each row of $A$ equals $s$, then $s$ is an eigenvalue of $A$. $ii)$Prove that if the sum of each column of $A$ equals $s$, then $s$ is an ...
0
votes
2answers
200 views

A step in finding the determinant of transpose of a matrix

The following question involves the permutation group, which I am horrible at handling. Any help will be greatly appreciated. Let $A$ be an $n \times n$ matrix with entries $(a_{ij})_{i = 1,2 \cdots, ...
2
votes
2answers
145 views

Orthogonal Decomposition

[Ciarlet 1.2-2] Let $O$ be an orthogonal matrix. Show that there exists an orthogonal matrix $Q$ such that $$Q^{-1}OQ\ =\ \left(\begin{array}{rrrrrrrrrrr} 1 & & & & ...
2
votes
1answer
61 views

Real Part with Singular Values

[Ciarlet 1.2-4] Lea $A$ a real matrix of order $n$. Show that a necessary and sufficient condition for the existence of a unitary matrix $U$ of the same order and of a real matrix $B$ (of the same ...
3
votes
2answers
102 views

Linear Algebra Matrix Question

I am having trouble showing that $e^AX = Xe^A$ for all $n$ by $n$ matrices $X$ where $A$ is an invertible $n$ by $n$ matrix iff $AX = XA$ for all $X$. Any help will be appreciated. Thank you
2
votes
1answer
469 views

Rank of matrices multiplication

Matrices $m_1$ and $m_2$ are over a finite field($GF(2^{8})$ for example). $m_1$ is a $m\times n$ matrix($n > m$) with $rank(m_1) = m$, and $m_2$ is a $n\times c$ ($c > n > m$) matrix ...
1
vote
2answers
61 views

RREF form of a Matrix

Regarding the definition of a RREF matrix.We dont need the LEADING ( first non-zero element from the left ) to be a pivot ( 1 ), right ? We can have a messy matrix in RREF as long as it has column ...
1
vote
1answer
41 views

Is $VV^T + D$ a submanifold?

If the positive definite matrix P forms a manifold, is that the subset that {P: P = V V^T + D} where V is a low rank matrix and D is a positive definite matrix a sub-manifold? This idea is ...
5
votes
1answer
111 views

What indexes do the subgroups of $\mathrm{GL}_n(\Bbb C)$ have?

Let $B_n\subset\mathrm{GL}_n(\Bbb C)$ be the group of invertible upper-triangular matrices. What is the index $[\mathrm{GL}_n(\Bbb C):B_n]?$ (By index I mean the cardinality of a coset space.) In ...
1
vote
0answers
77 views

Finding a basis which is orthonormal with respect to symmetric bilinear form

Let $X=\{a\in M_4(\mathbb{C}): a^T=-a\}$ be the $6$-dimensional vector space of matrices. Define the Pfaffian as $$ \mathrm{Pf}(a)=a_{12}a_{34}-a_{13}a_{24}+a_{14}a_{23} $$ There is an associated ...
3
votes
1answer
45 views

Correct term for “minor matrix”

If I get it right, the minor $M_{i,j}$ for an element $a_{i,j}$ of a matrix A is the determinant of the matrix created from $A$ by excluding the $i^{th}$ row and $j^{th}$ column. But what is a proper ...
1
vote
1answer
175 views

Generalization of the matrix concept

It has been some time since I left university... In a not too formal language, an $n$-dimensional vector is an indexed set of numbers $\{i_1, ..., i_n\}$. A $n\times m$ matrix is a set of numbers ...
-1
votes
2answers
69 views

Prove $f(A^T)=f(A)^T$ for a matrix $A$ [closed]

As the title says, I need to prove $f(A^T)=f(A)^T$ for a matrix $A$. (where $T$ is the transpose) I believe the proof involved the fact that an interpolation polynomial $r(A)=f(A)$ and then I must ...
1
vote
4answers
59 views

Finding determinant of a simple matrix [duplicate]

Can someone please explain how to compute the determinant of $J_n - I_n$ where $j_n$ it a matrix of ones? E.g. for $n=5$ we get the following matrix $$\left(\begin{array}{ccccc} 0 & 1 & 1 ...
0
votes
3answers
1k views

Why is only a square matrix invertible?

Can anyone give a very simple proof (or explanation) as to why only square matrix can possibly be invertible?
5
votes
2answers
83 views

Is $\mathrm{GL}_n(\mathbb C)$ divisible?

A group $G$ (possibly non-abelian) is divisible when for all $k\in \Bbb N$ and $g\in G$ there exists $h\in G$ such that $g=h^k.$ Is the group $\mathrm{GL}_n(\mathbb C)$ divisible? Or more precisely, ...
2
votes
2answers
172 views

Link between the norm $1$ of a matrix and its biggest eigenvalue

I am working on a set of matrices for a project, studying their highest eigenvalue, let's call it $\lambda_{1}$. I was curious and plotted the norm 1 of the matrix, ie $ \frac{1}{n^{2}}\sum_{i,j} ...