Questions tagged [linear-algebra]
For questions about vector spaces of all dimensions and linear transformations between them, including systems of linear equations, bases, dimensions, subspaces, matrices, determinants, traces, eigenvalues and eigenvectors, diagonalization, Jordan forms, etc. For questions specifically concerning matrices, use the (matrices) tag. For questions specifically concerning matrix equations, use the (matrix-equations) tag.
121,373
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Question about Hermitan and Positive Definite
I am asked to prove if A & B belong to Hermitian, and both are positive definite, I need to show that all eigenvalues of BA are real and positive.
How can I proceed?
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1
vote
0
answers
12
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System of linear equations with a free variable.
I am new to linear algebra. The following matrix $A_n$ is an $(n-1) \times n$ matrix. I need to solve $A_nX = 0$, where $X = (x_1, \cdots, x_n)$.
\begin{equation}
A_n = \begin{bmatrix}
b_2 & a_2 &...
- 921
1
vote
0
answers
10
views
Is it possible to get the exact solution of a large-scale convex quadratic programming problem (quadratic objective, linear inequality constraints)?
For a convex quadratic programming problem with the quadratic objective and linear inequality constraints. Theoretically, we can solve them using the KKT condition to get the exact solutions. For ...
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0
votes
1
answer
27
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How to use Cramer's rule for $AX=0$?
For a general system of equation $AX=b$, Cramer's rule states to obtain $x_j$, we need to replace the $j$th column of $A$ with $b$, let us name this matrix as $D_j$ then calculate
$$
\dfrac{\det(D_j)}{...
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1
vote
1
answer
36
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Showing the differential of a map $f:S\to\Bbb R^m$ is linear?
I've been asked an exercise where I have to prove that the differential, at a point, of a map defined on surface, is a linear map. I have this definition and then this lemma where they prove this in ...
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16
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Linear Algebra Smallest Subspace
smallest subspace containing generator.
I am not getting what will be the Smallest subspace containing generator.. IS it L(S)=V(F)
0
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0
answers
24
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I don't understand this question and how I should go about this
Let M be an m X m matrix of rank r of the form M = $ \begin{bmatrix} A & B \\\\ C & D \end{bmatrix} $ where A is an r x r matrix of rank r (so that A is an invertible matrix) and B is r x (m - ...
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1
vote
1
answer
33
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Finding a Basis to a Vector Space of Functions
I'm having some difficulty devising how one finds a linearly independent set of vectors to span this space:
$$ V = \{y~|~y''=0~~\text{and}~y \in C^2(- \infty , \infty )\} $$
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-2
votes
1
answer
35
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Linear algebra. Finding three matrices in Vectorspace $V$
I have this vector space
$$V = \left\{\begin{bmatrix}a & b \\ c & -a\end{bmatrix}\, \middle|\ a, b, c \in \mathbb{R}\right\}$$
and am supposed to find three matrices ($A$, $B$ and $C$) in it ...
- 1
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0
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28
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Find dimensions and bases of $\text{Im}(T)$, $\text{Ker}(T)$ for $T$
The question is:
Find dimensions and bases of $\text{Im}(T)$, $\text{Ker}(T)$ for $T: V \to W$, where $V = P_2$ (polynomials of degree 2 or less), $W = \mathbb{R}^2$, and $T (f) = (f (1) − f (−1), 2f (...
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1
answer
22
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Prove 2D rotation by $\theta$ is a linear transformation using trig identities for $r\cos(\alpha) + s\cos(\beta)$ and $r\sin(\alpha) + s\sin(\beta)$
Theorem
The rotation by $\theta$ function $T:\mathbb{R}^2 \to \mathbb{R}^2$ defined for $\vec{x}%=\begin{bmatrix}x_1 \\ x_2\end{bmatrix}
=\begin{bmatrix}r\cos(\alpha) \\ r\sin(\alpha)\end{bmatrix}$...
- 3,190
4
votes
1
answer
27
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Why does the semigroup of matrices form an epigroup?
An epigroup is a semigroup $S$ in which every element has a (positive) power that lies in a subgroup of $S$. (The subgroup may depend on the element). Note that if $x^n\in G$, where $G$ is a subgroup ...
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0
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28
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Eigenvalues of a restriction/interpolation?
The restriction operation (maps a $2n$ vector into an $n$ vector) is defined as:
$$R = \frac{1}{4}\begin{bmatrix}
1& 2& 1 \\
&&1 & 2 &1\\
&& \vdots\\
&&&...
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0
votes
0
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25
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Can I split this limit into two like this? [closed]
Let $f:\mathbb{R}^{n} \rightarrow \mathbb{R}$, and $f \in C^{1}$ (continuous and derivative continuous). Let $p,e,v \in \mathbb{R}^{n}$.
This is supposed to be true (is part of the proof of ...
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0
answers
25
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How to find eigenvalues of very complicated matrix product.
I have the following:
$$S\big(I - PA_{2h}^{-1}RA\big)S$$
Where $A = -D_2 + I$ with $D_2$ being the second derivative finite difference approximation. $A_{2h}$ is the same matrix but for a grid with ...
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votes
0
answers
24
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Choose a matrix to ensure a product is symmetric
Let $ A \in \mathbb{R}^{m \times n} $, $ B \in \mathbb{R}^{n \times m} $.
Is it possible to choose an $ A $ such that $ AB = (AB)^{\text{T}} $ for any $ B $ where $A \neq [0] $?
The reason I am asking ...
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5
votes
1
answer
42
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Improve exposition of this proof: Matrix multiplication is associative, due to commutativity of underlying field
I'd like to request tips on improving proof writing by taking a standard proof in linear in algebra which is nonetheless difficult to write well, and asking for verification and improvements.
I also ...
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0
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11
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Given a set of geometrically independent points if we add a point outside of the plane spawn the set the new set is still gometrially independent
If ${a_0, ..., a_n}$ is geometrically independent, and if $w$ lies outside of the plane that these points spawn, then ${w, a_0, ..., a_n}$ is geometrically independent.
I tried doing this assuming $w$ ...
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votes
3
answers
68
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Let $A$ be a square matrix such that $A(I-A)=I$, prove $A^2-I$ is invertible
Let $A$ be a square matrix such that $A(I-A)=I$, prove $A^2-I$ is invertible.
$A^2-I=(A-I)(A+I)$, clearly $A-I$ is invertible, but how do I prove that $A+I$ is also invertible?
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1
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30
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Proof check that $\|A\|_{F}^{2} = \sigma_{1}^{2} + \sigma_{2}^{2} + \dots + \sigma_{k}^{2}$
Consider $\|A\|_{F} = \sqrt{\langle A, A\rangle)}$ where $\langle X, Y\rangle = \text{Tr}(X^{*}Y)$ where '*' denotes the conjugate transpose of a matrix.
I was asked to prove the following question:
...
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1
vote
1
answer
34
views
Eigenvalues in commutative diagram
Let $K$ be a field.
Suppose, we have linear maps $f: K^n \mapsto K^n$, $g: K^m \mapsto K^m$ and $h: K^n \mapsto K^m$ such that $h$ is surjective and $h \circ f = g\circ h$. Then every eigenvalue of $g$...
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votes
2
answers
74
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Good materials on calculus and linear algebra [closed]
I have a master's degree in distributed networks and applications, and I graduated with a bachelor's degree in applied mathematics. I wouldn't say that I studied well, because my knowledge of calculus,...
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0
votes
0
answers
25
views
row rank = column rank, confusion about wikipedia proof using linear combination
The fifth line: Now, each row of $A$ is given by a linear combination of the $r$ rows of $R$.Therefore, the rows of $R$ form a spanning set of the row space of $A$
I think the first half only implies ...
- 95
0
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0
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29
views
How to find an equation of the plane
Find an equation of the plane passing through the point $P(1, 0, 1)$ and containing the line $[x, y, z] = [0, 1, -1] +t [2, 0, 1]$
0
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1
answer
20
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How to show that a map is $K$-multilinear
Hey I have this exercise where I have some questions
For a field $K$, let $x =\begin{pmatrix}x_1\\x_2\\x_3\end{pmatrix}, y = \begin{pmatrix}y_1\\y_2\\y_3\end{pmatrix}, z = \begin{pmatrix}z_1\\z_2\\z_3\...
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0
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27
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How to calculate a $n \times n$ even-order antisymmetric determinant [closed]
such as
$$
\begin{bmatrix}
0 & a & b & c \\
-a & 0 & d & e \\
-b & -d & 0 & f \\
-c & -e & -f & 0
\end{bmatrix}
$$
-1
votes
1
answer
18
views
Property of linear multidimensional function? [closed]
Is it true that if $f(x,y)$ is a linear on both x and y
$f(ax,by) = af(x,by)+bf(ax,y)$?
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0
votes
0
answers
32
views
Fréchet derivative of a similarity transformation
Suppose $h(Q) = Q^{T} A Q$, then the Fréchet derivative is given by $D_{h} [Q] (H) = H^{T} A Q + Q^{T} A H$. I am bit unsure about this so-called Fréchet derivative is obtained.
I would have just said:...
3
votes
1
answer
20
views
How to find optimal points through a series of circles? [closed]
So if you have n circles with radius and center. Let's call these circles C_1, C_2, ..., C_n.
Then how could I find the series of points that is the shortest path from the perimeter on C_1, to the ...
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0
votes
0
answers
14
views
The difference between the sum of the squares of the diagonal elements of WAW (for Wishart matrix W) and matrix A （for any A）
For any symmetric matrix ${ A}\in R^{K\times K}$, we can compute ${ B} = { W}{ A}{ W}$ where $W\in R^{K\times K}$ is a Wishart matrix with $N$ degrees of freedom (N>K).
I want to bound term $\sum_{...
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1
vote
0
answers
56
views
Proof verification: $[L_1L_2:K] \leq [L_1:K][L_2:K]$
Let $K \subseteq L_1,L_2 \subseteq L$ be fields. I want to show the following:
$$[L_1L_2:K] \leq [L_1:K][L_2:K]$$
My attempt:
The definition of $$L_1L_2:=L_1(L_2):=\bigcap \{F:F\text{ is a subfield ...
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1
vote
1
answer
29
views
Partial isometry, Isometry and Gram matrix
Suppose I have a set of matrices $\mathcal{H}$ s.t. $\forall H\in\mathcal{H}, \ H = CU$ where $C$ is a fixed matrix and $U$ is an orthonormal matrix. We know that this set can be characterized ...
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0
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1
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15
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How does multiplying a row and a col with a constant affect semi-positive definite property?
Sorry if this question is obvious....
Assuming $A$ is semi-positive definite (and symmetric). Now, if we want to multiply all elements on the $i$-th row AND all elements on the $i$-th column of $A$ by ...
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-1
votes
0
answers
26
views
Up to similarity, there is a unique 3 × 3 matrix with minimal polynomial (x−1)^2 (x−2) [closed]
Is this true. Because I can't find any invertible P, such that M = P D P^-1, so that the similarity satisfies. Can any one guide me regarding my approach to the problem?
Thanks @Gerry Myerson, my ...
0
votes
0
answers
15
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Exponentials of stochastic band matrices
This may be a duplicate, but I've done some searching and I can't find exactly this problem setting, probably due to not knowing the right terminology for how to refer to the transition matrix.
I'm ...
- 1,460
0
votes
3
answers
56
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Find orthogonal vectors in relation to span
Consider $R^3$ as an inner product space in relation to scalarmultiplication. Find all vectors in the subspace
$$Span\bigg(\bigg[\begin{matrix}1\\2\\1
\end{matrix}\bigg],\bigg[\begin{matrix}3\\4\\1
\...
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6
votes
1
answer
66
views
How to determine if a subspace of $ \mathbb{R}^n $ has an integer basis
Let $ W $ be a sub vector space of $ \mathbb{R}^n $. How can we determine if $ W $ admits an integer basis?
This is equivalent to asking how to determine if $ W \cap \mathbb{Z}^n $ spans $ W $.
...
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votes
2
answers
71
views
What is a sufficient and necessary condition for a system of linear equations to have a unique integer solution?
Bézout's theorem states that a linear diophantine equation $ax+by=c$ has integer solutions if and only if $\gcd(a,b)|c$. Then is there a theorem regarding the existence of a unique integer solution ...
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2
votes
0
answers
30
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Question on showing that $\frac{1}{n}\sum_{k=0}^{n-1}U^kf$ converges in norm to the orthogonal projection $Pf$ to the space $\{f\in H: Uf = f\}$
Edit: The reference I am reading is Yves Coudène's Ergodic Theory and Dynamical Systems, chapter 1, proof of theorem 1.1 on page 5.
Let $H$ be a Hilbert space and $U:H\to H$ a bounded linear operator ...
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0
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40
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Name of symmetric multilinear function?
What's the name (if any) for the multilinear symmetric function that maps a square matrix over some commutative ring $R$ with $1$ to a value in $R$? It's basically like the determinant, but symmetric....
- 8,810
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0
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31
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Find an orthogonal basis to minimize the norm
Let $A \in R^{n\times n}$. Find O, an orthogonal basis of $R^{n\times n}$, to minimize $∥A−O∥_F$.
$∥⋅∥_F$ is the Frobenius Norm.
It is similar to this question, except that O is not an orthogonal ...
- 1
0
votes
1
answer
48
views
using computer to solve a system of nonlinear equations
I was wondering if there is a program or website that solved a system of nonlinear equations
I was trying a system that looks like this
$$a+b = cd , a+c = bd ,a+d = bc$$
, ...
it's like a pattern
that ...
- 21
0
votes
0
answers
22
views
Determinant of involving adjacency matrix of subgraph
Let $A$ be the adjacency matrix of a graph $G$ labelled $\{x_1,\cdots,x_n\}$. Let $B:=(I-\frac{1}{2d}A)^{-1}$ where $d$ is a positive integer.
Then let $A'$ be the adjacency matrix of the graph $G\...
- 1,437
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vote
2
answers
54
views
$u_1, u_2$ is linearly independent if and only if the family $u_1 + u_2, u_1 − u_2$ is linearly independent.
I have this exercise where I want to check my solutions. Can someone help me?
Let $u_1$ and $u_2$ be elements of $V$. If $1 + 1 ≠ 0$ in $K$, then the family $u_1, u_2$ is linearly independent if and ...
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votes
0
answers
8
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Show that there exists exactly one $l_A$-invariant $K$-subspace of dimension $1$ in $K^2$.
So I have this exercise where I want to check my solutions. Can someone help me?
Let $A = \begin{pmatrix} a &b \\ 0& d \end{pmatrix} ∈ Mat_{2,2}(K)$ with $a, b, d ∈ K$, where $a ≠ 0$ and $d ≠ ...
- 388
3
votes
1
answer
69
views
Homogeneous equations for conic sections
The equation of a "standard" circle is $x^2 + y^2 = r^2$. That equation is not homogeneous and does not include the origin; we can homogenize it by adding a $z$ term of degree 2, getting $x^...
- 2,103
3
votes
1
answer
55
views
Left operator on $M_n(\mathbb{F})$ diagnolizable iff A is diagnolizable
Question: let $A\in V = M_n(\mathbb{F})$,left translation operator on V is defined as $T:V\rightarrow V,B\rightarrow T(B)=AB$, prove that $T$ is if and if only A is diagnolizable.
I have found a ...
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5
votes
1
answer
111
views
If we expand the definition of the general quadratic $ax^{2}+bx+c=0$ to include the case $a=0$, can we arrive at a general solution?
Through a simple mathematical substitution, I have stumbled upon an alternative formula for solving a quadratic equation:
$$x=\frac{2c}{-b \pm \sqrt{b^{2}-4ac}}$$
(Please refer to my formula ...
- 51
1
vote
1
answer
33
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Where is the gap in the proof of uniqueness of the row reduced echelon form?
The author of the book I'm reading says that the following proof has a gap,
but I don't know where it is.
Here is the proof which the author claims that it has a gap.
First, The author defines that '...
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5
votes
1
answer
64
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The Definition of Orthogonal Complement
In Linear Algebra Done Right by Axler, the author defines the orthogonal complement as follows:
If $U$ is a subset of a vector space $V$, then the orthogonal complement of $U$, denoted by $U^\perp$, ...
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