Linear algebra is concerned with 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 ...

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Sylvester's law of inertia for generic matrices.

By Sylvester's law of inertia, the positive and negative indices of a symmetric matrix $A$ are also the number of positive and negative eigenvalues of $A$. I was wondering if a similar result is known ...
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3answers
46 views

Inverse of partitioned matrices [on hold]

A matrix of the form $$A=\begin{bmatrix} A_{11} & A_{12}\\ 0 & A_{22} \end{bmatrix}$$ is said to be block upper triangular. Assume that $A_{11}$ is $p \times p$, $A_{22}$ is $q \times q$ and ...
2
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1answer
37 views

How to solve the convex combination problem of matrix?

Let $A \succeq B$ denote matrix $A-B$ is positive semidefinite, and here is the definition of redundant(all the matrix dimensions are $N\times N$ ): Given a set of matrix $\{B_i\}_{i=1}^{l}$, if ...
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0answers
39 views

Vector spaces and nontrivial subspace. [closed]

Give an example of a subset of $\mathbb{R}^2$ that is a nontrivial subspace of $\mathbb{R}^2$? $\mathbb{R}^2$ as $\{(a, b) \mid a, b \in \mathbb{R}\}$
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0answers
55 views

Why we define the adjoint operator

Suppose in vector space $A: X\rightarrow Y$ is a linear map, the adjoint operator $A^{'}: Y^{'}\rightarrow X^{'}$ is defined as: $f(Ax)=(A^{'}f)(x)$. As I can understand, the adjoint operator just ...
2
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0answers
61 views

Help me to prove the determinant formula

Actually it is about the question of n-linear function, but it is so relevant to the determinant formula. Here is the notation of the theorem. If $n>1$ and $A$ is an $n \times n$ matrix over $K$, ...
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2answers
30 views

In an inner product space, if the matrix is symmetric, is an eigenspace necessarily orthogonal to the range space?

Say I have 3 distinct eigenvalues for a symmetric matrix. By the Spectral Theorem, the three eigenspaces are mutually orthogonal. But, if I just wanted to compute the first eigenspace, ...
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1answer
44 views

Separating vectors from linear combination

Suppose I have a linear combination of vectors as follows $ \mathbf{s} = \alpha_1\mathbf{x}_1 + \dots + \alpha_m\mathbf{x}_m + \beta_1\mathbf{y}_1 + \dots + \beta_n\mathbf{y}_n $ where $\alpha_i, ...
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0answers
34 views

Wedge product of maps: functorial vs. exterior algebra

Suppose that $V$ and $W$ are finite-dimensional vector spaces over $\mathbb{F}$. If $\varphi, \psi \in \hom(V,W)$, there are at least two interpretations of the symbol $\varphi \wedge \psi$: It is ...
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1answer
15 views

Stability of non-homogeneous and non-autonomous first-order difference equation

I am seeking to analyze the stability of steady points in a system of $n$ variables $x_1(t), ..., x_n(t)$. With discrete time $t$ the system is described by \begin{eqnarray*} x_i(t+1) = ...
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1answer
29 views

Identification between wedge product and its dual

Let $\mathbb{F}$ be a field, and let $(e_i)$ be the usual elementary basis of $\mathbb{F}^n$. Let $\varphi_{ij}: \mathbb{F}^n \wedge \mathbb{F}^n \to \mathbb{F}$ be such that $v \wedge w \mapsto ...
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2answers
46 views

Is this a circulant matrix?

It's symmetric, but I'm not sure whether it is circulant. In a question that I had asked on MSE a couple of weeks ago, several commenters had said that this is a circulant matrix, and to study the ...
2
votes
3answers
56 views

Producing lower bounds for $\text{trace}(A^2)$ for a positive semidefinite, symmetric matrix $A$

Are there any lower bounds on $\DeclareMathOperator{trace}{trace}$ \begin{align*} \trace(A^2), \end{align*} where $A$ is positive semi-definite and symmetric? I am aware of the inequality $$ ...
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0answers
49 views

Find the solution to the following LPP by solving its dual. [closed]

Minimize : $ Z = 300X_1 + 110X_2$ Subject to : \begin{align*} 30X_1 + 5X_2 &\geq 6 \\ 20X_1 + 10X_2 &\geq 8 \\ X_1, X_2 &\geq 0 \end{align*}
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1answer
28 views

matrix multiplication manipulation

a,b $\in \mathbb{R^n}$ and C $\in \mathbb{R^{nxn}}$. I have $ab^TCab^TC$. I try to manipulate this multiplication into: $b^TCaab^TC$. I need help.
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2answers
25 views

Determining all scalars $a \in \mathbb{R}$ for which a matrixrepresentation is orthogonal?

Problem: Let $a \in \mathbb{R}$ and \begin{align*} T: \mathbb{R}^{n \times n} \rightarrow \mathbb{R}^{n \times n}: A \mapsto aA. \end{align*} Determine all $a \in \mathbb{R}$ for which the matrix of ...
7
votes
3answers
441 views

Rule for squaring arbitrary powers?

This is a really simple question, but I don't know how to phrase it well enough for Google. I'm going through a proof and don't understand how: $$ (q^{2^{n+1}})^2 = q^{2^{n+2}} $$ I thought it would ...
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2answers
25 views

Linear algebra: proving transformation matrix between orthogonal basis is unitary

The vector space $V$ is equipped with a Hermitian scalar product and an orthonormal basis $\{e_1,\ldots,e_n\}$. A second orthonormal basis $\{e_1',\ldots,e_n'\}$ is related to the first one by ...
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0answers
19 views

How to multiply the elements within a vector using matrix operations (e.g., dot product)?

Suppose a vector $\vec{v}^T=(v_1, v_2, \ldots, v_n)^T$. To sum the elements within the vector, I can use the dot product with a column vector of ones, $\sum_i v_i = \vec{v}^T \cdot \vec{1}$. My ...
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1answer
26 views

Show that there exists a non-negative integer $r$ s.t. $ker(T^r) = ker(T^{r+1})$.

Question: Let $V$ be an $n$-dimensional complex vector space, let $T: V \to V$ be a linear transformation. Show that there exists a non-negative integer $r$ s.t. $ker(T^r) = ker(T^{r+1})$. My ...
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3answers
32 views

Finding Null Space Basis

let $v$ be a vector $v=(1,-1,1)$, find $Ker(v)$ or $v*x=0$ I have approached it this way $(y-z,-y,z)=(y,-y,0)+(-z,0,z)=y(1,-1,0)+z(-1,0,1)$ But the answer $(1,1,0),(-1,0,1)$ Where am I wrong?
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2answers
47 views

Show that $T$ is normal

Let inner product space $V$ (finite) above $\mathbb{C}$. Let the operator $T:V\to V$ s.t. $$T^2 = \frac{1}{2}(T+T^*)$$ Prove that $T$ is normal $(T^*T = TT^*)$ $T^2 - T = 0$ So I've tried the ...
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0answers
20 views

Consider the ordered basis for the vector space V of lower 2x2 lower triangular matrices with zero trace.

Consider the ordered bases $$ \mathcal{B} =\left\{ \left[ \begin{matrix} -4 & 0\\ 0 & 4 \\ \end{matrix}\right]; \left[ \begin{matrix} 0 & 0\\ ...
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1answer
14 views

What's the difference between these two spaces?

In the finite element method, $Q1$ element is defined by $\textrm{span} \{1, x, y, xy\}$. And $\textit{rotated } Q1$ element is defined by $\textrm{span}\{1, x, y, x^2-y^2\}$. Please tell me what ...
3
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1answer
35 views

Prove $T|_{V_\lambda}$ is diagonalizable

Let $V$, an $n$-dimensional vector space and let $T, S:V\to V$, two diagonalizable linear operators. Show that if $TS=ST$ then every $V_\lambda$ of $S$ is $T$-invariant and the restriction, ...
0
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1answer
36 views

Linear transformation: Change of basis

I am given the following linear transformation $L$: $A=\begin{bmatrix}1&2\\0&3\end{bmatrix} \in \Bbb R^{2 \times 2}$ $L: \space \Bbb R^{2 \times 2} \longrightarrow \Bbb R^{2 \times 2}; ...
2
votes
2answers
40 views

Sign of eigenvalues of $A$ by $\det(A-\lambda I)=\lambda \det(B+D-\lambda I).$

Let $A$ be a $n\times n$ matrix, $B$ be a $(n-1)\times (n-1)$ matrix and $D$ be a $(n-1)\times (n-1)$ diagonal matrix with all entries positive. We assume that $$\det(A-\lambda I)=\lambda ...
2
votes
1answer
20 views

properties of the solution to a non-homogeneous matrix equation with a non-singular M-matrix

I have a matrix equation $Ax=b$, where $A$ is a $4\times4$ non-singular M-matrix ($A$ has negative off-diagonal and positive diagonal entries) and $b$ is a strictly positive vector. Let $x=(x_1, x_2, ...
3
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1answer
34 views

Understanding a simple proof about minimal polynomials

Let $T \colon V\to V $ be a linear operator, where $V$ is a vector space over $F$. Suppose that the minimal polynomial $M(t)$ of $T$ can be factored into the product of two coprime and monic ...
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1answer
25 views

An equality for the dimension of the sum of subspaces (in the non-degenerate case)

This post is a sequel of An inequality for the dimension of the sum of subspaces, inspired by this famous answer on $\dim(U+V+W)$. The inequality $$\dim(\sum_{i = 1}^{n} U_i) \le \sum_{r=1}^{n} ...
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1answer
42 views

Finding the Jordan form and basis failing

Let $$A = \left(\begin{array}{cccc} 3&4&-1\\0&-2&0\\1&-4&1 \end{array}\right)$$ Find the Jordan form $J$ and $P$ such that $P^{-1}AP = J$. So here's what I did: $f_A(x) = ...
1
vote
1answer
46 views

Are $A$ and $A^\top$ similar? [duplicate]

Let $K$ be a field and $A$ a square matrix with entries in $K$. Then A and $A^\top$ have the same characteristic polynomial. What do we know about similarity? Do you have an example where $A$ and ...
0
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2answers
32 views

Simple matrix derivative identity

Is the following correct, and is there some kind of similar identity when $x$ and $y$ are matrices? For $A \in \mathbb{R}^{n \times n}$, $\nabla_A x^T A y = x y^T$. And my proof: ...
4
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3answers
59 views

$A$ is a symmetric postivie definite matrix. Prove that $A^k$ is also a positive deinite

Let $A\in M_n(\mathbb{R})$, a symmetric positive-definite matrix. Prove that for every $k\in\mathbb{N}$, $A^k$ is also positive definite. So since $A\in M_n(\mathbb{R})$ is symmetric and positive ...
2
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1answer
34 views

$\left\| A \right\| \le \varepsilon \Rightarrow \left\| {\mathop A\limits^{\_\_} } \right\| \le \varepsilon$

Suppose $A \in {C^{n \times n}}$ $\left\| A \right\| \le \varepsilon$ such that $\left\| . \right\|$ is matrix norm subordinate to the euclidean vector norm. Is this true that $\left\| {\mathop ...
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0answers
39 views

Is There a Basis Free Definition of the Pfaffian

$\DeclareMathOperator{\pf}{pf}$ I recently came across a delightful fact that: The determinant of a $2n\times 2n$ skew-symmetric matrix is a the square of a certain polynomial called the pfaffian. I ...
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1answer
24 views

Adjoint and Adjugate are same or different?

The notions of adjoint and adjugate, which I saw, are as follows: (1) Let $T:V\rightarrow W$ be a linear map. Then there is a corresponding linear map between the duals of these spaces: ...
5
votes
1answer
86 views

Largest eigenvalue of a Hermitian matrix

I have two Toeplitz positive semi-definite Hermitian matrices $\mathbf{R}_1, \mathbf{R}_2 \in \mathbb{C}^{M \times M}$. They are in fact covariance matrices satisfing the following conditions: (1) ...
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1answer
39 views

Proof for $\mathbf{M}$ unitary if $||\mathbf{M}\mathbf{v}|| = ||\mathbf{v}||$

Let $\mathbf{M} \in \mathbb{F}^{n, n}$. Then $||\mathbf{M}\mathbf{v}|| = ||\mathbf{v}||$($\mathbf{v} \in \mathbb{F}^n$) implies that $\mathbf{M}$ is unitary. My question is, how to prove this ...
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5answers
43 views

Is it possible to solve for $m$ in a linear equation without knowing $b$?

Suppose you know certain points on a line say $(5,2)$ up to $(8,10)$ but you don't know exactly where the $y$ intercept would be being somewhere down there at like $-25$ area. How would you solve for ...
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1answer
55 views

Which matrices diagonalizes a diagonal matrix? [closed]

I think the answer is the set of all diagonal matrices but I am not sure. Can anyone give the answer with a proof?
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4answers
146 views

Do row operations change the column space of a matrix?

I know that (i) row operations do not change the row space (ii) column operations do not change the column space and (iii) row rank = column rank (but this is sort of unrelated, I think). But, ...
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2answers
70 views

Prove that similar matrices have the same nullity.

How do I approach this? I'm assuming it might have something to do with $B = P^{-1}AP$.
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0answers
50 views

Practice Exam question need help! [closed]

For vectors $f,g \in C[-\pi,\pi]$, we use the inner product $\langle f,g \rangle = \displaystyle \int_{-\pi}^{\pi} f(x)g(x)\,dx$. Then, $S=\{1/(2\pi)^{1/2},\sin(x)/\pi^{1/2}\}$ is an orthonormal set ...
2
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2answers
66 views

Finding Eigenvalue det(λI - A);

I want to know if what I'm doing to derive equation (2) from (M2) is correct or not; usually, before moving onto the next row in Guass-Jordan elimination we turn a_11 into a leading one or whatever ...
0
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1answer
34 views

Problem about dual of $W = V \oplus V'$

Let $V$ by finite dimensional, let $W = V \oplus V'$, and prove that the correspondence $(x,y) \rightarrow (y,x)$ is an isomorphism between $W$ and $W'$. (The direct sum is defined as the set of ...
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2answers
40 views

Help With Finding A Basis

I came up to the following matrix: $$\begin{pmatrix} 3 & 1& 3& -4\\ 0 & 0 & 0& 0\\ 0 & 0 & 0& 0\\ 0 & 0 & 0& 0\\ \end{pmatrix}$$ I know that ...
2
votes
2answers
50 views

Do linear operators that map one space into a different space have a Jordan canonical form?

I know that this answer is most likely "yes", and that, in the setting of matrices, all matrices are similar to its Jordan form, which is unique (up to the ordering of the Jordan blocks.) But what ...
0
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3answers
21 views

Finding Rank And Eignvalues Of Vectors Multiplication

Let $v=(3,1,3,-4)$ and $A=v^tv$, Find: the rank of $A$ $Null(A)$ eigenvectors and eigenvalues Is there a way to approach this without finding $A$ explicitly?
0
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1answer
13 views

Question about row operations and row-echelon form,

If I have a matrix, with, say, the first two columns consisting of all zeroes, then is the first entry of the third column, which is non-zero, my first pivot variable, so that when solving Ax=b, for ...