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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5
votes
2answers
603 views

Largest eigenvalue of a symmetric positive definite matrix with rank-one updates

I have a $n \times n$ symmetric positive definite matrix $A$ which I will repeatedly update using two consecutive rank-one updates of the form $A' = A + e_j u^T +u e_j^T$ where $\{e_i: 1 \leq i \leq ...
20
votes
3answers
622 views

Old AMM problem

I am working on an old AMM problem: Suppose $A,B$ are $n\times n$ real symmetric matrices with $\operatorname{tr} ((A+B)^k)= \operatorname{tr}(A^k) + \operatorname{tr}(B^k) $ for every positive ...
5
votes
2answers
46 views

Largest eigenvalue of a Hermitian matrix

I have two 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) ...
1
vote
1answer
25 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, ...
0
votes
0answers
25 views

Frequency response of unstable systems

Frequency response theorem's corollary ensures that the theorem can be applied to unstable systems as well, as long as you find a proper initial condition x(0). Is that right? Now, if I've an ...
1
vote
1answer
437 views

Diagonally Dominant Matrix Preserved after Gaussian Elimination (with a modification)

Prove or disprove: If a matrix has the property $0 \neq |a_{ii}| \geq \sum_{\substack{j=1 \\ j \neq i}} |a_{ij}| $ then Gaussian Elimination (without pivoting) will preserve this property. I assume ...
0
votes
0answers
18 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\\ ...
2
votes
2answers
61 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
votes
1answer
49 views

I need help with a simple proof for the associative law of scalar multiplication of a vectors.

I need help with a simple proof for the associative law of scalar multiplication of a vectors. If $$(rs)X =r (sX)$$ Define the elements belonging to $\mathbb{R}^2$ as ...
35
votes
0answers
2k views
+50

Limit of sequence of growing matrices

Let $$ H=\left(\begin{array}{cccc} 0 & 1/2 & 0 & 1/2 \\ 1/2 & 0 & 1/2 & 0 \\ 1/2 & 0 & 0 & 1/2\\ 0 & 1/2 & 1/2 & 0 \end{array}\right), $$ ...
-3
votes
0answers
28 views

Vector spaces and nontrivial subspace. [on hold]

Give an example of a subset of $\mathbb{R}^2$ that is a nontrivial subspace of $\mathbb{R}^2$? $\mathbb{R}^2$ as $\{(a, b) \mid a, b \in \mathbb{R}\}$
0
votes
2answers
17 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, ...
3
votes
0answers
35 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 ...
1
vote
0answers
34 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 $K$, ...
3
votes
1answer
129 views

Asymptotic behavior of the minimum eigenvalue of a certain Gram matrix with linear independence

Consider the density matrices with the following spectral decompositions: $$\rho=\lambda_1|\nu_1\rangle+\lambda_{2}|\nu_2\rangle$$ and $$\sigma=\gamma_1|\omega_1\rangle+\gamma_2|\omega_2\rangle$$ such ...
10
votes
1answer
132 views

Generalizing Cauchy-Schwarz for more than two vectors

For a complex inner product space, $X$, Cauchy-Schwarz inequality states $$ | \langle x,y \rangle |^2 \leq \langle x,x\rangle \cdot \langle y, y\rangle , $$ for any $x,y \in X$. Equality holds if and ...
1
vote
2answers
38 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 ...
0
votes
0answers
29 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 ...
0
votes
0answers
6 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) = ...
1
vote
1answer
21 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 ...
2
votes
2answers
49 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 ...
2
votes
3answers
41 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 $$ ...
2
votes
2answers
112 views

How to generate $3\times3$ matrices with integer eigenvalues?

I am looking for an easy way to generate non-trivial (i.e not just diagonal) examples of 3x3 matrices whose entries are integers and whose eigenvalues are also integers. I know how to do this for 2x2 ...
0
votes
1answer
56 views

Calculation of $trace(L^THL)$, L is lower triangular, H is symmetric.

I am working on a problem where I had to find the following expression: $$ l = Tr({P'HP})$$ I already modified my model formulation using cholesky decomposition for PSD matrices and came up with ...
-1
votes
0answers
17 views

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

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*}
0
votes
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?
2
votes
1answer
37 views

How do I extrac the anisotropic part of a tensor?

Given the elements $\chi_{ij}$ of a tensor in cartesian coordinates, with \begin{pmatrix} \chi_\bot& 0 &0 \\ 0 & \chi_\| &0\\ 0&0 & \chi_\| \end{pmatrix}, where the ...
0
votes
1answer
25 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.
0
votes
2answers
21 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 ...
1
vote
2answers
23 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 ...
0
votes
1answer
66 views

If $A$ is an invertible $n\times n$ complex matrix and some power of $A$ is diagonal, then $A$ can be diagonalized

Prove or provide a counter-example: If $A$ is an invertible $n\times n$ complex matrix and some power of $A$ is diagonal, then $A$ can be diagonalized. If $A^n$ is a diagonal matrix, then clearly ...
2
votes
2answers
35 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 ...
1
vote
1answer
14 views

What is the “Cumulative Distribution of the magnitude of the N-dimensional standard gaussian”

I am confused by this line from a paper: "Let $F_1(x)$ be the cumulative distribution of the magnitude of an $n$−dimensional standard Gaussian random variable and $F_2(x)$ be the cumulative ...
7
votes
3answers
414 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 ...
1
vote
0answers
29 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) = ...
0
votes
0answers
15 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 ...
4
votes
2answers
38 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 ...
1
vote
1answer
21 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 ...
1
vote
1answer
12 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
votes
1answer
26 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, ...
2
votes
1answer
17 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, ...
0
votes
1answer
27 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}; ...
3
votes
1answer
28 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 ...
0
votes
0answers
18 views

SVD of partitioned matrix where all cells except one are zero

Let $A$ be a real valued matrix of size $n \times n$. Let the SVD of $A$ be $$A= UDV^T.$$ I am interested in $$Q=VU^T.$$ Now assume we expand $A$ with zero rows and columns to get the block matrix ...
0
votes
2answers
31 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
votes
3answers
51 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
votes
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 ...
1
vote
1answer
38 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
votes
2answers
578 views

Change of basis matrix for orthogonal bases

I am trying to show that if $B_1$ and $B_2$ are orthonormal bases for $\mathbb{R}^n$, then the change of basis matrix $P$ from $B_1$ to $B_2$ is an orthogonal matrix. I'm a bit stuck. I started with ...
0
votes
1answer
89 views

How many coefficients are needed to reconstruct exactly vectors

How many coefficients are needed to reconstruct exactly vectors (a) $S(\{2,4,6\}) \subset l^2(\mathbb{Z}^9)$ and (b) in $S(\{1,3,4\}) \subset l^2(\mathbb{Z}^{12})$? Find the missing coefficients $x$ ...