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 ...

learn more… | top users | synonyms

4
votes
1answer
32 views

Determinant inequality and positive definite matrix

Let $B$ and $C$ be $n\times n$ hermitian matrices, with $B$ positive definite and $C$ positive semi-definite. (1) Show that $B+C$ is positive definite (2) Show that $\det(B)\leq \det(B+C)$. What ...
-3
votes
0answers
13 views

E as an expectation of a quadratic form [on hold]

if E(expectation of quadratic form) is an operator, show that E(AB+C) = AEB + EC. where b and c are variables.
-2
votes
0answers
12 views

Let T be the bounded operator and T* be the adjoint operator of T.Show the following. [on hold]

Let T be the bounded operator and T* be the adjoint operator of T. Show the following. 1.||TT||=||T|| 2.||TT||=||TT*||=||T||^2 3.(T+S)=T+S* 4.(αT)=α ̄T (α∈C) 5.(TS)=S T* 6.(T* )* =T
2
votes
1answer
25 views

Simple question - represent vector with respect to a basis

Basic question here, I've always been weak at this stuff. Suppose that we have a situation $U=WX$ where $U,W,X$ are matrices that are known to us. You can suppose that $U$ is invertible. I want to ...
-2
votes
0answers
17 views

System of Equations for 3-digit number [on hold]

This is a rare word problem where I've had trouble: Find system of equations and use elimination. The sum of the digits of a three-digit number is 9, and the tens digit of the number is twice the ...
1
vote
2answers
947 views

Gauss-Seidel method convergence algorithm

From Wikipedia: The convergence properties of the Gauss–Seidel method are dependent on the matrix A. Namely, the procedure is known to converge if either: ...
1
vote
1answer
38 views
+50

Asymptotic series of a matrix-valued function.

Consider the following matrix $$f(\lambda)=\left( \frac{\lambda-1}{\lambda + 1} \right)^{\nu \sigma_3} \ \ \ \lambda \in \mathbb{C} \setminus [-1,1]$$ where $\sigma_3=\begin{pmatrix} 1 & 0 \\ 0 ...
2
votes
1answer
59 views

A question about matrix algebras

Let $A,B \in M_n$, $n \geq 2$. If $A$ and $B$ do not share a common eigenvector, why is $\mathcal{A}(A,B) = M_n$? Notation and definitions: $M_n$: the set of $n \times n$ matrices over ...
1
vote
1answer
85 views

When and why can functions “take on” the role of vectors in defining vector spaces?

In what I call "advanced" linear algebra, we examine the properties of vectors in a vector space like an inner product space by checking that they satisfy e.g. the Cauchy-Schwarz inequality, the ...
0
votes
3answers
45 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 ...
3
votes
1answer
329 views

Vector Project onto Subspace

So the question is: Let S be the subspace of $\mathbb{R}^3$ spanned by the vectors $ u_2 = \begin{pmatrix} \frac{2}{3}\\\frac{2}{3}\\\frac{1}{3}\end{pmatrix} u_3 = \begin{pmatrix} ...
-3
votes
1answer
52 views

Verify the following assertion: [on hold]

Suppose that $U=\{(x,x,y,y)\in F^4:x,y\in F\}$ and $W=\{(x,x,x,y)\in F^4:x,y\in F\}$.Then $$U+W=\{(x,x,y,z)\in F^4:x,y,z\in F\}.$$ Not sure how to add these subsets. Please provide explanation.
-3
votes
0answers
20 views

How can I tell if two lines will cross using vectors [on hold]

things I need 1. a visual recpinatation 2. A explanation on how to solve it and 3. a problem to do
-1
votes
0answers
30 views

Don't understand about how to solve the first equation with Gauss Elimination

Please take a look at the picture. My quesion is: How to solve the first equation with Gauss Elimination that is displayed in the picture. I don't understand HOW.
2
votes
0answers
27 views

Eigenvalues of the product of two “incidence” matrices

I am trying to solve the following problem. Let the following incidence matrix of an undirected graph with four nodes $$ B = \begin{bmatrix}1 & 0 & -1 & 0 & 0 \\ -1 & 1 & 0 ...
0
votes
0answers
27 views

Linear independent vectors

I am stuck in an argument. Is the following true: Let $\{v_1,\ldots,v_k\}$ and $\{u_1,\ldots,u_k\}$ be sets of linear independent vectors. Set $u:=\sum_{i=1}^k \alpha_i u_i$, for some $\alpha_i$. Now ...
1
vote
1answer
23 views

On the projection onto the image set of an $m\times n$ matrix

I came accross as statement that: "If $K$ is the image set of an $m\times n$ matrix $A$ with full column rank, then $$P_Kx=A(A^TA)^{-1}A^Tx."$$ How do I verify this? I know that the inequality ...
8
votes
5answers
721 views

Show that if $T_1$, $T_2$ are normal operators that commute then $T_1+T_2$ and $T_1T_2$ are normal.

Let $V$ be a finite dimensional inner-product space, and suppose that $T_1$, $T_2$ are normal operators on $V$ that commute. How to show that $T_1+T_2$ and $T_1T_2$ are then normal? It is clear if ...
2
votes
2answers
353 views

If $V$ and $W$ are subspaces of the same dimension such that $V$ meets $W^\perp$, then $W$ meets $V^\perp$

I'm having a little difficulty understanding the proof for orthogonal complements. I kind of understand orthogonal complements, but I cannot seem to find a logic to this. I'm trying to follow along ...
1
vote
2answers
31 views

Solving system of nonlinear equations

Say I have a system of 4 equations, 4 unknown (A,B,C,D), how would you solve it analytically, assuming a, b, C1, C2, C3, C4, C5, C6, F, G, H, I are just some constants? If using Gaussian Elimination, ...
3
votes
1answer
28 views

A linear functional on the space of transformations is basis independent

I've been working on this problem for a bit and am not sure how to proceed: let $V$ be an $n$ dimensional $\mathbb{ R } $-vector space, and denote by $\mathcal{L}(V)$ the space of linear operators $V ...
3
votes
8answers
87 views

For two vectors $a$ and $b$, why does $\cos(θ)$ equal the dot product of $a$ and $b$ divided by the product of the vectors' magnitudes?

While watching a video about dot products (https://www.youtube.com/watch?v=WDdR5s0C4cY), the following formula is presented for finding the angle between two vectors: For vectors $a$, and $b$, ...
2
votes
5answers
14k views

Correlation matrix from Covariance matrix

This is for a project which I've been trying to find some information for Covariance matrix and correlation matrix. I understand that for a $n \times n$ matrix $A, AA^T$ will give me the covariance ...
3
votes
4answers
63 views

Does an $n\times n$ matrix $A$ only have an inverse if $rank(A)=n$? If so, why?

I'm currently learning about the rank and inverses of matrices, and every time I attempt to find the inverse of a matrix with a rank smaller than it's number of rows, I find I am unable. One example ...
0
votes
0answers
18 views

How to compute the Eigenvectors for a Markov matrix?

I have the following matrix for which I want to get the Eigenvectors. I know how to compute the Eigenvalues, but when I compute the vectors in the null space of the matrix, I get the wrong answer. ...
1
vote
1answer
41 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
4answers
48 views

Points on 3d line

Say we have $2$ points on a 3d line, point $A(x,y,z)$ and point $B(x,y,z)$. If we want to get the coordinates of a third point, beyond point $B$ but a certain distance from point $A$, how would we do ...
0
votes
3answers
1k views

Eigenvalues of normal matrix

I want to show that $\lambda$ is an eigenvalue of a normal matrix $A$ if and only if $\overline{\lambda}$ is an eigenvalue of $A^{*}$. I am trying to show it for a while and I guess there are some ...
0
votes
1answer
64 views

Matrix Algebras: Generator

Problem Given the algebra $\mathcal{M}_\mathbb{C}(2)$. Denote the normals: $$\mathcal{N}:=\{N\in\mathcal{M}_\mathbb{C}(2):N^*N=NN^*\}$$ And their calculus: ...
6
votes
2answers
38 views

minimum eigenvalue for difference of two matrices

Let $A$ a symmetric positive definite matrix, and $B$ a matrix constructed from $A$ by setting all its off-diagonal elements to zero. Then is there a way to see for which values of positive scalars ...
1
vote
2answers
65 views

Matrix invertibility in terms of elementary operations

There is a theorem that an $n \times n$ matrix $A$ is invertible if and only if $A$ is row equivalent to $I_n$, and in this case, any sequence of elementary row operations that reduces $A$ to $I_n$ ...
6
votes
4answers
10k views

The transpose of a permutation matrix is its inverse.

This is a question from the free Harvard online abstract algebra lectures. I'm posting my solutions here to get some feedback on them. For a fuller explanation, see this post. This problem is from ...
0
votes
1answer
22 views

Iterative solutions of linear systems

I do not understand that why $M$ must be invertible for $x^{(k+1)}$ to be uniquely specified in equation below: $$ Mx^{(k+1)} = Nx^{(k)} + b \quad (k=0,1,\ldots).$$ Why $M$ must be invertible? And ...
2
votes
0answers
58 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 $K$, ...
1
vote
1answer
30 views

Prove there exists a self-adjoint transformation $C$ s.t. $CA=B$ if $A$ and $AB$ are self adjoint

If $A$ and $B$ are linear transformations such that $A$ and $AB$ are self adjoint and such that $Ker(A)\subset Ker(B)$, then prove there exists a self-adjoint transformation $C$ s.t. $CA=B$
3
votes
1answer
33 views

Prove the theorem of ideal (about g.c.d)

If $p_1,\ldots,p_n$ are polynomials over a field $F$, not all of which are $0$, there is a unique monic polynomial $d$ in $F[x]$ such that (a) $d$ is in the ideal generated by $p_1, \ldots, ...
1
vote
2answers
24 views

a symmetric bilinear form has a basis such that it's matrix with respect to it is diagonal

I'm reviewing a proof regarding $f$, a symmetric bilinear form having a basis such that it's matrix with respect to this basis is diagonal. Here's a summarization: For $n=1$ there's nothing to ...
5
votes
1answer
81 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) ...
1
vote
1answer
14 views

Columns Of The Diagonalization Matrix

After finding the eigenvectors, we can create a matrix $Q$ such that $Q^{-1}\cdot A \cdot Q=D$ when $A$ is a matrix and $D$ is a diagonal matrix with the eigenvalues on the diagonal. In which order ...
1
vote
1answer
32 views

does the following matrix update law, converge? if so, to what?

Assume $A$ and $B$ to be arbitrary matrices (but you can assume some conditions on their norm), We have $X_{i+1}=AX_{i}A^T+B$, We are looking for $X_{\infty}$ (if it exists). Does it converge, if ...
0
votes
0answers
22 views

Eigenvalues of integrals over similar matrices

Let $\rho = \rho(x)$ be a $2\times2$ matrix (don't know if it is necessary, but $\rho$ is a density operator) and $I$ be the (two-dimensional) identity matrix. I have two matrices $A$ and $B$, where ...
0
votes
1answer
51 views

Symmetric matrix problem

$A$ is a symmetric matrix and has a eigenvalue $\lambda$ of order $m$. Why $\lambda$ has $m$ independent eigenvectors? I want to prove $A$ is diagonalizable by proving it has $n$ independent ...
0
votes
1answer
51 views

Linear Algebra: Question about determinants

The following matrices are $4 \times 4$ matrices. $$A=\begin{bmatrix} 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1\\ 1 & 1& 1 &0\\ 1 &1 &0 &0 \end{bmatrix}\\ B= ...
0
votes
1answer
18 views

Characterising Adjugate(adjoint) of a matrix

If $A$ is an $n\times n$ matrix over a field, then adj$(A)$ is an $n\times n$ matrix (obtained from $A$) such that $$\mathrm{adj}(A)\,A=A\,\mathrm{adj}(A)=\mathrm{det}(A)I_n.$$ Question: If $B$ is ...
1
vote
1answer
64 views

Preferred way to write elements of the direct sum of vector spaces

Suppose $V$ and $W$ are vector spaces over the same field and $V\oplus W$ is their direct sum. Reading through the literature I found essentially two ways of writing elements of $V\oplus W$. 1.) We ...
2
votes
4answers
145 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, ...
0
votes
1answer
16 views

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 ...
5
votes
2answers
606 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
641 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 ...
1
vote
1answer
33 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, ...