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

1
vote
1answer
145 views

Lower bounds on eigenvalues of a symmetric matrix based on the diagonals

A symmetric matrix $A$ always has real eigenvalues. If I know the elements on the diagonals, is it possible to have a lower bound on the smallest eigenvalue? How sharp would this bound be? For now I ...
0
votes
1answer
41 views

Matrix Solution

I have matrix integral equation of the following form ${f^{'}(x)}_{1 \times 1}A_{3\times 3}=P_{3\times3} (1-x)+Q_{3 \times 3}x \tag 1$ . All dimensions are indicated in equation itself. " ' " ...
1
vote
2answers
46 views

What is the Linear Transformation for this Matrix?

$$ A=\pmatrix{3&5&-4\\-3&-2&4\\6&1&-8}\pmatrix{x1\\x2\\x3}=\pmatrix{b1\\b2\\b3} $$ Given the above matrix A, there exists a linear transformation known as "T", what is this ...
0
votes
1answer
342 views

Prove that the multiplicative inverse of $a$ modulo $m$ exists if and only if $a$ and$m$ are coprime.

Prove that the multiplicative inverse of $a$ modulo $m$ exists if and only if $a$ and $m$ are coprime. Can someone help me with this?
0
votes
1answer
56 views

how to compute the distance between a matrix and its lower rank approximation?

I have a matrix $X$ and $Z$ a lower rank approximation of $X$ obtained using only few of the columns of $X$. I would like to have a measure of how distant are $X$ and $Z$. In particular I would like ...
5
votes
4answers
569 views

What is the non-trivial, general solution of these equal ratios? [closed]

Provide non-trivial solution of the following: $$\frac{a}{b+c}=\frac{b}{c+a}=\frac{c}{a+b}$$ $a=?, b=?, c=?$ The solution should be general.
3
votes
1answer
88 views

how to find $(I + uv^T)^{-1}$

Let $u, v \in \mathbb{R}^N, v^Tu \neq -1$. Then I know that $I +uv^T \in \mathbb{R}^{N \times N}$ is invertible and I can verify that $$(I + uv^T)^{-1} = I - \frac{uv^T}{1+v^Tu}.$$ But I am not able ...
0
votes
1answer
40 views

How do get eigenvalues of a matrix B if add a row/column pair of a matrix A?

I have a matrix of size N×N of the form: where and A is N-1 x N-1 matrix, a=0. I known the eigenvalues of A. Any possible for getting eigenvalues of B from eigenvalues of A?
0
votes
1answer
72 views

Eigenvalues and eigenvectors of a non-symmetric matrix which is a product of 2 symmetric matrices?

I have a non symmetric matrix $AB$ where $A$ and $B$ are symmetric matrices. How can I find the eigenvectors and eigenvalues of $AB$? In a paper( Fisher Linear Discriminant Analysis by M Welling), ...
42
votes
15answers
8k views

Why learn to solve differential equations when computers can do it?

I'm getting started learning engineering math. I'm really interested in physics especially quantum mechanics, and I'm coming from a strong CS background. One question is haunting me. Why do I need ...
-1
votes
1answer
29 views

Normal Matrices Unitarily Diagonazible

Disclaimer: This thread is just meant to record (Q&A). Are the unitarily diagonazible matrices precisely the normal ones? Surely, every normal matrix has an eigenbasis. Now I got asked by a ...
1
vote
2answers
51 views

Median of a triangle.

If ${A,B,C}$ be the position vectors of the vertices A,B,C of the triangle ${ABC}$,show that the three medians concur at the point ${\frac{1}{3}( A + B + C)}$, called the centroid. Note : I don't ...
1
vote
1answer
114 views

Jordan-Chevalley decomposition of $T$ acting on $k[T]/(\pi(T)^e)$

Given an algebraically closed field $K$, a f.d. vector space $V$ over $K$ and $A\in{\rm GL}(V)$, we can view the space $V$ as a $K[T]$-module, where $T$ acts by $A$. Using the fundamental theorem of ...
0
votes
2answers
116 views

Space of matrices that commute with a given matrix

Let $A$ be an $n\times n$ complex matrix, and $C(A)$ be the vector space of all matrices that commute with $A$. I have to determinate if the dimension of $C(A)$ is greater or equal than $n$, or not. ...
0
votes
1answer
36 views

Non-Unitarily Diagonalizable Matrices

When searching for matrices that are similar to a diagonal matrix but not in a unitary way then a first hint would be to exclude the normal ones. But apart from that is there a general form for such ...
0
votes
1answer
101 views

Null space of a matrix mcq

Let $M$ be the set of all $m\times n$ matrices with real entries. Which of the following statement is correct? There exists $A$ of order $2\times 5$ belonging to $M$ such that the dimension of the ...
0
votes
2answers
51 views

Give an example of an operator on a finite dimensional vector space.

Give an example of an operator on a finite-dimensional real vector space such that 0 is the only eigencvalue of T but T is not nilpotent. I've been stuck on this problem for a while, the main example ...
2
votes
2answers
29 views

Equality in the sequence of increasing ranges.

Suppose $T \in L(V)$. Let $n = \dim V$. Prove that $\text{rangeT}^n = \text{rangeT}^{n+1} = \text{rangeT}^{n+2} = \dots$ I need help finishing this proof. This is what I have so far: First prove ...
1
vote
1answer
29 views

If $v\in\mathbb{R}^N\setminus\mathbb{R}^{N-1}$, what is the projection $\pi_v$ with kernel $\mathbb{R}v$?

I going over a lemma for the Whitney Embedding theorem which shows that an injective immersion of an $n$-manifold into $\mathbb{R}^N$ can actually be immersed in a lesser dimensional Euclidean space ...
1
vote
3answers
114 views

non-symmetric matrix with orthogonal eigenvectors

Given that a symmetric matrix with real entries has orthogonal eigenvectors, is the converse true? That is, if a matrix has orthogonal eigenvectors, does it have to be symmetrical and real?
0
votes
1answer
38 views

Power of matrix expansion

We know about the expansion $(a+b)^n\tag 1$,for scalar variables. What will be the equivalent when we want to find $(A+B)^n \tag 2 $, when A and B are square matrices? Can we treat it as same as ...
2
votes
1answer
45 views

Prove that spec$(f(A)) = f$(spec$(A)).$

Can someone please explain this proof to me? Thanks! Let $A \in \mathbb{C}^n$ and let $f(x)$ be a polynomial. Prove that spec$(f(A)) = f$(spec$(A))$ (where if $S \subseteq \mathbb{C},$ $f(S) :=$ $\{ ...
3
votes
0answers
86 views

Matrix partwise multiplication

I am working on an artificial intelligence application that (among other things) combines "opinions" of several "experts" who each have access to different aspects of a "situation". I can build this ...
3
votes
1answer
81 views

When does the set of integer linear combinations of 3 vectors in the plane form a dense subset?

Is there an easy to check if and only if condition for when the set of integer combinations of 3 vectors in the plane will form a dense subset of the plane? It seems like having no vector being a ...
0
votes
1answer
30 views

Unique subspace that contains all vectors that are orthogonal to 2 independent vectors?

It's a problem from MIT 18.06 14 Spring Exam1. The vectors are all 4 dimensional. I suppose that there are infinitely many such space orthogonal to that fixed subspace spanned by those 2 independent ...
0
votes
2answers
58 views

How does a matrix change the magnitude of a vector?

I have the following problem: $z=Ax$, in which $z$ and $x$ are $N\times 1$ vectors and $A$ is a $N\times N$ matrix. I am interested in how the magnitude of $x$ changes after applies $A$ on it. Is ...
2
votes
1answer
95 views

Can integral transforms be viewed as change of basis formulas?

Forgive any lack of rigor. If you have a countable orthonormal basis $B$ for a Hilbert space $H$ , then any function $f \in H$ can be expressed as $$ f(t) = \sum\limits_{g \, \in \, B} \langle f, ...
2
votes
0answers
47 views

Show that $(Au,Bv)=(u,A^tBv)$

Let $ A, B $ be matrices of order $ n $, and $ \vec{u}, \vec{v} $ vectors from euclidean space $ \mathbb{R}^n $, then $ (Au,Bv) = (u,A^tBv) $ pd. $(\cdot ,\cdot)$ is my notation for inner product, ...
2
votes
1answer
70 views

Method to simplify this long expression

How can I simplify this long expression: $-a^3(d-b)(d-c)(c-b)+b^3(d-a)(d-c)(c-a)-c^3(d-a)(d-b)(b-a)+d^3(c-a)(b-a)(c-b)$ I know that it is equal to $(d-a)(d-b)(d-c)(c-a)(c-b)(b-a)$ but i have no idea ...
3
votes
1answer
97 views

Question about existence of unitary matrices with certain properties

We are given a set of $d$ normalized vectors on a $d$-dimensional complex vector space: $e_1$, $e_2$... $e_d$, where $$\langle e_j,e_j\rangle=1$$ for all $j$. These are not necessarily mutually ...
0
votes
2answers
75 views

Matrix Exponent - equivalent of a rotation matrix

Every Rotation Matrixcan be represented as a power of e with exponent a skew symmetric matrix. In particular, if we have a rotation matrix ${R}\in\mathbb R^{3 \times 3,}$ then there will be a skew ...
0
votes
0answers
16 views

$w \in \operatorname{Span}(T) \leftrightarrow [w]_b\in \operatorname{Span}[T]_b$?

$$w \in \operatorname{Span}(T) $$ lets apply coordinate function on both sides $$[w]_b=\alpha_1t_1+\cdots+\alpha_nt_n=[t_1,\ldots,t_n]_b$$ $$\operatorname{Span}(T)=\sum \beta_it_i=\left[\sum ...
0
votes
0answers
29 views

Find the reordering of a matrix rows based on vector.

Let $A\in\mathbb{R}^{m\times n}$ and $\mathbf{u}=(u_1,\ldots,u_n)^T\in\mathbb{R}^n$. Is there any map $f\colon\mathbb{R}^{m\times n}\times\mathbb{R}^n\to\mathbb{R}^{m\times n}$, $$ f(A,\mathbf{u})=B, ...
1
vote
1answer
33 views

Projective Spaces which are not Vector Spaces

I'm studying Projective Spaces, I've collected a few books and most of them define Projective Spaces in terms of Vector Spaces, that is, they define a 'projective space structure" in the vector space ...
0
votes
1answer
37 views

The Complex Spectral Theorem Explanation.

Consider the normal operator $T \in L(C^2)$ whose matrix with respect to the standard basis is $$ \left( \begin{array}{ccc} 2 & -3 \\ 3 & 2 \end{array} \right)$$ As you can verify, ...
1
vote
1answer
67 views

Comparable elements and classes. Linear Algebra - Shilov.

Given a subspace $\mathbf L$ of a linear space $\mathbf K$, an element $x\in\mathbf K$ is said to be comparable with an element $y\in\mathbf K$ (more exactly, comparable relative to $\mathbf L$) if ...
3
votes
3answers
53 views

How to get the two eigen vectors for eigen =1

I have to find the eigen vectors for this matrix. \begin{pmatrix} 1 & 0 & 1\\ 0 & 1 & 0\\ 0 & 0 & 0 \end{pmatrix} I end up with this matrix to plug in the eigen values. ...
0
votes
1answer
49 views

How to prove the uniqueness of linear functional

$\textbf{Theorem}$ If $V$ is a $n$-dimensional vector space, if $\{x_1,.,.,., x_n\}$ is a basis in $V$ and if $\{\alpha_1,\cdots \alpha_n\}$ is any set of $n$ scalars, then there is one and only one ...
1
vote
1answer
79 views

Convergence of square root operators

Let $Q_n$ and $Q$ be compact positive and symmetric operators. Let $A_n = {Q_n}^{\frac12}$ and $A=Q^{\frac12}$. Given $Q_n$ converges to $Q$ w.r.t. operator norm. Does $A_n$ converges to $A$? Thanks. ...
3
votes
1answer
81 views

Is det(A) maximal, if det(A+E) is maximal?

Let A be a binary matrix of size n x n and E be the matrix of the same size with all entries $1$. Proof or disproof : If det(A+E) has the maximal possible value, then det(A) also has the maximal ...
1
vote
1answer
120 views

On the decomposition of stochastic matrices as convex combinations of zero-one matrices

Let "stochastic" matrix be the matrix whose rows sum to one and deterministic matrix be a stochastic matrix whose all rows consist of a one and zero. For example $\left [ \begin{array}{ccc} 1 & ...
3
votes
2answers
46 views

Find orthogonal projection on basis of V

I'm not sure what to do in part three of this question. To be clear, I know how to get the solution of part 2. I imagine part three is trivial?
0
votes
1answer
73 views

Relation between eigenvectors after transforming a nonsymmetric matrix to symmetric?

I need to find eigenvectors and eigenvalues of a matrix which is product of 2 symmetric positive definite matrix(SwInverseSbProd=SwInverse*Sb). Since SwInverseSbProd is non-symmetric and calculation ...
0
votes
1answer
36 views

Where is this proof wrong (orthogonal projector)?

Assume you have a projector $P:V \rightarrow V$, $P^2=P$ and you incorrectly prove the equation which is valid only in the stronger case of an orthogonal projector $x = Px + (1-P)x$, $\quad \forall ...
0
votes
1answer
73 views

Matrix-valued differential equation $A'(t)=A(t)B(t)$

How to solve matrix-valued differential equations of type $$A'(t)=A(t)B(t) \tag 1$$ All the given functions are square matrices of dimension $3$ and only $A(t)$ is invertible (not $B(t)$ or ...
1
vote
1answer
188 views

How can I prove that the span of an a subspace and it's orthogonal complement is the whole vector space?

The book Linear and Geometric Algebra explains the following theorem in a way that I haven't been able to figure out: If $\mathbf{A}$ and $\mathbf{B}$ are subspaces of a vector space $\mathbf{B}$ ...
1
vote
0answers
72 views

Matrix exponent form

We have an equation of matrix exponent $ Ae^{Ax}R-e^{Ax}R (P_1 +P_2 x) = Y \tag1$ Given condition $A,R,P_1,P_2,Y$ are constant $3 \times 3 $ matrices. R is invertible,orthonormal,determinent ...
0
votes
1answer
108 views

matrix differentiation - derivative of matrix vector dot product with respect to matrix

Given the function $$f(N) = x_1^T M x_2 $$ where $x_1 = Nv_1 $ $x_2 = Nv_2 $ $x_1, x_2, v_1, v_2$ are vectors with dimension $n \times 1$ $M$ and $N$ are matrices with dimension $n \times n$ ...
0
votes
1answer
40 views

Doubt about subspaces being vector space

Whenever i am saying $V$ is a $n$-dimensional vector space, it means it has $n$ basis vectors each with n elements, right. So when i am proving some theorems or relations involving some ...
1
vote
0answers
31 views

How to convexify (relax) this L0 eigenvalue optimization problem?

Let $C_1,\dots,C_L$ be $N\times N$ hermitian matrices. Let $d<0$ be a given negative constant. Then consider the optimization problem \begin{align} \max_{r\in \mathcal{R}^{L\times 1}} &\mid\mid ...