Questions tagged [linear-algebra]

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 concerning matrices, use the (matrices) tag. For questions specifically concerning matrix equations, use the (matrix-equations) tag.

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2
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1answer
20 views

Matrix representation of basis with $n\times n$ dimension

The teacher has posted a solution for the following: Let $A \in \mathbb{K}^{n \times n}$ and $\Phi_A: \mathbb{K}^{n \times n} \to \mathbb{K}^{n \times n}$ defined by $\Phi_A(X) := AX$. Show that $\...
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1answer
34 views

About positive definite matrices

Suppose one is given matrices $A_1, A_2, \dots, A_w \in \mathbb{R}^{r \times n}$ and matrix $B \in \mathbb{R}^{n \times n}$. When and how can one construct matrix $C \in \mathbb{R}^{r \times n}$ such ...
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3answers
36 views

Counterexample: $V \neq \ker(T) \oplus T(V)$ if $T^2 \neq T$

We could show that Let $V$ be a finite-dimensional vector space over a field, and $T: V \to V$ be a linear transformation from $V$ to itself such that $T(V) = T^2(V)$, then $V = \ker T \oplus T(V)$...
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0answers
16 views

Show that image of error vector $L(f-f_d)$ belongs to orthogonal complement of $V_d$

Problem Let's observe in interval $[a,b]\subset \mathbb{R}$ defined real valued and continuous maps which form vector space $\mathcal{F}([a,b],\mathbb{R})$. Let $\langle ., . \rangle$ be some inner ...
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1answer
47 views

A linear system of equations $A\mathbf{x} = \mathbf{b}$ has two different solutions $\mathbf{x} = \mathbf{u}$ and $\mathbf{x} = \mathbf{v}$

A linear system of equations $A\mathbf{x} = \mathbf{b}$ has two different solutions $\mathbf{x} = \mathbf{u}$ and $\mathbf{x} = \mathbf{v}$. 1. Show that $\frac{1}{3}\mathbf{u}+\frac{2}{3}\mathbf{v}$ ...
2
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2answers
40 views

What is “Direction Cosine” in Linear Algebra?

My professor wrote the following thing on the board: $$\text{Direction Cosine: } \cos(x, y) = \dfrac {(x, y)}{||x|| \cdot ||y||}$$ $(x, y)$ represents an inner product. This was in the context of ...
1
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2answers
45 views

Why is the solution set of the reduced row echelon form of A equal to the solution set of A?

One way of solving a system of linear equations is to express it in an augmented matrix. Then, we can perform elementary row operations in order to bring the matrix into RREF (reduced row echelon form)...
2
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4answers
82 views

How to quickly compute the determinant of given matrix

I need to find the determinant of given matrix : $\begin{bmatrix} 1&0&0&0&0&2\\ 0&1&0&0&2&0\\ 0&0&1&2&0&0\\ 0&0&2&1&0&0\...
0
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4answers
51 views

Range of $f(x)=[x^2 - 3x]$ [on hold]

I need to find the range of this function $$f(x)=gif (x^2 - 3x.)$$ I have no idea how to start solving this. Can anyone help?
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0answers
35 views

Simplex optimization: how to project to nearest feasible point, with box constraints and total constraint

I am working in a simplex-based constrained optimization setting, wherein several texts suggest that algorithms such as Nelder-Mead can be modified for constraints by projecting an infeasible point ...
2
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0answers
21 views

Defective Matrices Transforms

Defective Matrix:$\begin{bmatrix}1&1\\0&1\end{bmatrix}$ I am unable to wrap my head around what kind of transforms yield defective matrices or matrices where geometric multiplicity is less ...
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1answer
22 views

Strong separation of symmetric rank $1$ matrices

Let $V$ be the inner product space of $n \times n$ symmetric matrices where the inner product is given by $\langle A, B \rangle = trace(AB)$ and let $A, B \in V$ be such that they have eigenvalue $1$ ...
3
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2answers
65 views

Same Eigenvalues = Similar Matrices?

While watching Strang's lecture on similar matrices he stated that if any two matrices,$A$ and $B$, have the same eigenvalues then they can be put in the form $A=P B P^{-1}$ . This is very easy to see ...
2
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1answer
27 views

A question on Matrix Norms

Suppose $\mathbf{A}\in\mathbb{R}^{n\times n}$, and $\mathbf{B}\in\mathbb{R}^{n\times n}$ be a matrix with strictly positive entries. Define ${\Vert\mathbf{A}\Vert}_{\mathbf{B}}=\max\limits_{1\leq i,j \...
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1answer
16 views

Kronecker delta identity for orthonormal vectors

I'm reading a textbook and I came upon the following identity: $\delta_{ij} = n_i n_j + m_i m_j + p_i p_j$ where m,n and p are orthonormal vectors. Can someone help me prove this? I've tried using ...
0
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2answers
21 views

How to find the transformation matrix from input/output?

I need some help with the following question - I'm not sure how to approach this problem. Can someone point me in the right direction? Let $T$: ${\rm I\!R}^3$ -> ${\rm I\!R}^2$ be a linear ...
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2answers
105 views
+50

Every matrix of the centralizer of the centralizer of a matrix is a polynomial in that matrix

Let $V=M(n,\mathbb C)$. For a subset $S \subseteq V$, let $C(S):=\{A \in V | AB=BA, \forall B \in S \}$ . How to prove that for every $A\in V$, we have $C(C (\{A\})) \subseteq \{ p(A) | p(t) \in \...
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2answers
34 views

If $\operatorname{Tr}(H)\ge 0$, is $\operatorname{Tr}(PH)\ge 0?\ $ $H$ is Hermitian, $P$ is positive definite.

Old: Given $\operatorname{Tr}(DH)\ge 0$ for positive real diagonal $D$ and Hermitian $H$, is $\operatorname{Tr}(H)\ge 0$? Let $D$ be positive real diagonal and $H$ be Hermitian such that $H$ may have ...
0
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0answers
29 views

Polynomial in a $2$-dimensional Euclidean space

I am reading about the problem of moments and get some troubles in they way one defines the measure on some $k$-dimensional Euclidean space by polynomial. Could you please help me? More precisely, I ...
0
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1answer
33 views

Whether there is always an intersection between Column Space and Nullspace [on hold]

Suppose there is a matrix $A$ that satisfies $Ax=b$, and it have two solutions $c$ and $d$. Then we can know $Ac=b$ and $Ad=b$, so $A(c-d)=0$. But $c$ and $d$ are the two column vectors of matrix $A$, ...
3
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2answers
42 views

Is set of matrices form $M_{3\times 3}(\mathbb R)$ with rank 1 is closed or open?

Is set of matrices form $M_{3\times 3}(\mathbb R)$ with rank 1 is closed or open ? I know that $M_{3\times 3}(\mathbb R)$ can be viewed as $\mathbb R^9$ with the euclidean norm on it. But I do not ...
0
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2answers
49 views

Can a polynomial have repeated complex roots?

Can a polynomial have repeated complex roots - or is it only possible that it can have repeated real roots? If so, can you please also provide an example of a polynomial with complex roots where you ...
0
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1answer
57 views

Can you factor equations which has $e^{-x}$ and $x$ together?

I am working on a math investigation, and I got a function: $y = 5x - 2e^{-4x} + 2$. The problem is, I want to change this equation such that $x$ is in terms of $y$. Is there a way so? Or is it ...
0
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1answer
31 views

Solve set of linear equations subject to sign of results

I have a set of linear equations, of the form $Ax=b$. However, the value of $b$ depends on the sign of the solution of $x$, such that $b_i = c_i + d_i$ if $x_i > 0$, and $b_i = c_i - d_i$ if $x_i &...
2
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0answers
42 views

Given two quadratic forms, find an isometry

Let $q_1(x)$ and $q_2(x)$, be two quadratic forms on a vector space $V$ based on $\mathbb{R}$. In the base $\mathcal{B} = (e_1, e_2, e_3)$, the expressions of the two forms are: $$ \begin{align} &...
0
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3answers
39 views

Is it okay to take basis from rref matrix as well as original matrix in both column space and row space?

Let $A$ be the given matrix of order $m \times n$. I want to find the basis for both row and column spaces of $A$. I transformed the matrix A in to its row reduced echelon form i.e., $rref(A)$. ...
2
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0answers
88 views

Find the numbers $\lambda_1, \lambda_2, \lambda_3, \lambda_4$?

I have a matrix a matrix $\mathbf{A}$ with its $\mathbf{QR}$ factorization: $$ \mathbf{A} = \left[\begin{array}{rrrr} 1&-3&8&-1\\ -4&-3&-2&-2\\ 2&4&1&3\\ 2&4&...
2
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1answer
40 views

Is there a formula to calculate the euclidean distance of two matrices?

Wiki gives this formula ${\displaystyle {\begin{aligned}d(\mathbf {p} ,\mathbf {q} )=d(\mathbf {q} ,\mathbf {p} )&={\sqrt {(q_{1}-p_{1})^{2}+(q_{2}-p_{2})^{2}+\cdots +(q_{n}-p_{n})^{2}}}\\[8pt]&...
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3answers
45 views

Proof that a certain set of polynomials is linearly independent [on hold]

For $c_0, \ldots, c_n$ pairwise distinct complex numbers. Show that the polynomials $((X-c_i)^n)_{0 \leq i \leq n}$ are linearly independent. I need to prove it without induction. Any help would be ...
0
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1answer
32 views

Eigenvalues for fingerprint classification

i acquired waveshare's (uart capacitive) fingerprint scanner which provides me with eigenvalues of a finger. these values are used to compare/classify fingerprints as per the scanners documentation. ...
0
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1answer
17 views

Orientation for a vector spaces determines a canonical orientation for the dual vector space

Suppose $V$ is an $n$-dimensional real vector space, with $n > 0$. Show that an orientation for $V$ determines a canonical orientation for $V^*$, the dual of $V$. The idea I had in mind to show ...
5
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3answers
108 views

Derivative of Dot Product as matrix multiplication

I've come across this definition when looking into how to differentiate parameter vectors in statistics. Given $ \pmb{x}^{T} \pmb{x}$ $$\frac{\partial (\pmb{x}^{T} \pmb{x}) }{\partial \pmb{x}}=2\ \...
0
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1answer
22 views

A Trace Bound Identity of Matrix Products (known for reals) in the Complex Space

Please refer to this beautiful paper on trace inequalities for matrix products. Theorem $3$ of the article (rephrased) states: For any real $n\times n$ matrix $A$ and any real symmetric $B$ of the ...
0
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1answer
19 views

Proof nonpositivity of a particular matrix

Let A be a nonnegative matrix and let h be its largest eigenvalue. Is it true that the inverse of (A - g I) is a nonpositive matrix if h < g?
4
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1answer
37 views

Existence of nonsimilar matrix with same properties.

I wanted to know Is there exist 2 nonsimilar matrices with all algebraic properties same? I think there exists such pair as otherwise there we necessary sufficient condition of diagonalisibily using ...
3
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1answer
56 views

Best straight-line approximation for $\sin(2x)$ on $(-\pi,\pi)$

I'm working on the second part of this problem from Strang's Linear Algebra, 4e (problem 3.4.21): What is the closest function $a\cos(x) + b\sin(x)$ to the function $f(x) = \sin(2x)$ on the ...
0
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1answer
21 views

What does Singular value represent in the context of Singular Value Decomposition

Regarding SVD: Why is it called Singular Value and what does that value represent? And where does the SVD play a part in applications? Thank you
1
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0answers
45 views

Inner product preserving mapping

Let $A \in \mathbb{R}^{n\times n}$ and $B \in \mathbb{R}^{m\times m}$ be positive definite symmetric (pds) matrices. For a matrix $M_1, M_2 \in \mathbb{R}^{n \times m}$, I would like to know that ...
1
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1answer
20 views

How to set positive-definite function to be equal to the length of inputs?

I want to prove or rebut the following: Given a full rank $n \times n$ matrix $\mathbf{M}$, the following equality is true: $\mathbf{v}^T \mathbf{M} \mathbf{v} = \mathbf{v}^T \mathbf{v}, $ for all $ \...
1
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1answer
18 views

Rank 1 Matrix Decompositions

Are there many ways to decompose a matrix, $\bf{A}\in \mathbb{R}^n$, into a sum of rank 1 matrices? I ask because I know if a matrix is diagonalizable and symmetric, $\bf{A}=\Phi\Lambda\Phi^{T}$, then ...
0
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1answer
25 views

Finding Linearly independent Matrices

I stumbled upon a problem which is as below Take the subset of 2 x 3 matrices consisting of matrices with first row entries adding up to zero and the second column sum equal to double the third ...
0
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3answers
49 views

Why must $M$ be diagonalizable?

I am trying to figure out why the answer to the following question is (a) $M$ must be diagonalizable based on the given information. I know that for $M$ to be diagonalizable the sum of the dimensions ...
0
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0answers
38 views

Points on two perpendicular lines [on hold]

I got in troubles. I have no idea how to work with 3D lines, just with 2D lines. My problem is this: I got a point on a line and a point on a perpendicular line in 3D space. I have to get the gradient ...
2
votes
2answers
81 views

What is *energy* in the context of linear algebra?

I have heard and read the term energy in the context of LA a few times now. e.g: The algorithm is based on the geometrical observations that the word embeddings (across all representations such ...
1
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3answers
42 views

Suppose $P$ is a non singular $m\times m$ matrix and $A$ be any $m\times n$ matrix,then how to interpret that $A$ and $PA$ have same row space?

I have recently encountered a theorem in linear algebra which states that if $P$ is an $m\times m$ matrix and $A$ is $m\times n$,then $A$ and $PA$ share the same row space.I have understood the proof ...
0
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1answer
29 views

Show that $\text{trace}(A^TA) \ge 0$ and $\text{trace}(A^TA)$ if and only if $A = O$ [duplicate]

Problem Show that $\text{trace}(A^TA) \ge 0$ and $\text{trace}(A^TA)=0$ if and only if $A = O$ when $A \in \mathbb{R}^{n \times n}, n \in \mathbb{N}$. Symbol $O$ denotes zero matrix. Attempt to show ...
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0answers
32 views

Solving linear system with inequality constraint

I have a linear system: $$\begin{bmatrix} -k_{1}-k_{2} & k_{2} & 0 & 0 & \dots & 0 & 0 & 0 \\ k_{2} & -k_{2}-k_{3} & k_{3}& 0 & \dots &...
0
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2answers
26 views

Clarification on the meaning of “uniqueness” in the (linked) Matrix diagonalization theorem

I came across this source yesterday, where they say: given a square real matrix S of size $nxn$, with n linearly independent eigenvectors, there exists an eigendecomposition such as $S=UDU^{-1}$. If ...
1
vote
1answer
50 views

importance of Krylov subspace

Can someone please explain to me why we are using Krylov subspaces for the CG-Method, GMRES-Method and Arnoldi. Unfortunately I do not see where the advantages are und why Krylov subspaces are so ...
1
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1answer
26 views

When is there an inner product making a given matrix unitary?

The unitarity of a given operator/matrix $A$ depends on the underlying inner product, so I guess it in principle possible for a given non-unitary $A$ to act as a unitary with respect to a different ...