Questions tagged [linear-algebra]

For questions about 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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16 views

$A = B+ C$ then $(B^{-1} - A^{-1})$ and $(C^{-1} - A^{-1})$ are positive semidefinite

Let $A$, $B$ and $C$ invertible and symmetric square real matrices of dimension $n$. I want to show that if $A = B+ C$ then $(B^{-1} - A^{-1})$ and $(C^{-1} - A^{-1})$ are positive semidefinite. For a ...
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1answer
25 views

Why is the determinant useful when finding eigenvectors?

I understand that the eigenvectors of a linear transformation are the vectors that, when transformed, are simply scaled by a factor of the corresponding eigenvalue. Therefore $$ A\vec{x}=\lambda\vec{x}...
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7 views

time derivative of jacobian of diffeomorphisms

Suppose $F_t$ is a 1-parameter family of diffeomorphisms. Let $J_t$ denote the jacobian matrix of $F_t$ and $H_t= det(J_t)J_t^{-1}$, i.e the inverse of the jacobian time's it's determinant. I'm trying ...
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1answer
20 views

Show $x^T \cdot A \cdot x=0 \implies x^T \cdot A=0$ [closed]

I want to show that $x^T \cdot A \cdot x=0 \implies x^T \cdot A=0$ (for A positive definite and symmetric) Anybody can help a brother out?
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1answer
14 views

Trace of sum of operator and adjoint

What is trace($M$) where $M \in \mathcal{L}(\mathcal{L}(V)$, $V$ is n-dimensional real space, and $M(S)=S^*+S$? To know this, we need to know something about the eigenvalues of $M$ but this seems not ...
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0answers
21 views

Converting a matrix to upper-triangular form

I need to convert the following Jacobian matrix to upper-triangular in order to find its eigenvalues: $$\begin{bmatrix} -u-k_{12} & k_{21} & 0 \\ k_{12} & -k_{21}-k_{23}-s & k_{32} \\ ...
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54 views

$tr(A)=tr(A^2)=tr(A^3) \implies A$ is Nilpotent

Let $A$ be a square matrix with $\mathrm{tr}(A)=\mathrm{tr}(A^2)=\mathrm{tr}(A^3)$. Prove that $A$ is nilpotent. The tip in the question is to find the matrix $B$ which is similar to $A$ and then $A^...
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1answer
8 views

Question on what maximum means in the phrase “maximum number of independent generalized $\lambda$-eigenvectors”

I was studying generalized eigenvalues, and I read the following property of the algebraic multiplicity of an eigenvalue $\lambda$: the algebraic multiplicity of $\lambda$ is the maximum number of ...
2
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1answer
24 views

Question in Proof of Gelfand's Formula

Gelfand's formula states that the spectral radius $\rho(A)$ satisfies $$\rho(A) = \lim_{n \to \infty} \Vert A^n \Vert ^{1/n}.$$ Multiple proofs of Gelfand's formula (including this one on Wikipedia) ...
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17 views

Davis-Kahan Theorem for PSD Matrices.

Let $A=XX^T$ and $B=YY^T$ be symmetric, PSD matrices, such that $\lambda_k(A)-\lambda_{k+1}(A)$ is large. Suppose $Y=X+E$, where $\|E\|$ is small, and $X$ and $Y$ are not necessarily symmetric. Is ...
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18 views

Find a basis $B$ for the domain of $T$ such that the matrix for $T$ relative to $B$ is diagonal.

Find a basis $B$ for the domain of $T$ such that the matrix for $T$ relative to $B$ is diagonal. $$T: R^2 \to R^2: T(x, y) = (6x + 3y, 2x + y).$$ I found the eigenvalues to be $0$ and $7$. I found the ...
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1answer
10 views

Show function involving Grassmannian is surjective

Let $\text{SO}_n$ denote the special orthogonal group and $G(n,k)$ be the set of all $k$-dimensional linear subspaces of $\mathbb{R}^n$. Show that $f:\text{SO}_n\to G(n,k)$ given by $f(O)=OL$, where $...
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19 views

Linear Transformation Linearity

This might be very straightforward, but if $S,T,R$ are all linear transformations in vector space V, is $(ST+RT) = (S+R)T$
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1answer
26 views

Properties of an orthogonal transformation $U$ with $U^{2} = -I$

From Linear Algebra Done Wrong: Let $U$ be an orthogonal transformation in a real inner product space, satisfying $U^{2} = -I$. Show that in this case dim$X = 2n$, and that there exists a subspace $E ...
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1answer
19 views

Example on basis change

I am reading the book Mathematics for Machine Learning by Marc Peter Deisenroth, A. Aldo Faisal, and Cheng Soon Ong. I have a problem understanding an example in this book: Example 2.23 (Basis change) ...
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1answer
43 views

Axler Linear Algebra Question

For question 2 from chapter 5A, it says that given ST = TS, prove that null $S$ is invariant over T. This was pretty straightforward by showing that $(S-\lambda I)Tv = T(Sv - \lambda v) = 0$ for $v \...
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0answers
34 views

How is this type of matrix called?

Let the matrix $M$ be defined as: $$M = \left(\begin{array}{rr} A & B \\ C & D \end{array}\right)$$ where $A = 0,B,C,D = 0$ are block matices and $B$ is a diagonal matrix and $C$ is a diagonal ...
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9 views

Upper bound of the largest eigenvalue of $DAAD$.

$H = DAAD$, where $D$ is a diagonal matrix and the elements on the diagonal are real numbers (could be positive and negative). $A$ is positive semi-definite and can be diagonalized into $A=U^T\Lambda ...
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0answers
6 views

Let A from $M3(R)$ be matrix of orthogonal projection onto plane which goes through origin

Find solutions of equation $Ax=0$ Ok first thing that came to mind was that all solutions are $(0,0,z)$ ( or $(0,y,0)$,$(x,0,0)$) but I feel like I'm missing something
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2answers
28 views

Rigid motion which is not a linear transformation

From Linear Algebra Done Wrong: Give an example of a rigid motion in $\mathbb{C}^{n}$, $T(\mathbf{0}) = \mathbf{0}$, which is not a linear transformation. where a rigid motion was defined as a ...
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0answers
19 views

Digit word problem

The unit's digit is twice less than the ten's digit. Find the number, if it's 18 more than the reversed number. Let x be the units digit
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2answers
27 views

Components of a bounded sequence of vectors are also bounded

If $\{x_k\}$ is sequence of vectors in $\mathbb R^n$ that is bounded: $\| x_k \| < D$. How to prove that its componentes are also bounded: $|x_k^{(1)}|, |x_k^{(2)}|, ... < M$ without assuming a ...
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15 views

How to find a profit function?

[enter image description here][1] [1]: https://i.stack.imgur.com/NcYwZ.jpg**strong text**
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0answers
27 views

Linear transformation and its eigenvalues and eigenvectors

I'm struggling with this problem: If one have the transformation $T:\mathbb{R}^2\rightarrow\mathbb{R}^2: T(\vec{x})=k\vec{x}$ where $k\in\mathbb{R}$ and $\vec{x}\in\mathbb{R}^2$, find the eigenvalues ...
3
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0answers
21 views

Dimension of certain matrix subspace

Let $V:= \mathbb{R}^{n \times n}$, $X \in V$ be given and $$ U_X := \lbrace A \in V: \mathrm{trace}(AX) = 0 \rbrace $$ a subspace. I want to find the dimension of $U_X$. If $X = 0$, then clearly $U_X =...
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3answers
29 views

Let $S_1$ and $S_2$ be convex sets. Show that the intersection $S_1 \cap S_2$ is a convex set

Found this exercise in "Introduction to Linear Algebra" by Serge Lang. It says: "Let $S_1$ and $S_2$ be convex sets. Show that the intersection $S_1 \cap S_2$ is a convex set" My ...
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0answers
29 views

Need help with proof of [closed]

Let {~v1, · · · ~vm} ⊂ Rn be a linearly independent set. Let L ∈ Rm×n . Prove that, if null(L) = {~0}, then {L~v1, · · · L~vm} ⊂ Rm is a linearly independent set.
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1answer
26 views

Sets that contain $2$ or $1$ different elements can we make them vector spaces?

Sets that contain $2$ or $1$ different elements can we make them vector spaces? Answer is for $1$ element $yes$ for $2$ element $no$. I know definition of vector space and it's $8$ axioms but don't ...
2
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1answer
50 views

Given a set $S$ of real numbers, find the smallest set $A$ such that every element of $S$ can be expressed as a sum of distinct elements of $A$.

I have been thinking about this problem and can’t come up with a solution for it. Perhaps someone here with a better quill may be able to! Given a set $S=\{s_1,s_2,...,s_N\}$ of $N$ real numbers, $s_i\...
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1answer
22 views

How to find the Log function only given the graph

What would the equation of this CURVE be?
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0answers
25 views

Find two disjoint subsets G1 and G2 of Z30

Question: Find two disjoint subsets G1 and G2 of Z30, each with 8 elements, such that both G1 and G2 are abelian groups with respect to multiplication.
1
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1answer
27 views

Least Square Methods

We know from linear algebra, the least square solution of linear equation system : $$Ax=b$$ always exists. That is, the equation $$A^TAx=A^Tb$$ always has at least one solution. $\bar x$ is the ...
0
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1answer
23 views

How to find the missing coordinate?

I have been given two points $P(0,1,3)$ & $Q(-1,5,2)$, both of them crosses a line in space. The third point is supposed to cross the line, but there is a missing coordinate $R(m,9,1)$. How do I ...
0
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0answers
9 views

Existence of matrix that satisfy a set of restrictions

Let $\Sigma$ be a $p\times p$ positive definite matrix, and $\mu$ a $p\times 1$ vector. I am asked to verify that there exists nonsingular $B$ such that $$B\mu = (\theta, \mathbf{0}')' \text{ and } B\...
4
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1answer
32 views

Why the number of linearly independent columns of a matrix doesn't change by if we apply Row operations

Can anyone please tell me why the number of linearly independent columns of a matrix doesn't change even if we apply row operations on the matrix? The column space does change by row operations but I ...
0
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1answer
21 views

Confusion about finding eigenvectors from matrix multiplication

I'm doing the following question here: Find the determinant of $A$=\begin{bmatrix} 6 & 2 & 2 & 2 &2 \\ 2 & 6 & 2 & 2 & 2 \\ 2 & 2 & 6 & 2 & ...
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0answers
24 views

Calculate the deteminant of the matrix $B$

Let $A\in \mathbb{R}^{3\times 3}$ with $\det(A)=13$. Let $a_1,a_2,a_3$ be the rows of the matrix $A$, $$A=\begin{pmatrix}\ldots a_1 \ldots \\ \ldots a_2 \ldots \\ \ldots a_3 \ldots \end{pmatrix}$$ ...
0
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1answer
22 views

How to solve $|1+x|\frac{|\alpha-\beta|}{|\alpha + \beta|} + \max\{1, |x|\} \leq 1$?

I am trying to solve $$|1+x|\frac{|\alpha-\beta|}{|\alpha + \beta|} + \max\{1, |x|\} \leq 1 \tag{1}, $$ with $\alpha<\beta<0$. Using $$\max(x, y) = \frac{x + y + |x - y|}{2}$$ in $(1)$ we get $$...
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0answers
12 views

the rank of the Hadamard product

For matrices $D\in C^{d\times p}$ and $E \in C^{d\times p}$ with $d> p$, if $D$ is a full column matrix, for what condition that $D\odot E$ is also a full column matrix where $\odot$ denotes the ...
3
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0answers
36 views

Writing Euler's equation for this minimization problem.

Let $A$ be a symetric and definite positive matrix of size $n \times n$. Let $\Sigma \subset \mathcal{P}( \{1, \dots, n\})$ a set of indices and consider the projection operator $P_{\Sigma} : \mathbb{...
2
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2answers
40 views

Prove that if AB=I_m, then rk(A)=rk(B). [closed]

If $A\in\mathfrak{M}_{m,n}(F)$ and $B\in\mathfrak{M}_{n,m}(F)$, prove that $AB=I_m$ implies $rk(A)=rk(B)$. Here $A\in\mathfrak{M}_{m,n}(F)$ means $A$ is a ($m \times n$)-matrix with entries in $F$. My ...
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2answers
25 views

Basis change qestion

Let $V$ be a vector space over a field $K$. Suppose we have two bases of $V$, $B=(e_1,...,e_n)$ and $B'(f_1,...,f_n)$. We can express $f_j$ in terms of the basis $B$: $$f_j=p_{1j}e_1+p_{2j}e_2+...+p_{...
0
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1answer
15 views

Which property will a composition of two mapping show if two earlier functions initially had some property?

I've read in a reference book that mentions this: Let $f: A \to B$, $g: B \to C$ be two functions, then $g \circ f: A \to C$ is injective $\implies f:A \to B$ is injective $g \circ f: A \to C$ is ...
1
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2answers
26 views

Standard matrix for linear transformation without using least squares

I have the following question here: Let $W$ be the subspace of $\mathbb{R}^3$ spanned by $$\{(1,1,1),(-1,1,2)\}.$$ Let $T:\mathbb{R}^3 \rightarrow\mathbb{R}^3$ be the linear transformation given by ...
1
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1answer
32 views

Why normal equation can have infinite solutions?

The normal equation is $$ X^Ty = X^TX\hat{\beta}. $$ If $X^TX$ is invertible, then $\hat{\beta}$ has an unique solution which is $$ \hat{\beta} = (X^TX)^{-1}X^Ty. $$ However, if $X^TX$ is non-...
2
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2answers
23 views

The value/s of $a$ will the linear system have a unique, infinite and no solution?

For which value/s of $a$ will the linear system have a unique, infinite, and no solution? $\begin{cases} x+y-z=2\\ x+2y+z=3 \\ x+y+(a^2-5)z=a\end{cases}$ Can someone verify my solution? Thank you. My ...
1
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0answers
32 views

Related to Euclidean and unitary vector space

Let $V$ be a finite-dimensional Euclidean or unitary $K$-vector space. Show or refute the following statements: (i) $(f + g)^{*} = f^{*} + g^{*}$ for all $f, g ∈ \operatorname{End}(V)$ (ii) $(λf)^{*}$ ...
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0answers
21 views

prove that the set of all differentiable functions on (−∞, ∞) that satisfy f ′ + 2f = 0 is a subspace of F(−∞, ∞) [closed]

How can I prove that the set of all differentiable functions on (−∞, ∞) that satisfy f ′ + 2f = 0 is a subspace of F(−∞, ∞)
0
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1answer
19 views

Minimizing the norm of the difference of two vectors.

Let $\mathcal{H}$ be a a vector space of finite dimension. Let $\mathcal{H_1}$ be a subspace of $\mathcal{H}$. Considering some vector $|\phi\rangle \in \mathcal{H}$ i need to show that there exists ...
1
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3answers
28 views

On the linear dependence of three coplanar vectors

The general consensus seems to be that any three coplanar vectors are linearly dependent. Here's one source that says so. However, considering three vectors, of which two are collinear and the third ...

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