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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Bounding the spectral norm of a block random matrix

Suppose that zero-mean iid random matrices $A_1 ,A_2,\dotsc,A_n$ satisfy $$\mathbb{P}\left(\left\|A_i \right\|\geq t\right)\leq \phi\left(t\right),\tag{*}$$ for $t>0$, where ...
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48 views

The subspace sum of closed subspaces is closed [duplicate]

Given an arbitrary Hilbert space $\scr H$ and closed subspaces $A,B\subseteq\scr H$ with trivial intersection, is it true that $A+B=\{x+y:x\in A,y\in B\}$ is closed? So far, I have the following: Let ...
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40 views

Representative value of non-square matrix

First of all, I apologise if this question is inappropriate, I wish I could be more specific - but due to the nature of it, as I am actually asking for a suggestion of some technique, that's hard to ...
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1answer
54 views

Why is the first left and right singular vectos scale by the first singular values a good approximation of the original matrix

Conceptually, why is the first singular vector a good rank one approximation instead of something like the averaging of the total singular vectors? If you have $$A = U\Sigma V^T $$ why isn't ...
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44 views

If $U$ is orthogonal to itself why can you say $ U = U\Sigma V^T$ where $\Sigma = I_n $ and $ V = I_n$

How can you tell $$ U = U\Sigma V^T$$ $$\Sigma = I_n \:\:\:\:\:\: V = I_n$$ $$\text{if} \:\: UU^T = I_n$$ by inspection without solving for the singular values of $U$ From my understanding SVD ...
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40 views

Injective Linear Map

Let $V$ and $W$ be vector spaces over a field $K$ and $T : V \to W$ be a linear map. Suppose that $T$ is injective. Show that there exists a linear map $S : \text{Im } T \to V$ such that $S\circ T(v) ...
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79 views

How to prove “rank is not less than the number of non-zero eigenvalues”?

I know to prove this using core-nilpotent decomposition. But if it feels like using a big tool for a small problem, is there any other better, simple proofs?
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Why do we define characteristic equation only for square matrix?

Why do we define characteristic equation only for square matrix? I note that for otherwise in such a equation we can't put the given matrix in place of $x$ in terms like $x^2,x^3,...$ etc. Is it a ...
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1answer
29 views

Linear transformation from endormorphism to real number

For a finite dimensional vector space $V$, is there a linear transformation between its endomorphism and real number, please? I suspect that since the element of the endomorphism can be represented by ...
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48 views

Linear Algebra.

Let $V$ be a vector space over the field $K$, and $S\subseteq V$. Suppose $S = \{\vec{v}_1, \vec{v}_2,\dots,\vec{v}_n\}$ has the property that for each $v \in V$, there exist unique scalars $ a_1, ...
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23 views

Let $A \in Mat_{n,n}(\mathbb C)$ (diagonalizable) and $u_1, \ldots, u_k$ be the different eigenvalues of $A$. Show $(A-u_1I_n) \dots (A-u_k I_n) = 0$.

Let $A \in Mat_{n,n}(\mathbb C)$ (diagonalizable) and suppose $p_A(\lambda) = (-1)^n(\lambda -\lambda_1 ) \dots (\lambda - \lambda_n)$ (characteristic polynomial of $A$), where $\lambda_i$ is an ...
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1answer
53 views

Suppose $A^n = 0$ matrix for some $n > 1$. Find an inverse for $I - A$. [Lay P160 Ch 2 Sup Q4]

Solution: From P160 Supplementary Exercise 3, the inverse of $I-A$ is probably $I+A+A^{2}+...+A^{n-1}$. To verify this, compute $ (I \color{orangered}{-A} ...
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1answer
46 views

Is there always a vector $x$ with positive entries such that $Ax=b$ for $b$ positive $A$ positive definite?

For a positive definite $n\times n$ matrix $A$, does there exist an $n \times 1$ vector $x$ with all entries positive such that the vector $Ax=b$ has all entries positive ? I think there is a counter ...
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2answers
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Mustn't both left and right inverses be checked? [Lay P160 Ch 2 Sup Q4]

Question: Suppose $A^n = 0$ matrix for some $n > 1$. Find an inverse for $I - A$. Solution: From P160 Supplementary Exercise 3, the inverse of $I-A$ is probably $I+A+A^{2}+...+A^{n-1}$. To ...
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2answers
127 views

If $AX=XA$ for all $X$, then $A = \alpha I$ for some $\alpha$

Let $A$ be a $2 \times 2$ real matrix such that $AX=XA$ for all $2 \times 2$ real matrices $X$. Show that $A= \alpha I$ for some $\alpha ∈R.$ I am absolutely stuck, i thought $X$ and $A$ are ...
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2answers
78 views

Axis of symmetry of a binary image

I want to calculate the axis of symmetry of a binary image. In other words I have an image that has a black irregular shaped object with a white background. I want to find the best approximation of ...
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3answers
254 views

determinant inequality $\det(A^2+AB+B^2)\geq\det(AB-BA)$

$A,B$ are two $2\times 2$ real matrices, then $$\det(A^2+AB+B^2)\geq\det(AB-BA)$$ The inequality is equivalent to the following problem: Let $X=A+\dfrac{B}{2},Y=-\dfrac{B}{2}$ ...
2
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1answer
49 views

skew-diagonalization of a matrix

I think about the skew-diagonalization of a matrix, for example, let $A=\begin{pmatrix}a & b \\ c& d \end{pmatrix}\in SL(2,\mathbb{R})$ , if $trace(A)=0$, is it conjugate to $\begin{pmatrix}0 ...
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2answers
61 views

Find an orthogonal basis for $\mathbb R^4$

Find orthogonal basis for $\mathbb R^4$ that contains the vectors: $v_1=$ $\begin{bmatrix} 2 \\[0.3em] 1 \\[0.3em] 0 \\[0.3em] -1 ...
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73 views

Does this characterize the operator norm of the inverse?

Let $A$ be an invertible operator (bounded with bounded inverse). Then $$\frac{1}{\|A^{-1}\|} = \inf\left\{\frac{\|Av\|}{\|v\|} : v \neq 0\right\}$$ I believe I have a proof as follows, but I just ...
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4answers
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I need help with a proof showing $\|u\|^2 = \|\operatorname{proj}_v u\|^2 + \|u - \operatorname{proj}_v u\|^2 $

So, I am dealing with the 2-norm and the projection is defined as the standard orthogonal projection, so far I have $$\|u\|^2 = \|\operatorname{proj}_v u\|^2 + \|u - \operatorname{proj}_v u\|^2 ...
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38 views

How to solve this kind of linear system?

IF there is a linear system such as : y=-2x-2z+1 x=-2y-z+2 z=x-y I want a ...
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4answers
91 views

A matrix $M$ that commutes with any matrix is of the form $M=\alpha I$

I feel like this is probably a simple proof but I can't quite come up with it in an elegant way nor could I find it here. Prove that if a matrix $M$ commutes with any matrix then $M$ is of the form ...
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1answer
368 views

Help with proving that the transpose of the product of any number of matrices is equal to the product of their transposes in reverse

Specifically I am trying to show that (An)T = (AT)n where A is an mxm square matrix and n is a positive integer. This is where I'm stuck: To prove the theorem I would like to show that ((An)T)ij = ...
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1answer
51 views

dual of linear operator space

Let $U$ and $V$ be finite dimensional vector spaces. Let $L(U,V)$ be the space of linear maps $U\rightarrow V$. Let $A \in L(U,V)^*$, i.e., the dual space of $L(U,V)$. Is it true that $A : U^* ...
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1answer
42 views

How do I find the range of a transition matrix?

I am unsure as to how to find the range of a transition matrix. For example, suppose $A=\begin{bmatrix} 1& 1 & -1\\ 1 & 2 & 1\\ 0 & 1 & 0 \end{bmatrix}$ for a linear transform ...
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1answer
46 views

Hermitian and Diagonal Matrix Norm inequality

I have a matrix inequality that I think is true, but I can't prove. $D_1$ and $D_2$ are diagonal matrices with non-negative entries. $M_1$ and $M_2$ are positive definite matrices. I want to show ...
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Question regarding how to find the coordinate vector of a transition matrix

Given that A= $\begin{bmatrix} 1.5 & -1 & -.5\\ -.5 & 2 & .5\\ .5 & 1 & 5/2 \end{bmatrix}$ is the standrad matrix for $T: \mathbb{R}^3 \to \mathbb{R}^3$ relative to B and the ...
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Invariants of shape operators?

Let $S:V\longrightarrow V$ be a Linear Transformation, then the Characteristic polynomial of $S$ and therefore its Coefficient are invariant. Except the first and the last Coefficient that we know ...
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60 views

What is the difference between column reducing a matrix and computing its inverse?

I read in a paper that column reducing a matrix of polynomials takes $\tilde{O}(n^{\omega}d)$, where $d$ is the max degree of the polynomials, and $\omega$ is the exponent of integer matrix ...
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1answer
59 views

spectral decomposition of $H_u$

Let $V$ a complex vector space with dimension n and inner product ,$u \in V $ unit vector. Let $ H_u: V \rightarrow V$ defined by $ H_u(v) = v - 2 <v,u>u$ $\forall v \in V$. Then: a)$H_u(u) = ...
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3answers
110 views

Show that two linear transformations are equal

Let $\{v_1, v_2,....,v_n\}$ be the standard basis for $\mathbb R^n$.Prove for any two $m\times n$ matrices that their linear transformations are equal if and only if the two matrices are equal. I know ...
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a system of linear equations $x-y+z=0$

Yall are probably gonna think me a noob. But I am working on this eigenvector problem and I reduced the matrix to $x-y+z=0$ . How do I describe this solution set. I know how to do it if it's just ...
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1answer
26 views

Question regarding nullity and basis of a kernel

I was doing some exercises in a textbook and it required that I find the nullity of a linear transform. That I know how to do and you it by finding the kernel of that linear transform. It then asked ...
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116 views

Is there a way in matrix math notation to show the 'flip up-down', and 'flip left-right' of a matrix?

Title says it all - is there an accepted mathematical way in matrix notation to show those operations on a matrix? Thanks.
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1answer
23 views

Question regarding kernels and dimensions.

My professor gave us a question and I would like some help in it. The problem goes as follows: Let $T:P_3 \to \mathbb{R}$ such that: $T(a_0+a_1x+a_2x^2+a_3x^3)=a_0+a_1+a_2+a_3$ Find the rank and ...
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1answer
45 views

Skew-Symmetric after base change symmetric?

Are there invertible matrices $A,B \in \textrm{GL}(\mathbb{C}^3)$ such that for every skew-symmetric matrix $S \in \textrm{Mat}_{3 \times 3} (\mathbb{C})$ the matrix $A \cdot S \cdot B$ is symmetric? ...
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2answers
53 views

Computing bottom $k$-eigenspace of a matrix via top $k$-eigenspace of another matrix

Let $R$ be a full rank, symmetric matrix. Suppose one wants to compute the space spanned by the bottom $k$ eigenvectors of $R$. Of course one could compute the eigendecomposition of $R$ directly. My ...
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1answer
35 views

Proof the Similarity of Matrices

Suppose $A$ is a $3\times 3$ matrix with entries in a field $F$ of characteristic $0$, and assume $\operatorname{Tr}A = 6$, $\operatorname{Tr}(A^2)=14$, and $\det A = 6$. ($\operatorname{Tr}$ denotes ...
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Prove that if $ u \cdot v = u \cdot w $ then $v = w$

I've tried putting it up as: $$ [u_1 v_1 + \ldots + u_n v_n] = [u_1 w_1 + \ldots + u_n w_n] $$ But this doesn't make it immediately clear...I can't simply divide by $u_1 + \ldots + u_n$ as these ...
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1answer
644 views

Determining the ratios needed in gear reduction

I am trying to work out the math behind building a gear box for turning a gear a specific RPM from a small motor. Given that a typical DC hobby motor turning at 200 RPM, and a target in the final ...
5
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Decoding of Gabidulin code

Consider the space of matrices in $\mathbb{F}_q^{n \times m}$ where $\mathbb{F}_q$ is the finite field with $q$ elements. We can define a metric on this space, given by $d(A,B) := rank(A-B)$, called ...
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1answer
22 views

About a/the definition of plane.

Let $P$ be a point in 3-space and consider a located vector $ \overrightarrow {0N}$. We define the plane passing through $P$ perpendicular to $ \overrightarrow {0N}$ to be the collection of all ...
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2answers
47 views

Show $A$ hermitian $\iff v^tA\overline{v}\in\Bbb R$ for all $v \in \mathbb{C}^n$

Show that a matrix $A \in M(n \times n, \mathbb{C})$ is hermitian iff $v^tA\overline{v} \in \mathbb{R}$ for all $v \in \mathbb{C}^n$.
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Solving for eigenvectors of a $3\times3$ matrix

I got the eigenvalues which equal $4$, $-2$ and $-2$. The matrix is $$\begin{pmatrix}1 &-3& 3 \\3& -5 &3\\ 6 &-6 &4\end{pmatrix}$$ Now, usually, when I solve these, the ...
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1answer
17 views

How does $|iw|=3|1-w|$ go to $|w|=3|w-1|$?

Well the question speaks for itself really. I worked the problem down to $|iw|=3|1-w|$, where i is the imaginary number, but I don't understand how to get to the next step, which the mark scheme ...
2
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2answers
131 views

Proving that two right and left eigenvectors are not orthogonal.

Let $A$ be a square non-Hermitian matrix and $c$ be an eigenvalue of $A$ with algebraic multiplicity $1$. Let $Ax = cx$ and $y^{H}A = cy^{H}$ where $y^{H}$ is a conjugate transpose of $y$. Prove that ...
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1answer
48 views

Question regarding diagonalization and eigenvectors.

My professor gave us a question and I would like some help in it. The problem goes as follows: Let $T:\mathbb{R}^3\to\mathbb{R}^3$ be the linear transformation represented by: ...
3
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2answers
101 views

Eigenvalues of self-adjoint/orthogonal endomorphism $\pm 1$ proof

Let $f$ be simultaniously self-adjoint and orthogonal. Show that $\lambda = \pm1$ for all eigenvalues of $f$. Does this also apply for self-adjoint (respectively orthogonal or normal) endomorphisms?
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1answer
41 views

Question about $\infty$-norms

According to my textbook, a matrix $A \in \mathbb{C}^{n \times n}$ has $\infty$-norm equal to the maximum row sum of the matrix. Is there any way of gaining intuition for this fact?