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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System of equations with a unique solution, no solution or an infinite number of solutions

I was doing a past OCR Further Pure 1 Paper from January 2011, but came across the following question that I could not solve, even with the help of the mark scheme: Determine whether the ...
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35 views

Upper Triangular Matrices

Consider the set $V$ of upper-triangular $n\times n$ matrices with elements in some field $K$. I.e., if $A$ is such a matrix, $a_{ij} = 0$, for $i > j$ Show that non-degenerate upper-triangular ...
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QR decomposition: Same results for Classical Gram-Schmidt and Modified Gram-Schmidt

I am implementing QR decomposition (in Fortran) for a complex-valued matrix, using Classical Gram-Schmidt and Modified Gram-Schmidt (and Householder). I was expecting that the Classical Gram-Schmidt ...
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37 views

Bilinear transformation and eigenvalues

I have a proof to do and I am stuck on proving that if there exist a matrix $A$ with eigenvalue $\lambda_i$ and $B$ with eigenvalues $\mu$ such that $A = (B+I)(B-I)^{-1}$ then we have ...
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I don't understand how to define a linear transformation

I don't know how he defined the image. And letter C, I don't know what it's asking me to do. With those two I can solve the rest of the exercises.
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How do I calculate the new matrix from the new basis?

$\bf(1)$ Let $\mathbb{R}^{2\times2}$ denote the vector space of all $2\times2$ matrices with real number entries. Set $A=\pmatrix{1&2\\-2&-4}$. Define a linear transformation ...
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solve linear system using gaussian elimination

I want to solve a linear system of the form Ax=b. First of all I create the augmented matrix (A|b). I apply some elementary row operations and i obtain the REF form of A. After than, I do not know ...
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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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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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32 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
42 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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Cannot use alternative definition of “nowhere dense” to show space of real sequences with only a finite # of nonzero terms is NOT complete?

Suppose that I define my space $V$ to the the space of real sequences with only a finite number of nonzero terms. Then, I define $V_n = (a_1,a_2,\ldots,a_n,0,0,\ldots)$. Then, it is that $V$ has a ...
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40 views

Cauchy Schwarz inequality with an operator

The standard Cauchy-Schwarz inequality is given by, $|\langle\Phi|\Psi\rangle|^2\le\langle\Phi|\Phi\rangle\langle\Psi|\Psi\rangle$ But now I'm intressted in what happens to ...
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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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2answers
35 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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2answers
67 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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1answer
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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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38 views

Eigenvalues of the linear operator $T^*T$

Let $T$ be a linear operator $T: H_1 \mapsto H_2$, where $H_1$ and $H_2$ are both Hilbert Spaces. Suppose further that $T$ is bounded, but not self adjoint. Suppose I also know that for functions ...
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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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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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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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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
108 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
60 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
239 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 $X=A+\dfrac{B}{2},Y=-\dfrac{B}{2}$ ...
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1answer
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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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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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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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1answer
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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
88 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
146 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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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
40 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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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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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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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
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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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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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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 ...
2
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1answer
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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
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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 ...