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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Solving for X with matrices

Solve for $X$ given that $A$, $B$, and $C$ are invertible matrices. $$(X-B)A=BC$$
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Reduce occurrence of $x$, retain definition at $x=0$

I need to apply a gamma curve to render an output variable $(x)$, to make better use of screen real estate, and it has me scratching my head with what is probably a simple math question. For $0 <= ...
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1answer
197 views

Row reduction of augmented matrix with unknowns

I've been stumped on this question for the past few days. The question asks that the following augmented matrix be row-reduced to a 'goal' matrix: \begin{matrix} 1 & 2 & -1 &| -3\\ 3 ...
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5answers
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Derivative of nuclear norm

I'm trying to take the derivative of nuclear norm with respect to its argument. nuclear norm is defined in the following way: $$\|x\|_*=\mathrm{tr}(\sqrt{x^Tx})$$ I'm trying to calculate: ...
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2answers
56 views

How is the derivative with respect to vector is taken in linear regression?

In the book I am studying the author motivates that the sum of the distances of data points to the fitted line can be written in matrix form as $$ (t-X\beta)^T(t-X\beta) $$ where X is a matrix that ...
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2answers
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Finding the dimension and basis of an orthogonal space

I am trying to find the basis and dimensions of the the space orthogonal $S$ which is in $\mathbb R^3$. $$S = \begin{bmatrix}1\\2\\3\end{bmatrix}$$ So the dimension would be two because it is $3 - ...
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1answer
75 views

Lattice in a vector space of dim 2 over a valuated field.

I'm reading "Arbres, amalgames et SL2" of J.P. Serre, and something is not clear to me, but is to him :) Let $k$ be a field, with a discrete valuation $v$, ie a group epimorphism $v:k^\ast \to ...
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0answers
117 views

Real and complex vector spaces

Suppose that $V$ is a real finite-dimensional vector space and let $V_\mathbb{C}=V\otimes_{\mathbb{R}}\mathbb{C}$ be its complexification. Now let $W\subset V_\mathbb{C}$ be a complex subspace. ...
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1answer
45 views

psittacism: Fundamental Theory of Time

This question is in reference to the programming question found here. What method of approach should I be thinking of if I have a list of lectures A, B, and C, and discussions D, E, and F, that are ...
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1answer
48 views

Question about Gram-Schmidt & orthonormal basis

Suppose $\alpha_1, \dots, \alpha_n$ are vectors of norm 1 in some $\mathbf R^d$. Let $\beta_1, \dots, \beta_n$ be the orthogonal basis vectors from the Gram-Schmidt process, i.e. $\beta_1 = \alpha_1, ...
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0answers
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If the operator is linear but diverges on the basis vectors, how to find its matrix?

If the operator is linear, defined as an expression involving series but diverges on the basis vectors, how to find its matrix?
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1answer
39 views

Finding collinear vector given length

Given vector A= (1, 7, -4) find a vector U that has length equal to 8 in the same direction as A. Write answer in form (u1, u2, u3). I know that length is equal to magnitude of a vector so 8 = ||U|| ...
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31 views

Showing a certain span of complex matrices closed under multiplication.

Let $T$ be a complex $n\times n$ matrix. Consider $M=\text{span}(T^k:1\leq k \leq n^2)$. I'm trying to show that $M$ is closed under multiplication. I've of course tried just writing up two standard ...
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1answer
22 views

Checking validity of eigenvectors

I've written a program that finds the first $K$ eigenvectors of a matrix and would like to figure out if my solutions are truly valid eigenvectors. What is a good way of doing this?
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2answers
329 views

Matrix Exponential of Identity Matrix

I was just wondering what would the sum be of $e^{I_n}$ where $I_n$ is the identity matrix. I know the maclaurin series for $e^x$ is $1+\frac x{1!}+\frac {x^2}{2!}+...$. I know that $e^0$ is 1 right? ...
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1answer
24 views

Parametrizing a contingency table

Consider a $3\times 3$ contingency table $N = (n_{ij})$ which can be regarded as a matrix in $\mathbb{N}^{3 \times 3}$, whose row-sums and column-sums are restricted to be $$ \sum_{j} n_{ij} = n_i, ...
2
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1answer
63 views

Subspaces, dimensions and images

Let $L$ be a four-dimensional subspace to the five-dimensional vector space $V$, and $ A : V \to W $. Let $ A(L) $ be the subspace of W consisting of the vectors $ A(\bar{v}) $ in $W$ for which $ ...
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0answers
24 views

Regression factors and covariance matrix

I am trying to follow some notes. They have two matrices. One is called comfact (company factors). This is a $580 \times 5$ matrix. The $580$ rows represent $580$ different companies. The $5$ ...
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0answers
157 views

Condition for a block matrix to be positive semi-definite

Let's say we have two positive semi-definite matrices $A$ and $B$ and a negative real $\lambda$. What would be the conditions for the matrix $M$, defined as follows, to be positive semi-definite too ? ...
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0answers
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Is my understanding of an annihilator correct?

This is how I understand the annihilator now, but I feel like it might be incorrect. So for some $U \subset V$, the annihilator of $U$ is all of the linear functionals $t(v)$ in $V'$, such that ...
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3answers
37 views

About the Scalar product

These are the lecture notes of my teacher and I am getting confused how he reached at $V_1$.$V_2$= Re($z_1$$z_2$). Can anyone help me to understand this.
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0answers
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Any way to simplify this expression?

So I have a vector of asset allocation weights given by $x \in R^4$ and a covariance matrix of the asset returns $\Sigma \in R^{4,4}$. I know by the spectral theorem, $\Sigma = V DV^{-1}$ and the ...
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2answers
54 views

Using a contradiction to show something is not compact

Consider the set of $3$ by $3$ matrices in $\mathbb{R}$ that have nonzero determinant. I want to prove that this is not compact preferably with a contradiction. Attempt: Since its a determinant it ...
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1answer
112 views

Left Eigenvectors vs. Right Eigenvectors

Suppose we have a matrix $A$ and a symmetric invertible matrix $D$ such that $DA$ is symmetric. The right eigenvectors of $A$ are $v_1,\cdots,v_n$ with eigenvalues $\lambda_1,\cdots, \lambda_n$. Can ...
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3answers
356 views

zero matrix to the power of 0

Why $0^0=I$? I'd tried prove that considering $N^0$ where N is a Nilpotent matrix and then using the Cayley -Hamilton theorem Thanks in advance.
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1answer
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find an invertible matrix $P$ such that $P^T AP$ is diagonal

Let $A = \left[ \begin{matrix} 2 & -1 & 1 \\ -1 & 3 & 0 \\ 1 & 0 & 0 \end{matrix} \right]$. Find an invertible matrix $P$ such that $P^T AP$ is diagonal my solve is ...
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2answers
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For all square matrices A and B of the same size, it is true that (A+B)^2 = A^2 + 2AB + B^2

The below statement is a true/false exercise. Statement: For all square matrices A and B of the same size, it is true that (A + B)2 = A^2 + 2AB + B^2. My thought process: Since it is not a proof, I ...
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1answer
62 views

Linear Algebra - Determine if a linear transformation is one-to-one

I have been faced with this question: $T:\mathbb R^3 \to\mathbb R^3$ defined by $T(X) = AX$ where $A = \begin{bmatrix}1&-1&0\\0&1&2\\2&-1&1\end{bmatrix}$ How do I tell if ...
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1answer
68 views

Linear Algebra - Understanding how to determine if a transformation is linear

I'm new to linear transformations in linear algebra and I can't quit understand how to find out if a transformation is linear. Any help would be much appreciated! a) $T:\mathbb R^3 \to\mathbb R^2$ ...
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1answer
61 views

Where can I find good examples about Algebra (but not only): Usual counter-examples, but also limit cases, rare ones, etc [duplicate]

I recently discovered the importance of examples and couter-examples in mathematics. Where could I find good examples books or anything related to it ? I am particularly looking for rare limit-cases, ...
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3answers
43 views

derivative of quadratic function without transposes

I'm trying to solve an equation of the following form: $$ \frac{\partial}{\partial X} A'XA'X $$ where $X$ and $A$ are both equal-length column vectors (and so that $A'XA'A$ is scalar). From looking ...
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3answers
74 views

Vandermonde determinant for order 4

I'd like to show the case $n=4$ for the Vandermonde-determinant. It should look like this: $V_4 := \det \begin{pmatrix} 1 & 1 & 1 & 1 \\ x_1 & x_2 & x_3 & x_4 \\ x_1^2 & ...
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1answer
48 views

Cauchy-Schwarz on real vector space: $| \langle u, v \rangle | = \| u \| \cdot \| v \| \iff u =\frac {\langle u, v \rangle} {\langle v, v \rangle} v$ [duplicate]

Cauchy-Schwarz on real vector space: $| \langle u, v \rangle | = \| u \| \cdot \| v \| \iff u =\frac {\langle u, v \rangle} {\langle v, v \rangle} v$ Proving that $u =\frac {\langle u, v ...
3
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1answer
88 views

Clever way to square a matrix

How do you square a matrix $A$? Do you use any clever way to do it (i.e not using the standard matrix multiplication)? It can be useful 'considering' $A$ like a linear application?
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1answer
52 views

Basic concepts about matrices and their decompositions

I am studying the basics of linear algebra and I have some questions that I can not conclude them by my own. Let $A \in \Bbb R^{m \times n} $ $A$ can always be expressed as a LU decomposition? ...
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0answers
37 views

Quick relatively sharp upper bound for the largest singular value of $m \times n$ matrix $X$

Is there anything analogous to the Gershgorin Circle Theorem but for the singular values of an $m \times n$ matrix $X$? I'm interested in a relatively sharp upper bound for the largest singular value ...
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1answer
22 views

Proof x \in L \leftrightarrow det(…) = 0.

I just need some help with the following proof: Let $v = (v_1,v_2) $and $ w=(w_1,w_2)$ be two points in $K^2 , v \not= w$ and $L \subseteq K^2 $ a line through these two points. Show that ...
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2answers
86 views

Rigorous proof that any linear map between vector spaces can be represented by a matrix

I searching the internet in hope of finding a proof. However, most of what I have seen this relationship is defined informally and/or gloss off this. Would you kindly point me in the direction of a ...
3
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2answers
80 views

Prove $A^{2}=0$ iff $C\left(A\right)=R^{0}\left(A\right) $

I'm learning some Linear Algebra through a University Textbook and I've come across this question which I have a hard time solving: Let There be a square Matrix A. Prove that $A^{2}=0$ iff ...
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1answer
99 views

How can I find a matrix $\bf B$, with positive eigenvalues, such that its square $\bf B^2$ is another matrix $\bf A$?

I've been given a 2x2 matrix $\mathbf A$, its eigenvalues $\lambda_1$ and $\lambda_2$, its eigenvectors $\mathbf v_1$ and $\mathbf v_2$, and a diagonal matrix $\mathbf D = \text{diag}(\lambda _1, ...
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1answer
35 views

Hermitian matrices and their eigenvalues

Let $C=A+B$ where $A$ and $B$ are two hermitian matrices can I prove that $\lambda_{i,C}=\lambda_{i,A}+\lambda_{i,B}$ iff $x_{i,A}=x_{i,B}$? Where $x_i$ is the eigenvector related to eigenvalue ...
2
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1answer
116 views

Does $GL(n,K)$ act transitively on $1$-dim subspaces of $K$

If we let $K$ be a field and $GL(n,K)$ act by right multiplication on the $1$-dim subspaces of $K^n$. Then if we take $\langle v_1 \rangle, \ldots \langle v_n \rangle \in K^n$ distinct and $\langle ...
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2answers
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If $T$ is a linear transformation on $V$,then which is true?

Let $V$ be a vector space with finite dimension $n$ and $T:V\longrightarrow V$ is a linear transformation such that $T^{2}=0$. Then $rank(T)\leq\frac{n}{2}$ $n(T)\leq\frac{n}{2}$ $rank(T)\geq n(T)$ ...
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0answers
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Eigenvalues of sum of two particular matrices

Let $A$ be a matrix with real eigenvalues, its maximum eigenvalue is $0$ and it has sum for rows equals to zero. Let $B$ be a matrix $\mathrm{diag}([1\,0\, ...\, 0])$ and let $I$ be the identity ...
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0answers
36 views

eigenvalues of a symmetric matrix

I diagonalized an arbitrary 3x3 symmetric matrix using two SO matrices at respective angles and the diagonal elements are nothing but eigenvalues. I separately found eigenvalues with the help of ...
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0answers
175 views

Homology out of Smith normal form: simultaneous or independent diagonalization?

Let $R$ be a PID and $R^m\overset{A}{\longrightarrow} R^n\overset{B}{\longrightarrow} R^o$ matrices with $BA=0$ and Smith normal forms ...
2
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1answer
65 views

general expression for isomorphism of tensor product

(I am still waiting for an answer to the following question. Thank you.) While I was reading postings relating to tensors, I came across the following explanation from Tensors as matrices vs. Tensors ...
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2answers
779 views

Find tangent vector to surface given a point on the surface and its normal vector (for a sphere)

I need to know how to find a tangent vector to a point on the surface of a sphere if I am given the point P and the normal vector at that point N. I know that there are many possible tangent vectors ...
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4answers
80 views

Linear maps using Tensor Product

While I was reading some posts (Definition of a tensor for a manifold, and Tensors as matrices vs. Tensors as multi-linear maps), I encountered the following explanation: "To give a linear map $V ...
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1answer
76 views

If A = BC and B is invertible, then how does reducing “B to I” also reduce “A to C”?

If $A = B*C$, where $B$ is an inverse, use row-ops to reduces "$B$ to $I$" also shows that it will reduce "$A$ .. $C$". Big-Hint: Represent the row operations by a sequence of elementary matrices.