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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Linear Algebra - Clarifying the meaning of a lemma related to linear maps

I am reading through Jim Hefferon's "Linear Algebra" and have encountered a lemma which I have had some difficulty in understanding. Through reading various other resources, I believe I currently ...
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Change of Basis of Polynoms

Let $B=\{1,x,x^2\}$ and $B'=\{1/2(x-1)(x-2),-x(x-2),1/2x(x-1)\}$. Find the transition matrix from B to B' and the transition matrix from B' to B. I was able to find the transition matrix from B' to ...
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Proof involving eigenvectors/values of a linear map and polynomials.

Let $V$ be a vector space over a field $k$, and let $T:V\rightarrow V$ be linear, and let $f\in k[x]$. Suppose that $\lambda\in k$ is an eigenvalue of $T$ and let $v\in V$ be a corresponding ...
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2answers
109 views

Determinant of a $n\times n$ Matrix

Let $A = (a_{ij}) \in R^{n\times n}$. Find the determinant if: $$a_{ij}= |i-j|$$ So we have the symmetric matrix \begin{bmatrix} 0 & 1 & 2 & 3 & 4 & \dots & n-1 \\ 1 & 0 ...
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Simple definition of a positive definite matrix

Aside from what it is written in the books about the definition of a positive definite matrix, can anyone explain it in a more earthly method?
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If U is a suspace of a real vector space V is V\U a subpace of V?

The title says it all. I started off arguing that although if V is a vector space then by definition it is closed under addition and scalar multiplication. But, since the zero vector is in U (again ...
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3answers
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Finding minimum polynomial of $e^{i\pi/6 }$

Finding minimum polynomial of $e^{i\pi/6 }$: I know it satisfies $t^6 +1 = 0$. I factorized $t^6+1 = (t^2+1)(t^4-t^2+1)$ obviously it does not satisfy $t^2 +1$ so it must satisfy $t^4-t^2 +1$. How ...
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reasoning question - Two candles are of different lengths and thicknesses

Two candles are of different lengths and thicknesses. The short and the long ones can burn respectively for 3.5 hour and 5 hours. After burning for 2 hour, the lengths of the candles become equal in ...
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Find the Eigenvectors of $\begin{pmatrix}1&-P\\P &-Q\end{pmatrix}$

I want to find the eigenvectors of this matrix: $\begin{pmatrix}1&-P\\P &-Q\end{pmatrix}$ First I found the eigenvalues: $det(A-\lambda_1)$ ...
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1answer
46 views

A canonical isomorphism of Lie algebras

Let $g$ be a Lie algebra/$\mathbb{C}$. I would like to investigate the existence of a canonical Lie algebra isomorphism/$\mathbb{C}$ of the form $g\otimes_{\mathbb{R}}\mathbb{C}\rightarrow g\oplus g$. ...
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How to infer $f$ from this operator: $g(k\nabla f - \nabla,.)$ where $g$ is the Euclidean metric, $k > 0.$

I have the following operator: $g(k\nabla f - \nabla,.)$ where $g$ is the Euclidean metric, $k > 0$, and $f$ is unknown. It acts on vectors in $\mathbb{R}^n$. What kind of informations can I obtain ...
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Is $v^* H w= h^T (w \otimes (v^*)^T)$ in this specific case?

Let $v$ and $w$ be an $n \times 1$ and $m \times 1$ unit norm vectors, respectively. Also, let $H$ an $n \times m$ matrix. We denote by vectors $a^*$ and $a^T$ the conjugate transpose and transpose of ...
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Finding a basis of $\mathbb R^{4}$ containing specific vectors. How can different standard basis vectors can be added, where both result in a basis?

An exercise from my textbook asks me to find a basis of $\mathbb R^{4}$ containing $S = (u,v)$, where $u = (0,1,2,3), v = (2,-1,0,1)$. The method they describe involves adding vectors from the ...
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Isomorphism Between V and V** [duplicate]

I have seen similar questions on s.e. but I really can't seem to understand the proofs given. So the question is the following: Let $V$ be a finite-dimensional vector space. show that there is an ...
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6answers
52 views

The multiplicative conjugate of an invertible matrix is invertible

If $A,B,C$ are $n \times n$ (real) matrices and $A$ and $B$ are invertible, with $AB=BC$, prove that $C$ is also invertible. My attempted proof is $(B^{-1})(AB) = (B^{-1})(BC)$. Then ...
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Finding the eigenvector of a sparse stochastic matrix

I would like to ask whether there exists an analytical, or fast numerical, solution for finding the eigenvector, $\bf\pi$, associated with the eigenvalue $\lambda = 1$ of the Algebraic equation: ...
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1answer
17 views

What is the standard matrix of $T^{101}$?

Be $R$ the line from the equation $y=-x$ in $\mathbb{R}^{2}$ and be $T:\mathbb{R}^{2}\rightarrow\mathbb{R}^{2}$ the reflection respect to the line $R$: $T\begin{pmatrix} 1\\ -1 ...
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79 views

Calculating derivative of a special matrix exponential

I am trying to calculate the following derivative: Let $V,W$ be fixed $n \times n$ real matrices. Define $A(t)=e^{V+tW}$. What is $A'(0)$? I do not assume $V,W$ commute. (If they commute, it becomes ...
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58 views

Find two vectors orthogonal to $v = [1,2,0]$

Find two vectors orthogonal to $v = [1,2,0]$ Usually i see questions with asking you two find given two vectors find two orthogonal vectors for it. Then you would use cross product and then use the ...
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Let $u_1=[3,−1,2]$ and $u_2=[2,0,1]$. Find all vectors $v=[x,y,z]$such that $u_1⋅v=0$ and $u_2⋅v=0$

Let $u_1=[3,−1,2]$ and $u_2=[2,0,1]$. Find all vectors $v=[x,y,z]$such that $u_1⋅v=0$ and $u_2⋅v=0$ So for this question i would generally just do the dot product between the two and get the answer ...
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1answer
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How to express adj(A) in terms of A if A is singular?

I need to find out how to 'simplify' adj(AB) and adj(adj(A)) when A is singular? I have a hint to consider $A+\lambda I$. I tried calculating the adjugate of $A+\lambda I$ but this doesn't seem to ...
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How to normalize an eigenvector when it has $\sqrt{i}$ as an entry?

If $V=\mathbb{C}^2$ and $T(a,b)=(2a+ib,a+2b)$ I found $$[T]_\beta=\begin{bmatrix}2 & i \\ 1 & 2\end{bmatrix}$$ hence the eigenvalues are $2+\sqrt{i},2-\sqrt{i}$. So using this, eigenvectors ...
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107 views

Complete Pivoting VS Partial Pivoting in Gauss Elimination

I have a hard time understanding that when and under what conditions we can use Gauss elimination with complete pivoting, and when with partial pivoting, and when with no pivoting? (I mean what is the ...
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Is this true that ${\left| {{{(Ax)}^*}(By)} \right|^2} \le {(Ax)^*}(Ax).{(By)^*}(By)$?

Let $A,B\in M_n$ and $x\in C^n$. Is this true that ${\left| {{{(Ax)}^*}(By)} \right|^2} \le {(Ax)^*}(Ax).{(By)^*}(By)$?
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What does it mean “Distance between $k$-planes induced by the identification plane-projection matrix”?

I'm reading some parts of Functions of bounded variation and free discontinuity problems by Ambrosio, Fusco, Pallara. At the very beginning of page 82 there's written "Let $G_k$ be the complete ...
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38 views

Find the closest vector to an other in 2-norm

Find the vector $x$ in $S$ which is the closest in the 2-norm to the vector $c=[1,1,1,1]^T$. I have a projection matrix that looks like this: \begin{bmatrix} 2 & -1 & 1 & 0 \\ -1 ...
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Norm of matrix $M= u \otimes v^*$ if $u$ and $v$ are unit norm vectors

Let $u$ and $v$ be an $n \times 1$ and $m \times 1$ unit norm (L-2 norm) vectors, respectively. Let us define matrix $M$ (of dimension $n \times m$) as the kronecker product of $u$ and $v^*$ $$M= u ...
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4answers
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What exactly is a “vector-space structure?” (Linear Alg)

I am supplementing my linear algebra book with wikipedia, and I came across an interesting term that isnt mentioned precisely by that name in my textbook. The full sentence is, "Similarly as in the ...
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Connection between maximizing a quadratic form and maximal variance

In order to find the "directions of maximum variance" of $X$ one finds the eigen decomposition of the variance covariance matrix $X^tX$. I have seen the eigenvector problem cast as maximizing the ...
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1answer
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If $ A ∈ M_n$ be positive definite then $(x^∗ Ax)(y^∗A^{−1} y) ≥ |x^∗ y|^2$

Let $ A ∈ M_n$ be positive definite, and let $x, y ∈ \mathbb{C}^n$. Why does $(x^∗ Ax)(y^∗A^{−1} y) ≥ |x^∗ y|^2$?
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Prove that $S(2,\mathbb{R})$ is a vector subspace of size $2\times2$

I have the following problem statement: Be $M(2,\mathbb{R})$ the vector space of $2\times2$ matrices with real coefficients. Remember that square matrix $A$ is symmetric if $A^{t}=A$, and it is ...
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If $A\succeq{B}\succeq0$ and $C\succeq{D}\succeq0$ then $A \circ C\succeq B \circ D \succeq0$

Let $A, B,C,D \in M_n$ are Hermitian. We define $A \circ B = [{a_{ij}}{b_{ij}}]$. $A\succeq{B}$ Iff $A-B$ positive semidefinite. suppose $A\succeq{B}\succeq0$ and $C\succeq{D}\succeq0$. Why does ...
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Is there a solution to $Tx=y$ such that $\|x\|$ is minimal?

I recently used the method of least norms to solve an underdetermined system of linear equations for a problem at work. This got me thinking, if I were to think about this more generally, does such a ...
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1answer
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lower bound for the condition number

I have shown that if we have an invertible matrix $A \in \mathcal{M}_{N}(\mathbb{R})$ and $C \in \mathcal{M}_{N}(\mathbb{R})$ such that $A+C$ is singular then $cond(A) \geq \frac{\mid \mid A \mid ...
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2answers
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Determinant of multiplication of two nonsquare matrices

Suppose $A$ and $B$ are $n\times m$ and $m \times n$ matrices, respectively, where $n<m$. The determinant of the product of two rectangular matrices can be obtained by the "Cauchy–Binet formula". ...
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Relation between AB and BA

Let $A$ and B be $n\times n$ matrices over a field $F$. Then prove or disprove $(i)\, AB$ and $BA$ have same characteristic values. $(ii)\, AB$ and $BA $ have same characteristic polynomial. ...
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69 views

Linear Transformation Geometrically interpretation

Describe the following linear transformations geometrically: $L( u_1, u_2)=(-u_1 ,u_2)$ I don't know how to represent Mathematically. I just know how to prove it a linear transformation and have no ...
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The normal formal of integer matrix

We know that a complex matrix is conjugeted to the Jordan form, and for a real matrix we have a similar result. Now I want to know the situation for the integer matrix. More precisely, for a matrix ...
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Calculate $\mathbb{E}\{ H X H^*\}$

Let $X$ be an $n \times n$ matrix. Also, let $H$ be an $m \times n$ matrix with i.i.d. elements. My aim is to calculate the expectation $$\mathbb{E}\{ H X H^*\},$$ where $H^*$ denotes the the ...
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Linear transforming on cyclic space.

Let $T:W \to W$ be a linear transformation on a finite dimensional vector space $W$ over a field $F$. We say that $W$ is a cyclic space for $T$ if $W=\operatorname{span} \{v,TV,T^2v,...,T^kv,...\}$ ...
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Proof $\det(T - \lambda I_v) = \det([T]_{\beta} - \lambda I) $

Let $T: V \rightarrow V$ where $\beta$ is a basis for $V$ and $V$ is finite dimensional. Proof $\det(T - \lambda I_v) = \det([T]_{\beta} - \lambda I) $ I claimed that $T$ and $[T]_\beta$ are ...
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Show that for two orthogonal bases $V_1, V_2$ of the same subspacce $M$ of $C^n$, we have $V_1 V_1^T x = V_2 V_2^T x $

Show that for two orthonormal bases $V_1, V_2$ of the same subspacce $M$ of $C^n$, we have $V_1 V_1^T x = V_2 V_2^T x, \forall x$. This is what I have done so far: $V_1V_1^Tx=V_2V_2^Tx \rightarrow ...
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Bases for image and kernel of a linear map

Let $T$ be the linear map $\mathbb R^4 \rightarrow \mathbb R^3$ given by $(w, x, y, z)$$\rightarrow$ $(x + 2y + z, z − w, 2x + 4y + 2w)$. Give bases for its image and kernel.
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Linear transformation change of basis

Let $T:\mathbb{R}^3\to\mathbb{R}^3$ by $T(x,y,z)=(5x+3y-3z,-y,6x+3y-4z)$. $1$. Let $w_1=(2,0,1)$ and $w_2=Tw_1$. Show that $W=span\{w_1,w_2\}$ is a $2$-dimensional subspace of $\mathbb{R}^3$. $2$. ...
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Eigenvalue properties

Suppose that $A$ is a $3 \times 3$ matrix with eigenvalues $\lambda_1$ = 1, $\lambda_2$ = 2, and $\lambda_3$ = 3. a - Find the eigenvalues of $B = A^2 − 2A + I$. (I is the $3\times 3$ identity ...
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1answer
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Do $A$ and $A^{2}$ share eigenvectors if both are real and symmetric?

If $A$ and $A^{2}$ is a real symmetric matrix and $\overrightarrow{\lambda}$ is an eigenvector of $A^{2}$ with eigenvalue $\lambda$ then does that imply that $\overrightarrow{\lambda}$ is an ...
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32 views

Calculating polar decomposition

How do I compute the left and right polar decompositions of a matrix by hand? I understand the definition of the decomposition but not how it is calculated. For instance are the left and right polar ...
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2answers
61 views

Matrix diagonalizable over $\Bbb C$ but not invertible?

Is there a matrix that is diagonalizable over $\Bbb C$ but not invertible? I can only think of a matrix that is diagonalizable over $\Bbb R$ but not invertible.
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68 views

Let $U$ be a unitary matrix, show that $r(x,y) := x^*Uy$ is an inner product.

Let $U$ be a unitary matrix, show that $r(x,y) := x^*Uy$ is an inner product satisfying $(u,v) = \overline{(v,u)}$ $(u,u)> 0$ for $u\neq0$; $(u,u)=0$ for $u= 0$ $(u+sv,w)=(u,w)+s(v,w)$ for a ...