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72.876 linear-algebra questions.

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If $A\in \mathbb{C}^{n\times n}$ with eigenvalues $(\mu_1,\ldots,\mu_n)$, is there anything we can say about the eigenvalues of $T = \Re(A)$; let's call them $(\lambda_1,\ldots, \lambda_n)$? ...

Prove that if $A$ has linearly dependent rows or columns then $\det(A)=0$. $$\DeclareMathOperator{\sgn}{sgn} \det(A)=\sum_{\sigma\in s_n}\Bigl(\sgn(\sigma)\prod_{i=1}^{n}a_{i,\sigma(i)}\Bigr)=\sum_{\...

The normal equation for weighted linear regression looks like this: $$J(\theta) = (X\theta - y)^TW(X\theta - y),$$ where $X\in\Re^{m\times n}$, $\theta\in\Re^{n\times n}$, $y\in\Re^{m\times 1}$, $W\...

If a pair of $3\times 3$ rotation matrices $R_A$ and $R_B$ satisfies $$R_B = R_C R_A R_C^T $$ where $R_C$ is an unknown $3\times 3$ rotation matrix. Can $R_C$ be uniquely determined by the congruent ...

Let $A,B$ be any square $n \times n$ matrices. Define $A \leq_{S^{n}} B$ if $A-B$ is negative semi-definite. Given a matrix $Q$, is it true that $Q \leq_{S^{n}} 0$ if and only if $Q \leq_{S^{n}} sI$ ...

Suppose $\|.\|_2$ is induced by vector 2 norm s.t. $\|A\|_2 = \max_{\|x\|_2=1}\|Ax\|_2$. Suppose $P and Q$ are matrices with orthonormal columns(not necessarily orthogonal though). Say $P\in R^{p\...

Given an invertible matrix $A \in \mathbb{R}^{n \times n}$ and its polar decomposition $$ A = U H, $$ where $U$ is unitary, and $H$ is positive definite. I'm interested in the distance of $$ arg ( \...

I'm struggling to get the following recurrence relation into a closed form if possible: $$f(m,0)=f(0,n)=2$$ $$f(0,0)=0$$ $$f(m,n)=f(m-1,n)\cdot(2n+1)(2n+2) + f(m,n-1)\cdot(2m+1)(2m+2)$$ where $m$ ...

Can the matrix transpose be represented by $X^T = AXB$ for a given $A$ and $B$? I think it is possible, but please correct me if I am wrong--please see my attempt below. My thinking (not sure ...

I'm struggling to prove/disprove this notion. I've figured if such matrix exists, it has to be nilpotent and non-invertible, and the sum of its eigenvalues is 0. Can anyone chip in? Edit : Oh yeah, $...

Given $$A = \begin{bmatrix} -9 & 4 & 4\\ -8 & 3 & 4 \\ -16 & 8 & 7 \end{bmatrix}$$ I calculated eigenvalues $\lambda= -1,-1,3$. The algebraic multiplicity (AM) and geometric ...

I am asked to prove that for a Hermitian matrix with all eigenvalues negative, the inverse is given by $$A^{-1} = - \int_0^\infty e^{tA} {\text { d}}t.$$ (I have the corrected the missing minus sign ...

The question is, Let $M$ be an $n\times n$ matrix with real entries such that $M^3=I$. Suppose that $Mv$$\neq v$, for any nonzero vector $v$. Then which of the following statements is/are true? (a) $...

In the book of Linear Algebra by Werner Greub, whenever we choose a field for our vector spaces, we always choose an arbitrary field $F$ of characteristic zero, but to understand the importance of the ...

I kind of see the reason why I need dot product in here, but I don't know how to use the dot product to help me figure out this question.

Specifically, I'm considering the function $$ V(\mathbf{X}) = -\ln \det \mathbf{X}\,. $$ Following an exercise, I've been able to show that $$ V(\mathbf{X}+\varepsilon\mathbf{Y}) = V(\mathbf{X})-\...

Let $f:\mathbb{R}^n\to\mathbb{R}^d$ be a $C^1$ function with the property that there is an $x_0\in\mathbb{R}^n$ such that on an open neighborhood of $x_0$, then rank of $Df(x)$ is a constant, say $j$. ...

We know the Pauli spin matrices (along with identity matrix) are a basis for the C2 vector space. All of them are self adjoint. However, since they are a basis for the entire C2 operator space, linear ...

Let $G$ be a group. A linear map $f:V\rightarrow W$ of $G$-graded vector spaces is said to be homogeneous of degree $g$ if $f(V_{h}) \subseteq W_{g\cdot h}$ for all $h\in G$. We denote the space all ...

Let W1 and W2 be subspaces of a vector space V such that V = W1 ⊕ W2. Prove that for every subspace V′ of V containing W1, one has V ′ = W1 ⊕ (V ' ∩ W2).

Suppose A is a symmetric 0-1 matrix. When is $A$ positive semidefinite?

Let $B \in \mathcal M(n \times n)$ be some fixed matrix and let $$S = \{A \in \mathcal M(n \times n; \mathbb C): \text{Re} \lambda_i(B-A) < 0 \text{ for } i = 1, \dots n\}.$$ Let us fix a matrix ...

A complex structure on $V := \mathbb{R}^2$ is a linear transformation $J : V\rightarrow V$ satisfying $J^2 = -1$. If $B(\cdot,\cdot)$ is an inner product on $V$, let $SO(V,B)$ be the subgroup of $GL(...

Here is question 10,section 6.6,Hoffman and Kunze: Let $F$ be a field of characteristic 0.Let $V$ be a finite dimensional vector space over $F$.Suppose that $E_1,..,E_k$ are projections of V such ...

Given that T is a cyclic operator on real vector space V with generating vector v, and minimal polnyomial $\mu_{T,v}(x) = x^3 - 8$. I see a theorem in my book stating that every T-invariant subspace ...

I've seen the term $x^T A x$ come up in a bunch of different areas of linear algebra, where A is a square and usually symmetric matrix. Places I've seen it include defining the Raleigh quotient, ...

Given $L$ a semisimple Lie algebra over an algebraically closed field of characteristic $0$ and given $V(0)$ the (standard cyclic) irreducible $L$-module of heighest weight $0$, then I want to show ...

I did the first part correctly, but when I computed $B=(X^T X)^{-1} X^T Y$, I got the wrong answer. I double checked my calculations on my calculator, all was fine. I think it's because of the unknown ...

I have read the questions here asking for a proof for $(W^{\perp})^{\perp} = W$ Here is the link to the question: in a finite dimensional vector space Now, I am wondering, would the same result hold ...

I know : "The columns of a matrix A are linearly independent if and only if the equation $$Ax=0$$ has only the trivial solution." What if the equation has a nontrivial solution. Does this imply that ...

Assume $(V,\langle \ , \ \rangle_V)$ and $(W,\langle \ , \ \rangle_W)$ are finite dimensional inner product spaces and $T : V \rightarrow W$ is an injective linear transformation. Prove that $T^*T : V ...

I've tried satisfying the logistic function with several vectors but have difficulty finding ones which are also orthogonal. the problem begins with me being given a neural unit weight vector w$$ w =...

If I have $T\colon \mathbb{C}^{3}\to\mathbb{C}^{2}$ and $S\colon \mathbb{C}^{2}\to\mathbb{C}^{3} $ which are both linear maps and we know that $\mathrm{rank}(ST)=2$. How would I show that $T$ is ...

Consider the set of skew-symmetric matrices $AS(n)=\{M \in M(n,n,\mathbb{R}): M^T=-M\}$. How to prove that, to the inner dot product $\left\langle .,. \right\rangle$ of $\mathbb{R^n}$, if $W$ is ...

I found this problem, with $f\in\text{End}_\Bbb{C}(V)$ such that ${f}^{m}= Id_V$ for some $m$. I know that $f$ is diagonalizable because it can't be nilpotent because of the hypothesis. The problem is ...

I’m trying to learn convex optimization watching Ryan Tibshirani lectures. I sometimes have a hard time understanding basic math in his lectures. Like sets and their properties or how you define a ...

I'm not really sure how to solve the following problem: Let $ \gamma : \mathbb{R^2}~ x~ \mathbb{R^2} \rightarrow \mathbb{R}~$ be given by the matrix $M= \left( {\begin{array}{cc} 1 & -3 \\...

Suppose that $A$ is an invertible matrix and let $X$ be such that $AX+XA=2I$. Does this imply that $X=A^{-1}$? I have tried simple algebra manipulations but I have not been able to conclude. For a ...

I have a question: If a matrix is given to be invertible (it for example has infinitely many solutions), are you allowed to assume that the determinant of the matrix is not equal to 0? I am not sure,...

When I first took linear algebra, we never learned about dual spaces. Today in lecture we discussed them and I understand what they are, but I don't really understand why we want to study them within ...

What is a complete exhaustive classification of all geometric transforms on the $\mathbb{R}^2$ plane obtained with: 2x2 matrices$$A = \pmatrix{a & b \\ c & d}$$ applied to a point $X= (x, y)$....

I have a Hermitian matrix of size $n$, $H_{n\times n}$ with only off diagonal entries, actually all the entries are real in $H$ (very special case). Now my job is to find the anti-commuting matrix say ...

In a first year engineering linear algebra class at my institution, the students learn about general linear transformations. I understand that when working in general, understanding the general ...

I need to calculate the dominant eigenvalue of the square matrix $\begin{pmatrix} 15 & -4 & -3\\ -10 & 12 & -6\\ -20 & 4 & -2 \end{pmatrix}$ and also the corresponding eigen ...

If A and B are two nxn matrices, how would I prove that rank(AB)$\leq$ rank(A)?

For all $k\in\mathbb{N}$, let $x_k,\ y_k\in\mathbb{R}^n$, and assume that \begin{equation} y_{k+1}=\left(I_n-\frac{ax_kx_k^{\rm T}}{b+\|x_k\|^2}\right)y_k,~~~~~~~~~~~~(1) \end{equation} where $a\...

I need to find a new base where X1X2+X2X3+X3X4 Is the sum of squares( diagonal matrix) Im trying with Lagrange method but without success (row and column actions also accepted) If anyone can ...

"For $n≥4$, let $U=\{X∈ M_n(\mathbb{C}) | \operatorname{adj}X=I\}$. Which of the following statements are true/false? Justify: $U$ contains $n-1$ elements, $U$ contains only scalar matrices, if $X_1,...

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