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 ...

learn more… | top users | synonyms (1)

2
votes
1answer
38 views

Find the unit vector so that this condition is true.

Let $(X_1,X_2)$ be jointly normal with density $$\phi(x_1,x_2;\rho) = \frac{1}{2\pi\sqrt{1-\rho^2}}\exp\left(\frac{-1}{2\sqrt{1-\rho^2}}(x_1^2 - 2\rho x_1x_2 + x_2^2)\right)$$ Find unit vector ...
2
votes
1answer
63 views

Suppose that the $n$ Gersgorin discs of $A \in M_n$ are mutually disjoint. If $A$ is real, show that every eigenvalue of $A$ is real.

Suppose that the $n$ Gersgorin discs of $A \in M_n$ are mutually disjoint. (a) If $A$ is real, show that every eigenvalue of $A$ is real. (b) If $A \in M_n$ has real main diagonal entries and its ...
1
vote
1answer
61 views

How to demonstrate a set is a real vector space (set governed by nonstandard operations)

I am really not that familiar with questions that ask you to work with a operation vector space, even less with the English terms for it. I am... quite lost. How would you prove that it is a real ...
2
votes
1answer
38 views

Show that the intersection taken over the Gersgorin discs of all similar matrices of $A$ $=$ $\sigma (A)$

Show that $\bigcap_S G(S^{-1}AS)$ $=$ $\sigma (A)$; the intersection is taken over all nonsingular $S$, and $\sigma (A)$ is the spectrum of $A$. I'm lost as how to even begin to prove this fact. Any ...
0
votes
1answer
73 views

Find bases for orthogonal complement $S^\perp$ for the subspace $S$

I'm having a tough time understanding the textbook on how to answer this question? I'm not too sure what to do? Any help will be appreciated. $$ S=\operatorname{span}\left[ \begin{pmatrix} 1 \\ -3 ...
1
vote
0answers
61 views

Proving the existence of an invertible square matrix

Assume $A$ is a square matrix with real values. Show that there exist an invertible square matrix $B$ such that matrix $B^{-1}AB$ is block upper triangular with diagonal blocks either of size ...
0
votes
1answer
62 views

Every idempotent matrix is diagonalizable.

Show that every idempotent matrix is diagonalizable. What can you say if $A$ is tripotent ($A^3=A?$) What if $A^k=A?$ The first two cases is obvious since we can find the minimal polynomial to be ...
0
votes
0answers
64 views

Is the ratio trace problem convex?

I have a ratio trace problem described as follows: $\arg\max_{w} \text{trace}((w^tAw)\cdot \text{inv}(w^tBw))$, where A and B are full rank matrices. This problem can be solved via generalized ...
2
votes
1answer
98 views

Relationship between similarity and having the same minimal polynomial

Let $A$, $B$ $\in M_3$ be nilpotent, where $M_3$ is the set of all complex 3by3 matrices. Show that $A$ and $B$ are similar if and only if $A$ and $B$ have the same minimal polynomial. Is this true in ...
1
vote
1answer
42 views

Prove: the sum of simultaneously diagonalizable transformations is diagonalizable

Let $T, S$, linear transformations which are simultaneously diagonalizable. Prove that $T+S$ is diagonalizable. I need to rely on the the definition: $T,S$ are called simultaneously ...
1
vote
2answers
52 views

Proof that the kernel of an endomorphism to the power $n$ is a subset of the kernel of the endomorphism to the power $n+1$

I am expected to know how to prove the following but I can't seem to draw it out. Knowing that V is a Vector Space$$ T:V\to V $$ Prove the following $$ Ker(T^n)\subseteq Ker(T^{n+1}) $$ How ...
4
votes
2answers
205 views

Eigenvalues of a unimodular matrix

Let $U$ be a unimodular matrix, i.e. $U \in \mathbb{Z}^{n \times n}$, and $\text{det}(U) = \pm 1$. Do the real (or complex for that matter) eigenvalues of $U$ admit a special structure? Edit: It is ...
4
votes
2answers
117 views

Find Jordan form of a $3\times 3$ matrix

$$\left( \begin{array}{ccc} 0 & 1 & 2 \\ -5 &-3 & -7 \\ 1 & 0 & 0 \end{array} \right) $$ I figured out the eigenvalues are all -1 from the characteristic polynomial, but I'm ...
2
votes
0answers
170 views

interpertation of the determinant of an X'X product matrix (D-optimal design application)

As the title suggests, I have been looking into an application of D-optimal design. I read this thread What does it mean to have a determinant equal to zero? and found some of the answers ...
0
votes
1answer
35 views

Finding eigenvvalue and eigenspace

I am given a matrix $A= \bigg({} \matrix{10 & 7 \\-14 &-11} \bigg{)}$ and eigenvalue $3$. My elite mission is to find the treacherous basis for the eigenspace. I used the $(A -eI)=v$ where ...
1
vote
1answer
34 views

Using trigonometric identity to compute an inner product through an integral to form an orthogonal sequence of functions.

Consider the inner product space: $(C(0,L),\langle \cdot,\cdot \rangle),$ where: $\langle f,g \rangle = \displaystyle\int_0^L f(x)g(x)\, dx$. Use the trigonometric identity: $\sin(u)\sin(v) = ...
4
votes
2answers
42 views

Theorem with seemingly reduntant part

I encountered the following theorem in a linear algebra book: For any vectors $u, v$ in $R^n$ and any scalar $k$ in $R$: $u . u \geq 0$, and $u . u = 0 \iff u = 0$ I found the theorem in almost the ...
0
votes
2answers
26 views

Can I just use the following notation when proving a set is a vector space?

If given all functions of form $$f(x) = a + b \cos(x) + c \sin(x),$$ where $a,b,c$ are real numbers, would it be sufficient to use the notation "$f(x)$," when proving that the axioms hold and that ...
2
votes
2answers
45 views

Question about diagonalization and projections

Let a finite dimensional vector space $V$ above $\mathbb{F}$. Let $T:V\to V$ a diagonlizable transformation. We denote $a_1 \ldots a_r$ the $r$ different eigenvalues of $T$. By diagonalization, we ...
1
vote
1answer
21 views

Projection of vectors

Compute $:$ $proj_\vec y (\vec x)$ $\vec{x}_1=\begin{bmatrix} 2 \\ 3 \\ 4 \\ 5 \end{bmatrix}, \vec{y}_2 = \begin{bmatrix} 1 \\ 0 \\ -1 \\ 0 \end{bmatrix}$ Since the projection would be $:$ $(-2/0) * ...
1
vote
1answer
35 views

Solving linear differential equations system

Upon trying to solve this particular system , I've encountered a few problems. $$ y'=5y+4z $$ $$z'=-4y-3z$$ After solving for eigenvalues the quadratic yielded a double root at $\lambda=1$ . But I ...
3
votes
2answers
70 views

This answer is confusing $4\times 4$ eigenvalue calculation

Question: Find the rank and the four eigenvalues of the following matrix: $\begin{bmatrix} 1 & 0 & 1 & 0\\ 0 & 1 & 0 & 1\\ 1 & 0 & 1 & 0\\ 0 & 1 & ...
3
votes
1answer
108 views

Two questions about diagonalization

Let A = $\begin{bmatrix}1 & 1 & 4\\0 & 3 & -4\\0&0&-1\end{bmatrix}$. Is the matrix A diagonalizable? If so find a matrix P that diagonalizes A. Can you write A as a linear ...
1
vote
1answer
64 views

Applying Gram-Schmidt process to a set of vectors to find first three polynomials orthogonal with respect to inner product

$.$ $\langle f, g \rangle = \displaystyle\int_{-1}^{1} f(x)g(x)dx$ Apply Gram-Shmidt process to the set of vectors $:$ {1, x, $x^2$, ...} to find the first three polynomials orthogonal with respect ...
2
votes
2answers
78 views

How to prove that a set is a vector space

How does one, formally, prove that something is a vector space. Take the following classic example: set of all functions of form $f(x) = a_0 + a_1x + a_2x^2$, where $a_i \in \mathbb{R}$. Prove ...
1
vote
1answer
79 views

Quadratic form - vector/matrix

I have two very simple (stupid) questions about quadratic forms. Having any matrices $A,B$ and vectors $x,y$ (real/complex, singular/regular, rectangular, infinite size, etc.) with appropriate size ...
1
vote
2answers
35 views

Inner prodcut space for complex numbers including complex conjugation

$..$ Consider inner product space : $(C, \langle \cdot,\cdot\rangle)$: where for complex numbers $..$ $\langle z_1, z_2 \rangle = \sqrt(z_1 *\overline{z_2}$) Computing $..$ $\langle 2-3i, 2-3i ...
1
vote
1answer
77 views

ordered partition, block matrix given by $r_j \times r_j$ nilpotent Jordan blocks is nilpotent, rational canonical form, jordan canonical form

Let $F$ be a field. For an integer $n \ge 1$, and ordered partition of $n$ is a sequence $\underline{r} = \{r_1, \dots, r_m\}$ of positive integers such that $r_1 \le \dots \le r_m$ and $\sum r_j = ...
4
votes
2answers
71 views

Some questions about the Eigenvalues of this $4\times 4$ matrix

\begin{bmatrix} 1 & 1 & 1 & 1\\ 1 & 1 & 1 & 1\\ 1 & 1 & 1 & 1\\ 1 & 1 & 1 & 1 \end{bmatrix} The rank is $1$ as there is only $1$ linearly ...
3
votes
2answers
75 views

$A$ be a $10*10$ matrix with complex entries s.t. all eigenvalues are non negative real and at least one eigenvalue is positive.

Let $A$ be a $10*10$ matrix with complex entries s.t. all eigenvalues are non negative real and at least one eigenvalue is positive. Then which of the following statements is always false? A. ...
1
vote
2answers
39 views

Computing inner products with linearity in the first argument.

Consider the vector space $..$ $(\mathbb{P},\langle \cdot,\cdot \rangle)$ where the inner product is given by: $$$$ $\langle p(x),q(x) \rangle = \displaystyle\int_{-\infty}^{\infty} p(x)q(x)e^{-x^2} ...
1
vote
2answers
31 views

Question in regards to definition: finite dimensional

Do we denote a vector space as finite dimensional IF it has a basis, or do we say that it is finite dimensional if it's associated through an isomorphic transformation with a "number space", ie. ...
0
votes
1answer
48 views

Proof of this theorem: $ tr(A^-)=\sum_{i=1}^r \lambda_i^{-1} $

If $A$ is an $n\times n$ symmetric matrix with $r$ nonzero characteristic roots $ \lambda_1,\lambda_2,...,\lambda_r$ then $$ {\rm tr}(A^-)=\sum_{i=1}^r \lambda_i^{-1}. $$ Note: $A^-$ is generalized ...
0
votes
1answer
50 views

Trace distance between “weighted” Hermitian matrices

The trace norm for a matrix $\mathbf{A}$ (also known as Shatten 1-norm) is defined as follows: $\|\mathbf{A}\|_1=\operatorname{trace}[\sqrt{\mathbf{A}^*\mathbf{A}}]$. It yields a useful distance ...
0
votes
2answers
96 views

what is the convex hull of such a matrix cone?

A matrix cone is in the following form: $M: = \begin{pmatrix} 1 \\ x\end{pmatrix}\begin{pmatrix}1 & x^T\end{pmatrix}$ where $x\in F$ , let $F = \{x: x\in [l,u]^n\}$ How to express the convex ...
8
votes
4answers
769 views

A faster way of calculating this determinant?

I'm doing a problem involving Cramer's rule, and one of the determinants I have to work with is as follows: \begin{vmatrix} 1&1&1\\ a&b&c\\ a^3&b^3&c^3 \end{vmatrix} So I ...
0
votes
2answers
44 views

How to show a set spans a space?

I've just started working with abstract algebra, and while the theory makes some sense, I have a bit of trouble figuring out the actual methods to complement the theory. For example, a base for a ...
3
votes
1answer
76 views

Order of $\mathrm{GL}_n(\mathbb F_p)$ for $p$ prime [duplicate]

While proving some facts about matrix group operations on finite fields, I stumbled across the following question: What is the order of the group of invertible $n\times n$ matrices over a ...
0
votes
0answers
616 views

How to show a set of vectors does not span a vector space?

Let's say I am given a $4\times4$ matrix and I am to determine whether the columns of that matrix span $\mathbb R^4$. Please tell me if I'm correct: One way to determine that is to calculate the ...
2
votes
1answer
159 views

trace inequality of positive definite matrices.

Assume $A,B \in M_n(\Bbb{R})$ are positive definite matrices, show that $$\text{Tr}(AB)\leq \text{Tr}(A)\text{Tr}(B) $$ I only prove it for $n=2$, it is straightforward calculate.but when $n \geq ...
0
votes
0answers
37 views

Maps from 3-D to 2-D

What is the name for a function which maps from a Euclidian 3D space to a 2D image? Basically imagine you have some 3D modeling software. I am talking about the function that maps from the 3D model, ...
9
votes
4answers
559 views

covariant and contravariant components and change of basis

I encountered the following in reading about covariant and contravariant: In those discussions, you may see words to the effect that covariant components transform in the same way as basis ...
2
votes
1answer
61 views

Polynomial functions/basis

If I suppose $R \subset F$ and have polynomial functions $p_{k,j} : F \to F$ by $p_{1,0}(x)=(x-2)^3$ $p_{2,0}(x)=(x-1)$ $p_{2,1}(x)=(x-1)(x-2)$ $p_{2,2}(x)=(x-1)(x-2)^2$ and the polynomial ...
4
votes
1answer
94 views

$T:V\to W$, both has same basis

Suppose $W,V$ have the same basis $\{u_1,u_2\}$ and that $T:V \to W$ is a linear transformation. Give an example (not the identity transformation) of a) $T = T^{-1}$ b) $T= T^2$ for a) $T=T^{-1} ...
2
votes
0answers
33 views

Is there necessary and sufficient condition on whether a symmetric circulant matrix is non-singular?

Is there necessary and sufficient condition on whether a symmetric circulant matrix is non-singular? I found many example supporting that positive symmetric circulant matrices which has at least two ...
1
vote
1answer
49 views

Is $x^3$ in the null space of the transformation $p(x) \mapsto xp(x)$?

Let $h: P_3 \to P_4$ be given by $p(x) \mapsto xp(x)$. Is $x^3$ in the null space ? Or is it in the range space ? Also, I am having difficulty finding the null space and the range of this map, can ...
2
votes
2answers
63 views

Show that V=W1⊕W2

Let $L_1$, $L_2$ be linear operators where $$L_1=L_1^2, \quad L_2=L_2^2 \tag{a}$$ $$L_1L_2=0, \quad L_2L_1=0 \tag{b}$$ $$I=L1+L2\tag{c}$$ Show that $V=W_1 \oplus W_2$. ($W_1$ should be the range of ...
1
vote
2answers
80 views

Is this function continuous? (vector function)

Assume you have $k$ vectors: $\{v_1,\dots,v_k\}$ in $\mathbb{R}^n$, and $\lambda\in\mathbb{R}^k$. Look at the function: $F\colon\mathbb{R}^k\rightarrow \mathbb{R}^n$ where ...
2
votes
1answer
38 views

If a linear transformation $T$ has $z^n$ as the minimal polynomial, there is a vector $v$ such that $v, Tv,\dots, T^{n-1}v$ are linearly independent

Let $T: V \to V$ with a minimal polynomial $z^n$, prove there's a vector $v$ such that $v, Tv, T^2v, ..., T^{n-1}v$ are linearly independent? The way I did it orginally was not allowed. No Jordan ...
0
votes
0answers
45 views

Adjoint linear operators and inner products question; why does $\langle T(x),T(x)\rangle =\langle T^*T(x),x\rangle $?

I have seen this multiple times in my textbook; $\langle T(x),T(x)\rangle=\langle T^*T(x),x\rangle$; why is this true? I know the definition of adjoint is if $\langle x,T(y)\rangle=\langle ...