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

6
votes
2answers
71 views

Finding the number of symmetric,positive definite $10 \times 10$ matrices having…

I was looking at old exam papers and I was stuck with the following problem: What is the number of symmetric,positive definite $10 \times 10$ matrices having trace equal to $10$ and determinant ...
1
vote
1answer
44 views

How can I construct a matrix?

Construct a matrix whose one eigenvector is $(1,-1,1,-1)$. Here only one eigenvector is given and I need to construct a matrix with this. I don't know how to proceed. Please help.
1
vote
2answers
39 views

Find possible number of hours put in by the three workers

I'm having problems solving the following problem: Jan hires three types of laborers (I, II, III) and pays them \$8, \$6, and \$5 per hour, respectively. If the total amount paid is \$2,800 for a ...
1
vote
1answer
136 views

Hermitian Inner Product | Basis | Orthogonal Complement

If I say $X = \{x, x'\}\subset\mathbb{F}^3$ is a subspace, where $x$ and $x'$ are linearly independent (for some field $\mathbb{R}$ or $\mathbb{C}$), with $$\mathbb{F}^n := ...
3
votes
1answer
88 views

Hamiltonian Quaternions: A Call for Counterexample for ($AB=I_m \implies n≥m$)

How can I build a counterexample to $AB=I_m \implies n≥m$ in the ring of Hamiltonian quaternions? Notice that the vectors $(1,i)$ and $(j,k)=(1,i)j$ are linearly dependent as vectors of the right ...
2
votes
2answers
174 views

Matrices with at most one negative eigenvalue

Suppose a vector $y$ and a symmetric matrix $M$ are given. \begin{equation} \forall x; \quad x^Ty=0 \implies x^TMx \ge 0 \end{equation} Prove that $M$ has at most one negative eigenvalue.
2
votes
1answer
124 views

Proof of correctness of Putzers algorithm

I have a question regarding the proof (seen below) of Putzers algorithm for matrix exponentiation. It's written by our danish lecturer at the university, so I translated the important parts into ...
5
votes
2answers
211 views

Finding minimal polynomial

Let $A \in \mathbb{R}^{n \times n}$ be a symmetric matrix, such that $A$ is not of the form $A=c I_n, c \in \mathbb{R}$ and $(A-2I_n)^3 (A-3I_n)^4=0$. Find the minimal polynomial of $m_A(x)$of $A$. I ...
3
votes
2answers
175 views

Prove that a local min is also a global min

Let Q $\in \mathbb{R}^{d \times d} $ and A $\in$ $\mathbb{R}^{d' \times d} $ be two matrices. Let b $\in \mathbb{R}^d$ and c $\in \mathbb{R}^{d'}$ be two vectors. Suppose that d' < d. I want to ...
4
votes
3answers
374 views

Matrix Inverses

So in class we have been discussing matrix inverses and the quickest way that I know of is to get a matrix A, and put it side by side with the identity matrix, like $[A|I_{n}]$ and apply the ...
1
vote
2answers
48 views

Does a linear quotient map have sections

Suppose $V$ is a vector space with vector subspace $N$. Then there is a natural projection $$ \pi_N: V \to V/N $$ from the vector space $V$ to the quotient space $VN$ of $V$ modulo $N$. Does ...
0
votes
0answers
188 views

What does the partial trace of an operator tell me about the full operator?

I have a situation where I would like to know something about an operator when I know something about its partial trace. Let $A$ be a trace class operator on $H\otimes H$, nonnegative, of trace one, ...
2
votes
1answer
164 views

Eigenvectors of matrices which commute with a projection

Just a quick question. Cant seem to prove it or find any relevant references! Maybe it's really simple :\ Is the following statement true (for square matrices of the same finite dimension)? If there ...
3
votes
1answer
354 views

Evaluating the time average over energy

For more info see the article equations 37 Edit: The $\varepsilon ^3 $ has vanished due to time average. But how to get the 4th order? Let us define some function for scalar field $$\phi= ...
2
votes
2answers
74 views

A simple problem of linear algebra in infinite dimension

Let $E$ be a vector space (of infinite dimension) and $u : E \rightarrow E$. Suppose that $E/u(E)$ has finite dimension. Is it true then that this dimension is equal to $\dim \ker u$ ? In finite ...
3
votes
1answer
70 views

A proof on smooth function that I don't know what to proof.

Here's the question: Suppose $f: U \rightarrow V$ is a smooth map, for $U \subset R^k$ and $V \subset R^\ell$ open sets. That is, all partial derivatives (of all orders) of $f$ exist and are ...
4
votes
1answer
119 views

the rank of idempotent matrices

Let $B_i, i=1,2,\cdots,k$ be idempotent matrices, i.e. $B_i^2=B_i$. Can we prove that $\mathrm{rank}(I-B_1\cdots B_k)\leq \sum\limits_{i=1}^k \mathrm{rank}(I-B_i)$, where $I$ is the identity matrix?
1
vote
1answer
132 views

Can the diagonal elements of a precision matrix be 0

I have this confusion that why the diagonal elements of the precision matrix cannot be 0? Any suggestions will be much appreciated
-1
votes
1answer
50 views

sparse overdetermined linear system with noise

I'm trying to solve the problem of global alignment of panorama images (it seems it's called bundle adjustment). We have N images and we can calculate translation(dx,dy) between all pairs of images.(I ...
1
vote
1answer
42 views

Proving that $T^{-1}(V)$ is a subspace of $X$ when $V$ is a subspace of $X$?

I just came across a question which is as follows: Let $X$ and $Y$ be normed linear spaces and let $ T : X \rightarrow Y$ be a linear operator with domain $D(T) \subset X$ and range $R(T) \subset Y$. ...
3
votes
3answers
132 views

Basis of a basis

I'm having troubles to understand the concept of coordinates in Linear Algebra. Let me give an example: Consider the following basis of $\mathbb R^2$: $S_1=\{u_1=(1,-2),u_2=(3,-4)\}$ and ...
0
votes
1answer
540 views

Computing the Frobenius normal form

I was wondering whether someone could give me an example how one actually determines the Frobenius normal form of a given matrix. Further, it seems hard to find an example where the new basis is ...
2
votes
1answer
196 views

If $A,B$ are positive definite matrix then I need to prove so is $ABA^*$

If $A,B$ are positive definite matrix then I need to prove so is $ABA^*$, here is what I have done $$x^*ABA^*x=(A^*x)^*B(A^*x)=y^*By>0$$, is it okay? $y=A^*x$
0
votes
1answer
61 views

Bilinear form properties

Let us look at: $B:V\times W$ be a bilinear form. $T_{B} : W \to V^{*}$ $(T_{B}(w))(v) = B(v, w)$ and $S_{B} : V \to W^{*}$ $(S_{B}(v))(w)=B(v,w)$ I need to prove that: ...
11
votes
0answers
356 views

Determining the kernel of a Vandermonde-like matrix

The kernel of a Vandermonde matrix can be determined using this formula. The following type of matrix has a similar structure, and should also have a one-dimensional kernel. $V= \begin{bmatrix} 1 ...
-5
votes
2answers
294 views

$\mathbb{C}^3$: Orthogonal Complement

Let $S=\{(1,0,i),(1,2,1)\}$ in $\mathbb{C}^3$. What is the method used to find a basis for $S^{\perp}$? EDIT$^1$: I think this bit of literature from Gockenbach's Finite-Dimensional Linear Algebra ...
2
votes
2answers
123 views

Compute eigenvalues and eigenvectors problem

I really don't know how solve this problem: Let $V$ be the space of real functions spanned by $\cos(x)$, $\cos(2x)$ and $\cos(3x)$. Let $T\in\mathcal{L}(V,V)$ con $T(\cos(x)) = 3\cos(x) + 2\cos(2x) - ...
10
votes
2answers
181 views

Determinant of $4\times4$ Matrix

I tried to solve for a $4 \times 4$ matrix, but I'm unsure if I did this properly, can anyone tell me if I did this correct? Or if there were any mistakes where at? Also, I know this is an inefficient ...
1
vote
1answer
28 views

Problem from Roman: a lower bound for trace of |T|^2

I'm working on a problem from Stephen Roman's Linear Algebra text, #20 on p. 235: Suppose $\tau \in \mathcal{L}(\mathbb{C}^n)$ and let the characteristic polynomial $\chi_{\tau}(x)$ have roots ...
1
vote
2answers
216 views

How to prove a linear map to be the trace function,3x

Let $M_n(F)$ be the matrix space on the field $F$, $f$ be the linear map from $M_n(F)$ to $F$ such that $f(I)=n$, where $I$ is the identitiy matrix. Furthermore, $f(AB)=f(BA)$ for any $A,B\in M_n(F)$. ...
1
vote
1answer
463 views

$f(r,\theta)=(r\cos\theta,r\sin\theta)$ Then

$f(r,\theta)=(r\cos\theta,r\sin\theta)$ Then $1.$ $Df(r,\theta)$ is always nonzero for any $(r,\theta)\in\mathbb{R}^2,r\ne 0$ $2.$ $f$ is injective on $\{(r,\theta):r\ne 0\}$ $3.$ On any ...
2
votes
1answer
149 views

Determine invariant subspaces

imagine that a matrix of an endomorphism has the characteristic polynomial $(\lambda-2)^2(\lambda-3)$ now i was wondering whether all invariant subspaces can be determined by $0,V$ and $\ker(A-2)^2, ...
1
vote
1answer
278 views

Generalized eigenspace decomposition of vector space

I was curious whether the direct sum of generalized eigenspaces in a finite dimensional vector space is the largest decomposition of this space invector spaces that are invariant under the ...
1
vote
0answers
86 views

Simplifying expression

I am looking for a way to simplify this expression: $$ \sum_{i=0}^{n-k-1} \sum_{j=0}^{k-1} \left[ {n-k-1 \choose i} {k-1 \choose j} ((-1)^{k-1-j} - (-1)^{n-k-1-i}) \times {(n+0.5)! \over ...
3
votes
3answers
87 views

Determinant of unspecified matrices

Suppose A and B are $5\times 5$ matrices with $\det(A) = -1/3$ and $\det(B) = 6$, find the determinant of $ 2AB$. Solution: $$= \det(2AB) $$ $$= 2^5 \det(A)\det(B) $$ $$= (32)(-1/3)(6)$$ $$= -64$$ ...
0
votes
1answer
93 views

How to find change of basis matrices

Suppose $U$ has a basis $e_1$, $e_2$, $e_3$ and $V$ has a basis $f_1$, $f_2$ and that $T:U \rightarrow V$ is a linear transformation whose matrix $A$ with respect to the given bases is ...
0
votes
1answer
80 views

Problem to find the matrix of linear transformation

Let $B = \left(\vec{e}_1, \vec{e}_2, \vec{e}_3\right)$ a direct orthonormal basis of $V^3$ and let $\vec{a}$ and $\vec{b}$ be two nonzero vectors of $V^3$ such that ...
0
votes
2answers
164 views

Change of basis matrix for inner product space

If $B = (e_1,e_2,\ldots, e_n)$ is a basis for an inner product space $V$ and $B' = (f_1,f_2,\ldots,f_n)$ is an orthonormal basis of $V$. Is the change of basis matrix $P$ necessarily orthogonal?
1
vote
1answer
686 views

Cyclic vector space

In class we defined what it means that there is a creating element $v$ of a vector space, such that for an endomophism $A$ on $V$ we have: ${\rm span}(v,Av,...,A^{n-1}v)=V$. Also we said that if the ...
1
vote
1answer
54 views

Basis given the rank of A is equal to n

Let $A$ be an $m \times n$ matrix with columns $C_1,C_2,\dots,C_n$. If $\operatorname{rank} (A) = n$, show that $\{ A^T C_1,\dots,A^T C_n\}$ is a basis for $\mathbb{R}^n$. I'm really confused ...
1
vote
0answers
522 views

Eigenvalues of Block Anti-Diagonal Matrix

In line with this answer, I am trying to find the eigenvalues of: $\mathbf P\mathbf K\mathbf P^\top=\begin{pmatrix}& d_1 & & & & & & \\d_1 & & e_1 & ...
1
vote
3answers
17k views

Non-trivial solutions implies row of zeros?

If there exist non trivial solutions, the row echelon matrix of homogenous augmented matrix A has a row of zeros. True or False? I'm not sure where to begin as to see why this would be true or ...
0
votes
1answer
171 views

Decompose real positive-definite symmetric matrix

Can any $M\in \mathbb{R}^{n\times n}_{\text{sym}+}$ (real, positive-definite, symmetric matrix) be decomposed in $M=C^{T}.C$ with $C\in \text{GL}_n(R)$ and vice versa ($C\in \text{GL}_n(R)\implies ...
1
vote
1answer
54 views

biconditional matrices

So my professor gave me this question: Assuming that $A\in M_{m\times n}$ and $B\in M_{m\times n}$. We will say that $A,B$ are biconditional matrices if there exist invertible matrices $P\in ...
1
vote
2answers
342 views

How to prove two subspaces are complementary

To give some context, I'm continuing my question here. Let $U$ be a vector space over a field $F$ and $p, q: U \rightarrow U$ linear maps. Assume $p+q = \text{id}_U$ and $pq=0$. Let $K=\ker(p)$ and ...
0
votes
2answers
62 views

How to check if $x^TCx\geq0$?

I have the next 3x3 block matrix C, where each block is a square matrix. $$ C = \begin{bmatrix} 0 & A & B \\ 0 & A+K_1 & B \\ 0 & A & B+K_2 \end{bmatrix}, $$ where $K_i$ is ...
2
votes
1answer
23 views

Showing that the matrix transformation $T(f) = x*f'(x)+f''(x)$ is linear

I want to show that the following matrix transformation is linear. $T(f) = x*f'(x)+f''(x)$ I know I have to show that $T(f+g) = T(f) + T(g)$ but I don't understand what $T(f+g)$ will look like. Is ...
3
votes
2answers
324 views

properties of an alternating bilinear form it's coordinate matrix

I found that I lack many basic knowledge about linear algebra, so read the wiki article about Bilinear Forms. Especially this Paragraph. I tried to proof of "Every alternating form is skew-symmetric." ...
1
vote
1answer
19 views

Why $L$ is the eigenspace of $L_A$?

$A=\frac{1}{\sqrt{5}}\begin{pmatrix} 1&2\\2&-1 \end{pmatrix}$ Let $L_A$ be a reflection of $R^2$ about a line $L$ through the origin. Then $L$ is the one dimensional eigenspace of $L_A$ ...
3
votes
1answer
34 views

Calculate the matrices of $R$ and $R\circ R$ with respect to the basis $(e_1,e_2,e_3,e_4)=(1,x,\frac{1}{2}x^2,\frac{1}{6}x^3)$

I am unsure how to calculate the basis matrices of the linear map defined below. I appreciate your help. Let $V=\mathbb{Q}[x]_{\le3}$ be the set of polynomials over $\mathbb{Q}$ of degree at most ...