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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When is the product of two arbitray matrices symmetric?

Let $\mathbf{A}$ be a real $n \times m$ matrix. Let $\mathbf{B}$ be a real $m \times n$ matrix. How to solve the following matrix equation? $$\mathbf{A}\mathbf{B}=\mathbf{B^{t}}\mathbf{A^{t}}$$ ...
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3answers
75 views

Basic way to show for $n\times n$ matrices $A$ and $B$, that $(AB)^{-1} = (B^{-1})(A^{-1})$

In looking at matrix inverses, I know the following works (I is the identity matrix): If $AB$ are nxn matrices and are invertible, then $(AB)C = I$, and therefore $C = (AB)^{-1}$. I can show that ...
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1answer
266 views

Tutte matrix - Determinant

I'm trying to understand the proof of the "magic theorem" about the Tutte matrix which states: Let $T$ be the Tutte matrix of $G(V, E)$. Then, $$\det(T) = 0 \quad\Longleftrightarrow\quad G ...
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1answer
47 views

Computing covariance matrix in PCA

I am implementing PCA in matlab and I have to compute the covariance matrix. I am using 'cov' command from matlab to compute the covariance matrix. But it is very slow and takes a lot of time to ...
2
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0answers
54 views

Inequality with eigenvalues

Let matrix $ X $ is Hermitian and denote $ \lambda_1(X) \ge \lambda_2(X) \ge \ldots \ge \lambda_n(X) $ eigenvalues of matrix $ X $. Prove that $ \lambda_i(A + B) \le \lambda_i(A) + \lambda_1(B) $ I ...
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73 views

Find the eigenvalues of $A$. $A^2 = 1$ and $A\ne\pm1$

$A \in \mathbb{R}^{n\times n}$, with $A^2 = 1$ and $A\ne\pm1$ Show that the only eigenvalues of $A$ are $1$ and $-1$.
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Linear Algebra-invariant subspaces

Suppose $V$ is a real vector space and $T\in \mathcal L (V)$ has no (real) eigenvalues. Prove that every subspace of $V$ invariant under $T$ has even dimension.
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Finding all the invariant subspaces of a certain linear transformation.

Assuming I have given affine transformation $ \mathbb{R}^3\to \mathbb{R}^3 $ which has matrix representation $$ \left[\begin{array}{cccc} 3&2&-3&-10\\ 4&10&-12&-29\\ ...
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1answer
77 views

Let $trcA=0$.why $A=M+N$ where $M$ and $N$ are nilpotent matrices?

Let $A \in {M_n}$ and $trcA=0$.why $A=M+N$ where $M$ and $N$ are nilpotent matrices?
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1answer
64 views

Finding a kernel and an image of $T^2$

Let $T$ be a linear transformation $T: \mathbb{R}^4 \to \mathbb{R}^4$ that is defined by: $$T\begin{pmatrix}x\\y\\z\\u\end{pmatrix}=\begin{pmatrix}0\\z\\y\\x\end{pmatrix}$$ Find the kernel and image ...
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If $A$ can be written as a sum of nilpotent matrices why $trcA=0$?

Let $A \in {M_n}$. If $A$ can be written as a sum of two nilpotent matrices, why $trcA=0$?
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1answer
367 views

linear algebra questions about matrices

Given a a matrix $X$ where $$ X= \begin{bmatrix} 1 & 0 \\ 0 & p(x) \end{bmatrix} $$ where $p(x)$ is some polynomial of degree $3$ or 4 and different from $0$. I'm trying to ...
3
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2answers
99 views

Let $A,B \in {M_2}$ and $C=AB-BA$. Why is ${C^2} = \lambda I$ true?

Let $A,B \in {M_2}$ and $C=AB-BA$. Why does ${C^2} = \lambda I$?
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1answer
173 views

Examples of Parallel planes and others.

I need to draw this but I would like to add an extra mark there by putting some examples of these. They must be in a 3D cartesian plane. Any help? Three parallel planes Two parallel planes and a ...
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1answer
25 views

Determinant with one parameter, how to deal with this?

Let $t\in \mathbb R$ be a parameter, and $$|A(t)|= \begin{vmatrix} a_{11}+t &a_{12}+t &\cdots &a_{1n}+t\\ a_{21}+t &a_{22}+t &\cdots &a_{2n}+t\\ \vdots &\vdots ...
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Optimal Matching Distance

I'm stuck on problem II.5.9 from Bhatia's Matrix Analysis. The problem is as follows: Let $\{\lambda_1,\dots,\lambda_n\},\{\mu_1,\dots,\mu_n\}$ by two $n$-tuples of complex numbers. Let $$ ...
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2answers
41 views

Is $Q=V^TWV$ positive definite?

I have the real symmetric $k \times k$ matrix $Q = V^T W V$, where I know $V$ is a $n \times k$ orthogonal matrix (its columns are orthogonal) and $W$ is a $n \times n$ diagonal matrix with all its ...
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2answers
56 views

Prove boundedness of the matrix series

Suppose $A$ is a square matrix, such that all eigenvalues of $A$ has norm strictly less than $1$, can I say $\sum_{i=k_0}^kA^{k-i}$ is bounded for all large enough $k_0$ and $k$? From some other ...
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1answer
48 views

Principal Component analysis by eigenvalue decomposition.

I do know how to perform PCA by using SVD but I am unaware about how to use eigenvalue decomposition of X(transpose)*X matrix. I found a paper online which explains the approach to perform PCA by ...
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0answers
23 views

How to figure out whether PCA can be performed on a data set or not?

I do have idea on the way PCA works but I do not know how to figure out whether a high dimensional data set is suited for PCA compression. I googled for some algorithms but could find any. Are there ...
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2answers
88 views

Finding x'y' coordinates from xy coordinates with unit basis vectors

I wasn't really sure where to get started with this question as I don't fully understand what it's asking.. I can see that u1 is made up of i + j (or u2) and that x' is scaled for some scalar k from ...
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2answers
1k views

Prove the determinant is the product of its diagonal entries

Prove that the determinant of an upper triangular matrix is the product of its diagonal entries. What I have so far: We will prove this by induction for an $n$ $\times$ $n$ matrix. For the case of a ...
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42 views

Show that there is a vector $w$ in ${\rm ker}\ (T)$ such that $v=u+w$

Suppose $U$ and $V$ are vector spaces such that $T:U\rightarrow V$ is a linear map. Suppose also that $u$ and $v$ are vectors in $V$ such that $f(u)=f(v)$. Show that there is a vector $w \in \rm ...
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Linear algebra problem about projections

Let A,B be real matrices of order $n \geq 6$. Let $A + \alpha B$ be projection operator for any $\alpha \in \mathbb{R}$. True or false: if A is orthogonal projection, then $A \neq B$. ...
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1answer
86 views

Derivative of a trace function

Let $K$ be a Hermitian matrix, and $X$ be a positive one. What is the derivative of the trace function $$ \mbox{ Tr } X|e^{itK} - X|^3$$ with respect to $t$ at $t = 0$ ? There is a nice formula for ...
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determination of the volume of a parallelepiped

Here is a parallelepiped.I want to determine the volume of the parallelepiped. One of my friends said to me that the volume of the parallelepiped can be found out by the following formula. ...
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0answers
32 views

Basis of orthogonal complement subspace [duplicate]

Let $A$ be the matrix $$ \begin{pmatrix} 1 & 1 & -1&-1 \\ 1 & 2 & -2 & 1 \\ \end{pmatrix} ,$$ let $W$ = ker $A$ and let $W^\bot$ be the ...
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1answer
46 views

Proofs involving orthonormal basis

Suppose that $V$ is an inner product space. (a) Show that if $\{e_1, . . . , e_n\}$ is an orthonormal basis for $V$ , then $$||v||^2=\sum_{i=1}^{n}|\langle v|e_i\rangle|^2\quad \quad \text{for every ...
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1answer
63 views

Unique linear combination in matrix with skew-symmetric condition

Let $A$ be an $n\times n$ matrix with real entries such that the numbers in each column sum to $0$, and $a_{ij}\in\{0,1\}$ for all $i\neq j$, and $a_{ij}=0\leftrightarrow a_{ji}=1$ for all $i\neq j$. ...
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1answer
73 views

Spans of Orthogonal complements

Let $A$ be the matrix $$ \begin{pmatrix} 1 & 1 & -1&-1 \\ 1 & 2 & -2 & 1 \\ \end{pmatrix} ,$$ let $W$ = ker $A$ and let $W^\bot$ be the ...
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3answers
109 views

Im(AB) $\subset$ Im A

Let A,B be linear operators over $\mathbb{R}^n$, where $n > 2$. Which of the following statements is correct? Im(AB) $\subset$ Im A Im(AB) $\subset$ Im B Im(AB) $\supset$ Im A ...
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1answer
252 views

Proving that the matrix exponential map is surjective onto the general linear group

Let $M_n(\mathbb{F})$ be the set of all $n\times n$ with entries in $\mathbb{F}$ and let $\exp:M_n(\mathbb{C})\to M_n(\mathbb{C})$ be defined by $$ \exp(A)=\sum_{k=0}^{\infty}\frac{A^k}{k!},$$ for ...
3
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1answer
27 views

Unique linear combination in matrix with column sum $0$?

Let $A$ be an $n\times n$ matrix with real entries such that the numbers in each column sum to $0$, and all diagonal entries are non-zero. So, $A$ is non-invertible, and some linear combination of ...
2
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1answer
94 views

Is there an easier way to find the inverse of a 3x3 matrix?

I know the normal process is to do row operations to transform the matrix to get the identity matrix and then apply the same row operations in the identity matrix to get the inverse. But this process ...
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1answer
38 views

Finding eigenvectors through triangularization

I have an exam tomorrow and am working through notes. We derived the following stochastic matrix: $$P=\left[ \begin{matrix} 0.8 & 0.5 & 0 & 0\\ 0.2 & 0.5 & 0 & 0 \\ 0 & 0 ...
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1answer
40 views

Solution of system of equations in prime fields

In 'Algebra', Artin writes that the system of equation: $$8x+3y = 3$$ $$2x+6y = -1$$ have no solutions in $\mathbb{F}_2$ and $\mathbb{F}_3$ as the determinant (of the coefficient matrix) evaluates ...
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1answer
96 views

Find orthogonal Q given eigenvalue and eigenvector?

Given some upper Hessenberg matrix $H \in R^{n \text{x} n}$, i know how to find an orthogonal matrix which is a product of Givens rotations such that $P^THP$ is also upper Hessenberg, but I'm not sure ...
2
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0answers
67 views

Why is unit circle not sufficient to bound the partial sums?

I want to find vectors $\textbf{v}_1, \dots,\textbf{v}_n$ in $\mathbb{R}^2$ with that $\sum_{i=1}^n\textbf{v}_i=\textbf{0}$ and $\Vert \textbf{v}_i\Vert\leq 1$ for all $i=1,\dots,n$, such that for ...
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1answer
48 views

Proof that an $n \times n$ matrix is positive definite iff all of its eigenvalues are positive

I am trying to prove that an $n \times n$ matrix is positive definite iff all of its eigenvalues are positive. I know that if $\lambda$ is an eigenvalue then: $Ax = \lambda x$ for eigenvalues ...
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0answers
64 views

Draw the parallelogram spanned by the vectors.

I am trying to figure out how to solve the following generic problem: Draw the parallelogram spanned by the vectors $\begin{bmatrix} q \\ r \end{bmatrix}$ and $\begin{bmatrix} s \\ t \end{bmatrix}$. ...
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1answer
26 views

Is the set $V$ = { $([t], [g], [t], [j]): t,g,j∈$Z$,[2t+j] = [0]$} a subspace of vector space $(\mathbb Z_3)^4$?

Is the set $V$ = { $([t], [g], [t], [j]): t,g,j∈Z,[2t+j] = [0]$} a subspace of vector space $(\mathbb Z_3)^4$? I am inclined to think that it is a subspace. However, I cannot find any basis for the ...
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Co ordinate independent linear algebra over graphs

It is frequently said that Linear algebra is not correct until it is coordinate free or something to that effect and indeed, almost all the major results can be stated without picking a basis. ...
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2answers
34 views

Linear Applied Algebra | Verify the vectors

Question: My response: Math has never been a strength, particularly proofs, so I would appreciate any and every help. I am just not sure if I am following a proper procedure for the above ...
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1answer
46 views

Applied Linear Algebra | Linear Dependent Matrix

Question: My response: Am I solving the above question correctly? Or am I on the wrong path? Thank you for your help.
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2answers
46 views

Applied Linear Algebra |

Question: Determine whether or not any column in the matrix is a linear combination of other columns. Provide a general method for answering the same question for any n x n matrix A. My response: ...
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1answer
52 views

Prove that the set $\{ x^2 + 4x -3, 2x^2 +x + 5, 7x - 11\} $ does not span $\textit{P}_2$.

Could someone please explain how to prove this? Also, why is it that we must create a set of coefficients for every polynomial contained in S in order to prove it and why is rank so significant? ...
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1answer
70 views

Applied Linear Algebra | Prove the intersection of two subspaces

Question: Determine whether or not any column in the matrix is a linear combination of other columns. Provide a general method for answering the same question for any n x n matrix A. My response: ...
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2answers
60 views

When is the transpose of a square unitary matrix also unitary?

If I have a unitary square matrix $U$ ie. $U^{\dagger}U=I$ ( $^\dagger$ stands for complex conjugate and transpose ), then for what cases is $U^{T}$ also unitary. One simple case I can think of is ...
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1answer
56 views

Exercise on linear maps with a nilpotent one

sorry for asking to help me with this trivial problem. Unfortunately I'm in a very bad shape with linear algebra, being this the fourth exercise I'm not able to solve. I need some suggestion. ...
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1answer
56 views

Relationship between eigenvectors of two matrices

Suppose I have matrix $A \in R^{2n \text{x} 2n} $ given by $X^{-1} diag(W - iY, W + iY) X$ and matrix $B \in C^{n \text{x} n}$ and $B = W + iY$. Let $v$ be an eigenvector of $A$. How can I relate ...