For any topic related to matrices. This includes: systems of linear equations, eigenvalues and eigenvectors (diagonalization, triangularization), determinant, trace, characteristic polynomial, adjugate, transpose, Jordan normal form, matrix algorithms (e.g. LU, Gauss elimination, SVD, QR), ...

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0
votes
1answer
19 views

What matrix transforms $(1, 0)$ into $(2, 6)$ and tranforms $(0, 1)$ into $(4, 8)$?

In the last 2 lectures of linear algebra we have talked about linear mappings and other stuff, but I missed actually the last one and I am quite in bad situation. What matrix transforms ...
2
votes
1answer
28 views

Any square matrix is equivalent to zero diagonal matrix

Let $A$ and $B$ be two square matrices of dimension $n\ge 2$. We say that $A$ and $B$ are equivalent if there exist $P$ and $Q$ invertible such that $B=Q^{-1}AP$. Is it true that every square matrix ...
3
votes
3answers
83 views

Is there any geometrical interpretation as to why matrix product is not commutative?

Is there any geometrical interpretation as to why matrix product is not commutative? Similarly, is there any geometrical interpretation of matrix product when you have matrices $A$, $B$ such that ...
0
votes
0answers
12 views

Positive semi-definite Matrix, Schur complement

Let $\mathbb{R}^{n \times n} \ni C = C^\top \succ 0$. Let $A \in \mathbb{R}^{m \times n}$ with $\text{rank}(A) = m$, where $m \leq n$. How do I show that \begin{equation} C - CA^\top(ACA^\top)^{-1}AC ...
0
votes
0answers
13 views

Find the change of basis matrix P from S to S'.

Consider the following bases of $\mathbb{R}^2$: $$S=\left\{\begin{pmatrix} 1\\ -2 \end{pmatrix},\begin{pmatrix} 3 \\ -4 \end{pmatrix}\right\}$$ $$S'=\left\{\begin{pmatrix} 1\\ 3 ...
0
votes
0answers
9 views

Find the singular value decomposition

Find the singular value decomposition of : $$A=\begin{pmatrix} 1 &1 \\ 2& 2 \end{pmatrix}$$ I think the singular value decomposition is $A=P\Sigma Q^T$ right? $$K=A^TA$$ $$=\begin{pmatrix} ...
0
votes
1answer
12 views

relation between conformal and orthogonal matrices in 2D

I want to show that if a matrix $T \in \text{GL}(2, \mathbb{R})$ is conformal, i.e. $$ \text{arccos} \left( \frac{\langle Tv,Tw \rangle}{|Tv||Tw|} \right) = \text{arccos} \left( \frac{\langle v,w ...
0
votes
0answers
10 views

How close is Cartesian product of unit orthogonal bases of SVD to identity matrix?

If I have N unit orthogonal vectors of length N $\phi_{i,N\times 1}$ obtained from SVD of a $N\times M$ matrix $U$ : $$ U_{N\times M} = \sum_i^N \sigma_i\phi_{i,N\times 1}\times\psi_{i,1\times M}\\ ...
0
votes
0answers
21 views

Find the matrix representation of T relative to the basis

Let $T: \mathbb{R}^2\rightarrow \mathbb{R}^2$ be the linear operator defined by $$T\begin{pmatrix} x \\ y \end{pmatrix}=\begin{pmatrix} 2x+3y \\ 4x-5y \end{pmatrix}$$ Find the matrix ...
0
votes
1answer
12 views

Unique eigenvalue of maximal absolute value?

Let $A$ be an $n\times n$ matrix with $a_{ii}=0$ for all $i$, and $a_{ij}\in\{0,1\}$ for all $i\neq j$, and $a_{ij}=0\leftrightarrow a_{ji}=1$ for all $i\neq j$. Is it necessary that $A$ as a unique, ...
0
votes
1answer
10 views

Are there any general strategies to prove $K(x,y)$ is a machine learning kernel? (I.e. always defines a covariance matrix)?

So there are certain functions of two variables such as the standard Gaussian/radial function $K(x_i,x_j) = e^{-(x_i-x_j)^2}$ which are "kernels" as machine learning calls them, meaning that for any ...
1
vote
0answers
19 views

Largest circulant matrix without a non-zero vector in its kernel

Consider a $k$ by $n$ circulant matrix with entries that are either $-1$ or $1$. For some of these matrices there exists a non-zero vector $v \in \{-1,0,1\}^n$ in its kernel and for some no such ...
0
votes
1answer
29 views

Proving that an $n\times n$ matrix is positive definite iff the eigenvalues of that matrix plus its transpose are positive

I am trying to prove that an $n\times n$ matrix $A$ is positive definite iff the eigenvalues of $(A + A^T)$ are positive. So far I have: Let $x$ be an eigenvector of $(A + A^T)$ and let $\lambda$ be ...
1
vote
1answer
34 views

Does $\text{End}_K(K^n) \cong \text{Mat}(n\times n, K)$?

Let $K$ be a field, $K^n$ a vector space over $K$. Is the following true? $\text{End}_K(K^n) \cong \text{Mat}(n\times n, K)$ Does this change if $K$ is a ring, and $K^n$ a module over $K$?
0
votes
0answers
10 views

Diagonalization of Markov Matrices

If we're given a $ \displaystyle 2 \times 2 $ Markov Matrix (so all entries are non-negative and columns add to 1) M$(a,b)$ such that $$M = M(a,b) := \begin{bmatrix}1 - a & b\\a & 1 - b ...
0
votes
0answers
6 views

Markov chain ergodicity

$Xn$ is a discrete-time, time-homogenous Markov chain. I have have the following transition matrix and want to show whether the chain is ergodic. P = \begin{pmatrix} \frac{1}{2} & 0 & 0 ...
1
vote
1answer
42 views

How to prove that $I+A^{T}A$ is invertible [duplicate]

Let $A$ be any $m\times n$ matrix and $I$ be the $n\times n$ identity. Prove that $I+A^{T}A$ is invertible.
3
votes
1answer
21 views

Prove matrices are of equal rank

Suppose $P$ and $Q$ are $n \times n$ matrices of real numbers such that $P^2 = P$, $Q^2=Q$ and $I-P-Q$ is invertible, where $I$ is the $n × n$ identity matrix. Show that $P$ and $Q$ have the same ...
1
vote
1answer
30 views

Bound for eigenvalues of some special matrix

Let $Tridiagonal(a, c, b)= \begin{vmatrix} c & b & 0 & \ldots & 0 \\ a & c & b & \ldots & 0 \\ 0 & a & c & \ldots & 0 \\ \vdots & \vdots & ...
1
vote
2answers
29 views

Determinant of a square matrix in a field [duplicate]

\begin{array}{rrrrr|r} b & a & a & \cdot \cdot \cdot & a \\ a & b & a & \cdot \cdot \cdot & a \\ a & a & b & \cdot \cdot \cdot & a \\ ...
0
votes
0answers
8 views

Coppersmith-Winograd algorithm

I'm interested in algorithms to compute matrix multiplications. Is the Coppersmith-Winograd algorithm similar to the Strassen algorithm ? I have two other questions: 1) Are the multiplications done ...
0
votes
1answer
19 views

Finding the matrix representation of a linear transformation $ T: P_{3} \to \text{M}_{2 \times 2} $.

Define a function $ T: P_{3} \to \text{M}_{2 \times 2} $ by $$ T \! \left( a_{0} + a_{1} x + a_{2} x^{2} + a_{3} x^{3} \right) = \begin{pmatrix} a_{3} & a_{0} \\ a_{2} & a_{1} \end{pmatrix}. ...
0
votes
1answer
10 views

Eigenvector / eigenvalue pairs for a Markov Matrix

If we're given a $ \displaystyle 2 \times 2 $ Markov Matrix (so all entries are non-negative and columns add to 1) M$(a,b)$ such that $$M = M(a,b) := \begin{bmatrix}1 - a & b\\a & 1 - b ...
2
votes
0answers
18 views

Express a quadratic form as a sum of squares using Schur complements

So I was able to figure out the first part of this problem, but I have no concept of how it relates to Schur complements, so I'm not sure (no pun intended) how to proceed. The question is as follows: ...
0
votes
0answers
15 views

Find a basis and the dimension of the eigenspaces of the matrix

Find a basis and the dimension of the eigenspaces of the matrix $$ \left( \begin{array}{ccc} 1 & 0 & 2 \\ 0 & 3 & 0 \\ 2 & 0 & 1 \end{array} \right) $$ given that the ...
0
votes
1answer
10 views

Transitions of matrix

$T: \Bbb R^3 \to \Bbb R^3$ and $S: \Bbb R^3 \to \Bbb R^4$ are matrix transformations whose standard matrices are $$T=\begin{bmatrix} 1 & 0 & 2 \\ 2 & 3 & 4 \\ 1 & 5 & ...
0
votes
1answer
14 views

Finding eigenpairs for Markov Matrices

If we're given a $ \displaystyle 2 \times 2 $ Markov Matrix (so all entries are non-negative and columns add to 1) M$(a,b)$ such that $$M = M(a,b) := \begin{bmatrix}1 - a & b\\a & 1 - b ...
1
vote
0answers
17 views

why is the sum over even and odd permutations the same?

let $m$ be an $n \times n$ matrix (over $\mathbb{R}$,say) and for a permutation $\sigma \in S_n$ define the monomial: $$ P_\sigma(M) = \prod_{j=1}^n m_{j,\sigma(j)} $$ let $\tau$ be an odd ...
1
vote
2answers
27 views

Find the eigenvectors of $ A = \begin{bmatrix} 0 & 2 \\ 1 & 1 \end{bmatrix} $.

Find the eigenvectors of $$ A = \begin{bmatrix} 0 & 2 \\ 1 & 1 \end{bmatrix}. $$ I know you can solve $ \det(A - \lambda I) = 0 $ to find the eigenvalues of $ A $, but I keep getting no free ...
0
votes
1answer
14 views

Finding a non-zero vector in both the column space and the null space of a nilpotent matrix

$A$ is a $ \displaystyle 10 \times 10 $ matrix such that $A^{3} = 0$ but $A^{2} \neq 0$ and therefore, by definition, $A$ is nilpotent. Is there a non-zero vector that lies in both the column space ...
0
votes
1answer
19 views

Solving a variable in a matrix equation?

I am having trouble solving for a in the problem below. I've simplified it down to: $e^{14} = ln(e^e \cdot a)$. I'm not really sure where to go from here.
6
votes
4answers
536 views

Why are eigenvalues of nilpotent matrices equal to zero?

If $A$ is a $ \displaystyle 10 \times 10 $ matrix such that $A^{3} = 0$ but $A^{2} \neq 0$ (so A is nilpotent) then I know that $A$ is not invertible, but why does at least one eigenvalue of $A$ ...
-1
votes
2answers
26 views

Prove Determinant [on hold]

A is the standard matrix of a counterclockwise rotation about the origin in R2 through an angle theta , show that det(A) = 1. How would one go about in solving/proving this? I know I have to use [ ...
1
vote
0answers
20 views

Converting second order system into first order system (ODE)

The second order equation $\frac{d^2\vec{x}}{dt^2} = A\vec{x}\ + \vec{g}(t)$ models an earthquake's effect on a 7-story building. Let $x_j(t)$ be the displacement of the $j$th floor with respect to ...
-4
votes
0answers
22 views

Mathematics liner algebra [on hold]

Define what it means for two matrices $A$ and $B$ to be similar. Are the matrices $$A = \pmatrix{1&2\\-1&1}$$ and $$B = \pmatrix{5&3\\-6&-3}$$ similar to each other? Please ...
12
votes
11answers
230 views

Why represent a complex number $a+ib$ as $[\begin{smallmatrix}a & -b\\ b & \hphantom{-}a\end{smallmatrix}]$?

I am reading through John Stillwell's Naive Lie Algebra and it is claimed that all complex numbers can be represented by a $2\times 2$ matrix $\begin{bmatrix}a & -b\\ b & ...
0
votes
1answer
15 views

“distance” metric between two bases modulo determinant, rotation and chirality

I'd like some kind of metric that tells me how similar two complete, not necessarily orthonormal bases (represented by non-singular matrices $B_1, B_2 \in \mathbb{R}^{n \times n}$) are to each other, ...
0
votes
1answer
14 views

Diagonal factorization of upper triangluar matrix to unit uppper triangular matrix

How to compute Diagonal factorization of upper triangluar matrix to unit uppper triangular matrix. i.e U = D*M where U is upper triangular; D is diagonal; M is unit upper triangular.
4
votes
1answer
83 views

Schur decomposition of a matrix with distinct eigenvalues is almost unique

Let $M\in \mathbb C^{n,n}$ have $n$ distinct eigenvalues, and let $U_1, U_2$ be two Schur-forms of $M$. Show that if $U_1, U_2$ have equal diagonals, there is a hermitian diagonal matrix $Q$ such ...
1
vote
1answer
99 views

Find a $4\times 4$ matrix whose reduced row echelon form has two leading ones and whose row space intersects its column space only at the origin.

Find a $4\times 4$ matrix whose reduced row echelon form has two leading ones and whose row space intersects its column space only at the origin. Would $$ \begin{bmatrix}1& 0& ...
2
votes
2answers
555 views

What does Determinant of Covariance Matrix give?

I am representing my 3d data in convariance matrix. I just want to know what the determinant of Convariance Matrix gives. If the determinant is positive, zero, negative, high positive, high negative. ...
4
votes
0answers
122 views

Can the “inducing” vector norm be deduced or “recovered” from an induced norm?

Can the "inducing" vector norm be deduced or "recovered" from an induced (operator) norm? This question occurred to me after seeing this question. I'm hoping that perhaps there exists something like ...
4
votes
3answers
611 views

Column Space and SVD

I was reading Gilbert Strang's book and he says that if $A=USV'$ be the SVD of A ( assume square for the moment) then the nullspace of A is given by the last $n-r$ columns of V and the column space by ...
1
vote
3answers
847 views

Every positive definite matrix can be written as $B^TB$ for some invertible $B$

Let $A$ be a positive definite symmetrix matrix. Show that there exists an invertible matrix $B$ such that $A=B^TB$. [Hint: Use the Specral Theorem to write $A = QDQ^T$. Then show that D can be ...
12
votes
3answers
1k views

Block Diagonal Matrix Diagonalizable

I am trying to prove that: The matrix $C = \left(\begin{smallmatrix}A& 0\\0 & B\end{smallmatrix}\right)$ is diagonalizable, if only if $A$ and $B$ are diagonalizable. If $A\in\mathbb{C}^n$ ...
7
votes
1answer
3k views

Intersection of conics using matrix representation

I came across a very interesting section of a wikipedia article on conics: http://en.wikipedia.org/wiki/Conic_section#Intersecting_two_conics I am trying to work out a couple of examples to add to ...
0
votes
3answers
11k views

Image and Kernel of a Matrix Transformation

So I had a couple of questions about a matrix problem. What I'm given is... Consider a linear transformation $T: \mathbb R^5 \to \mathbb R^4$ defined by $T( \overrightarrow{x} )=A\overrightarrow{x}$, ...
55
votes
1answer
4k views

Is the following matrix invertible?

$$ \begin{bmatrix} 1235 &2344 &1234 &1990\\ 2124 & 4123& 1990& 3026 \\ 1230 &1234 &9095 &1230\\ 1262 &2312& 2324 &3907 \end{bmatrix}$$ Clearly its ...
49
votes
4answers
71k views

Is a matrix multiplied with its transpose something special?

In my math lectures, we talked about the Gram-Determinant where a matrix times its transpose are multiplied together. Is $A A^T$ something special for any matrix $A$?
9
votes
4answers
17k views

Similar matrices have the same eigenvalues with the same geometric multiplicity

Suppose $A$ and $B$ are similar matrices. Show that $A$ and $B$ have the same eigenvalues with the same geometric multiplicities. Similar matrices: Suppose $A$ and $B$ are $n\times n$ matrices ...