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
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1answer
24 views

Are all matrices of the following form Hermitian?

If I have a matrix $A$ (and it is square and nonsingular), is $A^* A $ Hermitian? Also, does $A $ have to be nonsingular for this to hold?
0
votes
1answer
187 views

Orthogonal columns imply orthogonal rows?

The original question is: Column Vectors orthogonal implies Row Vectors also orthogonal? A counterexample with zero entries is given in one post. However, my question is whether pairwise orthogonal ...
0
votes
1answer
52 views

Definition of onto for linear transformation

I had a question ask the following: "A linear transformation is onto if and only if the columns of its standard matrix form a generating set for its range." To me that seems true but the answer was ...
0
votes
1answer
816 views

If the null space contains only the zero vector, the map is one-to-one

How does finding out if the null space has only the zero vector prove one-to-one? One-to-one means that there are distinct images for each distinct vector input. $$\mathbb R^n \to \mathbb R^m$$ ...
0
votes
1answer
53 views

For an arbitrary symmetric matrix, the relation between the number of eigenvalues and the rank of the matrix?

For an arbitrary symmetric matrix $A\in \mathcal{S}^n$, $n$ symmetric space: what is the relation between the number of eigenvalues and the rank of $A$ ? If we know $rank(A) = r$, what is the ...
0
votes
1answer
88 views

Looking for proof of “ two congruent symmetric matrices with real entries have the same numbers of positive, negative, and zero eigenvalues ”

For any real square matrix $X$ let $P(X)$ denote the no. of its positive eigenvalues counting multiplicity . Let $A$ be a real symmetric $n \times n$ matrix and $B$ be a real invertible $n \times ...
7
votes
1answer
90 views

$GL_n(\mathbb{F})$ contains a copy of $\mathbb{F}^{n-1}$

It is a fact of matrix multiplication that $$\left( \begin{matrix} 1 & a & b \\&1&\\&&1 \end{matrix} \right) \left( \begin{matrix} 1 & a' & b'\\&1&\\&&1 ...
4
votes
3answers
624 views

Kernel Explanation

sorry for asking so many questions lately but our lecturer is doing a terrible job explaining things. Calculate $ker(A)$ given that: $f:\{\mathbb{R}^3→\mathbb{R}^3; r→ A\vec{r}\}$ $A= ...
0
votes
1answer
81 views

Solve Unknown Matrix Variables

I have a markov chain matrix with probabilities as such, on finding the steady state.. ...
0
votes
2answers
172 views

Maximum eigenvalue of product of two matrices

Let $A$ and $B$ be two Hermitian matrices. I wanted to know if there is any relation between the maximum eigenvalue of $AB$ and that of $A$ and $B$. Is the following relation true? ...
0
votes
1answer
36 views

Yet another n-Dimensional Rotation question: is there a definition for n-Dimensional rotation which is cosine-distance invariant? (TO CLOSE)

(Moved to Searching for a definition for n-Dimensional rotation which is cosine-distance invariant, flagged it to delete it) I'm wondering if there exists a rotation definition by which the vectors ...
-1
votes
1answer
134 views

What is the meaning of |⋯| notation for an index subset?

I am currently working on a research project. In the attached image what does the capital $|I|$ and $|J|$ mean ?
0
votes
2answers
38 views

Why $V_{ij} = \frac {1}{2}(v_iv_j^T + v_jv_i^T),$ is rank-2 if $i\neq j$?

Can someone help me figure out the following argument ? $V_{ij} = \frac {1}{2}(v_iv_j^T + v_jv_i^T),$ is rank-2 if $i\neq j$ where $v_i,v_j \in \mathbb{R}^n$, $v_i,v_j$ are linearly indepedent. ...
5
votes
1answer
2k views

are there any bounds on the eigenvalues of products of positive-semidefinite matrices?

I have real positive semidefinite matrices (symmetric) $A$ and $B$, both are $n \times n$. I am looking for upper bounds and lower bounds on the $m$th largest eigenvalue of $AB$, in terms of the ...
1
vote
1answer
140 views

Is the cone of rank one matrix convex and closed?

Define the following cone: $M=\{xx^T:x\in\mathbb{R}^n\}$ Is this cone convex and closed? How to prove? Thanks
1
vote
4answers
62 views

Why is this a valid definition of the dot product?

$(\vec{u},\vec{v})=u_1v_1+2u_2v_2+3u_3v_3$ I have never seen this definition before. I am used to the dot product looking something like this: $(\vec{a},\vec{b})=a_1b_1+a_2b_2+a_3b_3$ Where do the ...
2
votes
2answers
55 views

Confusion about a Linear Transformation question.

Let $\beta := [M_1, M_2, M_3, M_4]$ be the ordered basis of $R^{2×2}$ defined by: $$ M_1 := \begin{pmatrix} 1 & 0\\ 0 & 0 \end{pmatrix}, M_2 := \begin{pmatrix} 0 & 1\\ 0 & 0 ...
0
votes
0answers
61 views

Algorithm to efficiently compute$ A^k$

If a symmetric matrix $A$ has SVD $A=U\Sigma U^{\top}$, then $A^k=U\Sigma^kU^{\top}$. What would be the most efficient algorithm to compute $A^k$ such that the worst case time complexity is as low as ...
2
votes
1answer
171 views

Symmetric matrix with same diagonal elements

A paper I was reading made the claim that the eigenvector of a symmetric matrix with same diagonal elements is : $$i_n = {1, e^{jna}, e^{2jna}, ..., e^{j(N-1)na}}$$ $$ n =0, 1, 2 , ...N-1$$ $$a = ...
7
votes
2answers
74 views

Eigenvalues of $A(A+B)^{-1}$

Given a positive semidefinite matrix $A$ and a positive definite matrix $B$ of the same dimension. Can we show that each eigenvalue: $$ \lambda\{A(A + B)^{-1}\} < 1$$ (in the scalar case, this is ...
1
vote
2answers
31 views

If matrix $\sum_0^\infty C^k$ is convergent, how can I prove that $A(\sum_0^\infty C^k)B$ is convergent?

For an $n \times n$ matrix $C$ and If $\sum_0^\infty C^k$ is convergent, how can I prove that for two matrices $A$ and $B$, $A(\sum_0^\infty C^k)B$ is convergent? It seems quite obvious that you just ...
0
votes
3answers
63 views

A good source for linear algebra on matrices

I am studying for an Algebra qualifying examination to be taken in one month. I need a good source (a book) where I can find the most important theory and examples on linear algebra on matrices. Does ...
0
votes
2answers
51 views

How many orthogonal matrices map one vector to another?

Say you have two real vectors $u$ and $v$ and $\|u\| = \|v\|$ How many real square and orthogonal matrices $A$ are there such that $Au = v$? Assuming $u$ and $v$ are not parallel/antiparallel, there ...
1
vote
0answers
64 views

Does $SL(2, \mathbb F_p)$ always have a subgroup of order $(p-1)(p+1)$?

For $p$ prime, does $SL(2, \mathbb F_p)$ always have a subgroup of order $(p-1)(p+1)$? (It does seem to be the case for $p=3,5,7,11$) If yes, what are the generators? I guess this is simple, but my ...
4
votes
1answer
573 views

Det(AB)=0: what is the determinant of A and B

True or false. If the determinant of AB is zero, then the determinant of A is zero or the determinant of B is zero. I put true in my exam. After all det(A)det(B)=det(AB). Why was I wrong? The answer ...
2
votes
0answers
73 views

Inequality in inverse Laplacian

I have the following problem, which is motivated by geometric diffusion on a directed graph. Conjecture. Let $A \in [0,1]^{n\times n}$ be strictly substochastic - i.e. $\forall i ~ \sum_j A_{i,j} ...
4
votes
1answer
58 views

Show that $\|B^{-1/2}AB^{-1/2} - I\| >1/2$, when $\|A\|>2\|B\|$

Let $A,B$ be two positive-definite matrices. Suppose that $\|A\|>2\|B\|$. Is it possible to show that $\|B^{-1/2}AB^{-1/2} - I\| >1/2$, where $I$ is an identity matrix and the norm is the ...
3
votes
1answer
86 views

Simplifying the sum $\sum\limits_{i=1}^n\sum\limits_{j=1}^n x_i\cdot x_j$

How can I simplify the expression $\sum\limits_{i=1}^n\sum\limits_{j=1}^n x_i\cdot x_j$? $x$ is a vector of numbers of length $n$, and I am trying to prove that the result of the expression above is ...
0
votes
1answer
43 views

Matrix addition

How do I solve the following? [2x1 -3x2 + x3; 4x1 - 2x3] + [x1 +2x2; 0x1 - 2x2; 4x1 + x2]^T When I do the transpose of the second matrix and try to add them together I get lost. Should I consider x1 ...
0
votes
2answers
40 views

Symmetric matrices and orthogonality

I'm struggling to make any progress with this question. I have defined C as the standard n-dimensional identity matrix. As A is semidefinite, I believe the diagonal matrix D must have positive ...
1
vote
2answers
381 views

is this always identity matrix?

do you think the following matrix multiplication results in I? $R(R^TR)^{-1}R^T$= I or diag(I, O) R is not necessarily square and may not have an inverse.
0
votes
1answer
18 views

if L is the cholesky decomposition, then what is L' L?

Let $L$ be a lower-triangular matrix such that $LL^T=A$. Then $B=L^TL$ has the same eigenvalues as $A$, but different eigenvectors. 1) Why are the eigenvalues the same? 2) Is there an analytical ...
1
vote
0answers
547 views

Gram matrix of Gaussian kernel is not positive definite

I am developing a machine learning software, where I am trying to apply kernel methods. I have N uniformly sampled scalar values, $\{x_1,\dots,x_N\}$ from a given interval $[a,b]$. My aim is to ...
2
votes
5answers
411 views

Given a matrix, find a matrix that satisfies

Let A be a matrix (3x4) Prove that there does not exists a matrix X that satisfies $$ \begin{pmatrix} 1 & 1 & 2 & -1 \\ 0 & 2 & 1 & 3 \\ 1 ...
2
votes
1answer
44 views

Determining diagonalizability of a linear transformation defined by a matrix.

Suppose $A\in M_n(\Bbb C)$ satisfies $A^6-A^3+I=O$. Prove that if a linear transformation $T:M_n(\Bbb C)\rightarrow M_n(\Bbb C)$ is given by $T(B)=AB$, then $T$ is diagonalizable. How to prove it? ...
0
votes
1answer
235 views

Solve a viscous Burgers' equation with a Newton-GMRes method

I implemented a preconditioner for a GMRes method. To test this preconditioner I want to solve this one dimensional viscous Burgers' equation $$\partial_t u(x,t) + u(x,t) \partial_x ...
1
vote
0answers
232 views

Applying modulus to determinant

I'm have trouble understanding how to get the determinant of a matrix and apply a modulus to it. I have have $((6)(16) - (15)(5))^{-1} \mod29$ I have no idea how to break this down.
1
vote
0answers
118 views

Matrix norm induced by a vector norm.

All matrices are real. $A$ is a matrix of size $n \times k$ with $k < n$ and has independent columns. The function $v(x) = \|Ax\|_1$ is a norm. What is the matrix norm induced by $v$? Is it of ...
0
votes
1answer
39 views

$A , B$ square matrices of size $n$ with real entries with $B$ invertible , the does $\exists c \in \mathbb R$ such that $\det (A+cB)=0$?

Let $A$ be a $n \times n$ matrix with real entries and $B$ is an invertible $n \times n$ matrix with real entries ; then does there exist $c \in \mathbb R$ such that $\det(A+cB)=0$ ?
0
votes
2answers
84 views

Question about Involuntary matrix?

If A be a 2x2 matrix with real entries. If $A^2=I$, then which of the following statements are true?: If $A\ne\pm I_2$ then $|A|=-1$ I know a supporting example ...
0
votes
2answers
23 views

Divide elements of a matrix by row

Suppose I have a matrix that looks like this: $$A=\begin{bmatrix} 1 & 1 & 0 & 0 \\ 1 & 0 & 1 & 1 \end{bmatrix}$$ I want to divide each term by the sum of terms in that row, ...
0
votes
2answers
26 views

What's the spectrum of this element

I am reading that $$ x = {1 \over 2}\left ( \begin{array}{cc}3 & 2 \\ 2 & 1 \end{array}\right )$$ is not positive since it has a negative eigenvalue. I think that $x$ is positive because it ...
0
votes
1answer
75 views

Matlab Recursion Loop

I need to create a loop to answer part c of the following linear algebra question. I'm new to Matlab so this is extremely confusing. I'm not sure how to make a loop that "counts" from 0-25 in order to ...
0
votes
0answers
257 views

distinct primes in a matrix (Putnam problem, December 6 2014)

Let $A$ be a matrix that is $r \times s$ . And suppose that it has at least $r + s$ distinct primes among absolute values. How do I show that the rank of $A$ must be at least $2$ . I think this would ...
0
votes
1answer
51 views

Prove that $A_{ii}$ is similar to an upper triangular matrix iff $A$ is similar to an upper triangular matrix

Let a field $\mathbb{F}$ and $n_1,\ldots,n_l$, natural numbers. For all $1\le i\le l$ Let $A_{ii} \in M_{n_i}(\mathbb{F})$. Let $$A = \left( {\matrix{ {{A_{11}}} & {{A_{12}}} & \cdots ...
0
votes
0answers
19 views

Let $A = QR $ be a reduced QR factorization. Why is the null($R$) $\subset$ null($A$)?

In a proof I am reading it was glossed over as obvious, yet I fail to see why this is.
4
votes
1answer
171 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 ...
4
votes
4answers
773 views

Quick question: matrix with norm equal to spectral radius

For $A\in \mathcal{M}_n(\mathbb{C})$, define: the spectral radius $$ \rho(A)=max\{|\lambda|:\lambda \mbox{ is an eigenvalue of } A\} $$ and the norm $$ \|A\|=max_{|x|=1}|A(x)| $$ where |.| is ...
2
votes
1answer
126 views

Taking the Derivative: Power Rule with Respect to Vector

I'm trying to take the derivative of \begin{equation} \phi\left(\mathbf{x}\mathbf{\theta}\right)\mathbf{x}^{\top} ...
1
vote
1answer
55 views

Find orthogonal matrices

Let $A=\begin{bmatrix} 1 & -1/2&-1/2 \\ -1/2 & 1& -1/2\\ -1/2&-1/2 &1 \end{bmatrix}$. Is it possible to find explicitly orthogonal matrices $P, Q$ such that ...