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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2
votes
1answer
28 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? ...
1
vote
2answers
349 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
2answers
33 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 ...
0
votes
1answer
70 views

If $A^2 = O$, is $A = O$?

I think the answer is "no", but I'm trying to find the flaw in this reasoning: $A^2 = O \implies AA = O \implies A^{-1}AA = A^{-1}O \implies A = O$ This shouldn't be true, as far as I know, so what ...
2
votes
5answers
295 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 ...
0
votes
0answers
18 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.
0
votes
0answers
10 views

Rank Minimization

I have a n*m matrix, the rank of matrix (r) is near to min(m,n) I want to minimize the rank by removing some of the rows or columns to get r << min(m,n) The goal is to achieve least rank ...
2
votes
1answer
48 views

Solving equation with integer matrices as unknowns

I am currently working on a problem, where I need to know for square integer matrices, $A$ and $B$, whether or not there exists square integer matrices, $X$ and $Y$, such that $X(A-I)Y=B-I$, where $I$ ...
0
votes
1answer
18 views

Generalized Eigensystems

I am looking for solution algorithms for a second order generalization of the eigenvalue problem. A, B, and C are n-by-n matrices, I is the n-dimensional identity matrix, $\lambda_i$ is an unknown ...
0
votes
1answer
22 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
0answers
34 views

Represent $90$ degree clockwise rotation about the $z$-axis as a $3\times 3$ matrix

I honestly can't find anything regarding an issue I have with transformational matrices. I understand that this matrix: $$\begin{pmatrix} \cos 90&-\sin 90&0\\ \sin 90&\cos 90&0\\ ...
1
vote
2answers
27 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
2answers
21 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
23 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 ...
3
votes
1answer
57 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
0answers
16 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.
1
vote
0answers
41 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
0answers
12 views

strassens matrix multiplication for getting square of a matrix

Using the same approach as of strassen's only 5 multiplications are sufficient to compute square of a matrix. A[2][2]=[a, b, c, d]. the multiplications are a* a, d* d, b(a+d), c(a+d), b*c. If we ...
0
votes
1answer
23 views

Solve Unknown Matrix Variables

I have a markov chain matrix with probabilities as such, on finding the steady state.. ...
0
votes
2answers
19 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, ...
1
vote
1answer
21 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} ...
0
votes
2answers
32 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
25 views

Matlab code for creating matrix

I have a problem, I am working with numerical approximations. Now i have $-1=x_0<x_1 <...<x_n=1$ with $x_i=x_{i-1}+h$ and $h=2/n$. Now given a function $f$ , I want to find $f(x_i)$ for ...
1
vote
1answer
44 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 ...
1
vote
1answer
48 views

Can the equation $\mathbf{Av}=\mathbf{b}$ be solved as $\mathbf{v}=\mathbf{A}^{-1}\mathbf{b}$?

Say I have a $3\times 3$ matrix called $\mathbf A$ and a column matrix vector $\mathbf v$ and another column matrix vector $\mathbf b$. If I have the equation $\mathbf{Av}=\mathbf{b}$ where I know ...
0
votes
1answer
20 views

creating a tridiagonal matrix in scilab/matlab

I want to create a tridiagonal matrix in scilab/matlab such that it uses for loops. I dont know how to create the matrix but here is what i have started ...
4
votes
1answer
46 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 ...
0
votes
1answer
21 views

Vector and matrices question (Introduction to finite mathematics 3rd edition)

Stuck on this exercise for quite a while now; (u1+u2+u3+u4) multiplied by the matrix \begin{pmatrix} u1\\ u2\\ u3\\ u4 \end{pmatrix}= \begin{pmatrix} 1\\ 3\\ 5\\ 7 \end{pmatrix} Find u1,u2,u3,u4 ...
3
votes
1answer
46 views

Is there always a mapping from invertible $A$ to any $B \in M_n(\Bbb R)$?

Let $A, B$ be $n\times n$ matrices, then $1)$ If $A$ is invertible then for every $B$ exists a matrix $X \in M_n(\Bbb R)$ such that $AX = B$. $2)$ If for every $B$ there exists a matrix $X \in ...
0
votes
1answer
31 views

Matrices proofs

Let $A,B$ be a $n × n$ matrices $A = I + AB$ Prove: 1) A is invertible and AB = BA. 2) If B is a symmetric matrix then so is A. 3) $B^3 = 0$ if and only if $A = I + B + B^2$ OK for the first ...
3
votes
1answer
48 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 ...
0
votes
0answers
132 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
0answers
22 views

Transformation matrix of 2D image

I have 2D image with 256x256 pixels, while top left point is [0,0]. My task is to create transformation matrix, which will mirror this image by 10th row and then rotate it clockwise by the [5,5] ...
0
votes
1answer
20 views

Matrix multiplication proofs

Let $A,B$ be a $n × n$ matrices Prove or disprove: $(A + B)^2 = A^2 + 2AB + B^2$ $A^3− I = (A − I) (A^2 + A + I)$ I'm having trouble figuring this out, how can I prove the first one? the only ...
0
votes
2answers
30 views

What am I doing wrong in solving the following matrix easy equation?

If $$\begin{bmatrix}2&1\\7&4\end{bmatrix}A\begin{bmatrix}-3&2\\5&-3\end{bmatrix}=I_2$$ Then I think A will be: ...
0
votes
2answers
22 views

Proving an Equality involving Matrices

I have been thinking about this problem for a while and I still can't come up with a solution. Could you please point me in a direction? Here's the problem. ...
1
vote
1answer
33 views

Do I need to evaluate exact value of $A^9$ to find $Det.(2A^9B^{-1})$?

Do I need to evaluate exact value of $A^9$ to find $Det.(2A^9B^{-1})$? Actually $A=[\begin{matrix}32\\21\end{matrix}],B=[\begin{matrix}31\\73\end{matrix}]$
1
vote
1answer
20 views

representation of a matrix norm of the inverse of a matrix

Let $A$ be an $n$ by $n$ nonsingular matrix and suppose that a matrix norm $|||\cdot|||$ is induced by the vector norm $\lVert \cdot \rVert$ on $\mathbb{C}^n$. Show that $$|||A^{-1}||| = ...
3
votes
1answer
21 views

Terminologies for $nA=0$

Let $A$ be a matrix over a ring. Suppose $nA=0$ for some $n\in\mathbb{Z}$. I wonder if there are terminologies for such A and $n$.
0
votes
1answer
26 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
2answers
37 views

Prove that matrices commute

Let $I$ be the identity matrix, $S$ any stochastic matrix and $a \in (0,1)$. How can we prove that $$(I-S) \text { and } (I-aS)^{-1}$$ are commutative?
1
vote
2answers
50 views

How can I show that these matrices don't commute

I want to show that $A\in O(2) \setminus SO(2)$ and $B \in SO(2)$ don't commute. To prove it I wrote $$ B = \left ( \begin{array}{cc} \cos \theta & \sin \theta \\ - \sin \theta & \cos \theta ...
1
vote
1answer
23 views

How can I prove a formula for integration by parts for matrix functions?

Questions asks State and prove a formula for integration by parts in which the integrands are matrices functions. But, For any given matrices, we can differentiate and integrate by considering ...
2
votes
1answer
52 views

Prove that the real and imaginary parts of an eigenvector are linearly independent.

Say we have a 2 by 2 matrix $A$ with real entries and $A$ has a complex eigenvector $V = a+bi$ with corresponding complex eigenvalue $\lambda$. How do I prove that the vectors $\mathrm{Re}(V) = a$ and ...
0
votes
1answer
27 views

Determinant of a matrix of size n

I received a matrix for which I need to calculate its determinant. $$ A = \begin{pmatrix} 0 & 1 & 1 & \cdot & \cdot & \cdot & 1 \\ 1 & 0 & 1 & 1 & \cdot & ...
2
votes
1answer
34 views

Conditions for invertibility of $AA^t$

Let $A$ be a matrix whose rows are pmfs (i.e. nonnegative entries, each row sums to $1$). Are there any conditions on $A$ weaker than invertibility such that $AA^t$ is invertible?
2
votes
1answer
21 views

Proving Schur's Theorem can create both an upper and lower triangular matrix.

I've seen two very small distinctions in Schur's theorem, one is that for any $n\times n$ matrix $A, \exists$ a Unitary $U$ s.t. $U^*AU$ is upper triangular, the other is that $U^*AU$ is lower ...
7
votes
3answers
299 views

How to prove that $A^2$ is a symmetric matrix

Conjecture 1 : Let $A$ be a real matrix such that $A^5=A A^T A A^T A$. Then $A^2$ is a symmetric matrix. (here $A^T$ denotes the transpose of a matrix A). I guess that the following is also ...
0
votes
1answer
18 views

general idempotent matrix possible values of the determinant

If A is a general idempotent matrix, calculate the possible values of det (A) I caculated the det = o what other values can it equal?