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
60 views

Find the value of a so that the 2 x 2 matrix A is invertible

Find the value of a so that the $2 \times 2$ matrix $A = \begin{pmatrix} a-3 & 1\\ 2a+14 & a \\ \end{pmatrix}$ is invertible. do I just use the $\frac{1}{ad-bc}$ rule and solve for $a$, ...
2
votes
0answers
296 views

How to distribute 5-digit numbers in 5x5 matrices

I have 98000 5-digit numbers, from 00001 to 98000. I need to distribute these 98000 numbers in 14000 5x5 matrices. A matrix cell must contain only a digit from 0 to 9. Each matrix must receive 7 ...
1
vote
2answers
147 views

Fastest Gaussian Elimination Method?

I have this matrix and I want to know is there a method that I can always rely on to get the inverse without much trial and error. The matrix is; $$ \begin{bmatrix} 1 & 1 & 1\\ 0 & 3 ...
5
votes
4answers
1k views

Is every self-inverse matrix diagonalizable?

If $A=A^{-1}$, is there always a matrix C such that $C^{-1}AC$ is a diagonal matrix (containing only -1 and 1 in the main diagonal) ? How can I check with PARI/GP, if a given matrix is ...
0
votes
3answers
97 views

Linear Algebra Allowed Operations on Matrices

Can I get this matrix \begin{bmatrix} 1&1&0\\ 0&0&1\\ 0&0&0 \end{bmatrix} from the identity matrix $I_3$ like this: \begin{bmatrix} 1&0&0\\ 0&1&0\\ ...
3
votes
1answer
780 views

How do I find the initial state Matrix?

The question gives a $2\times2$ transition matrix: $$ \begin{bmatrix} 0.8 & 0.2\\ 0.3 & 0.7 \end{bmatrix}. $$ And then it gives me the initial state matrix but I'm wondering how do I find the ...
0
votes
1answer
82 views

Homework: canonical form of quadratic form

X=(x,y,z) Q(X) = $x^2 + 4xy + 6xz + 3y^2 +8yz +5z^2 $ I got by using completing the square method: Q(X) = $(x+2y+3z)^2 - (y+2z)^2$ so as I learned now I do: $u = x+2y+3z$ $v = y+2z$ $w = 0 $ ...
1
vote
1answer
33 views

If the first r columns of U are linearly independent, then so are the first r columns of A?

Let $U$ be a row echelon form of a square matrix $A$. If the first $r$ columns of $U$ are linearly independent, then should the first $r$ columns of $A$ be linearly independent? In my opinion, "Yes" ...
1
vote
1answer
56 views

Left shift operator $L: l^2 \rightarrow l^2$ on the sequence space $l^2$

$$L: l^2 \rightarrow l^2$$ is defined by $$b = (b_1,b_2,...) \mapsto Lb = (b_2,b_3,...)$$. $(Lb)_n = b_{n+1}$ respectively. How can I determine the adjoint endomorphism $L^*$? Kind regards George
0
votes
1answer
79 views

Spectral radius and Dominant Eigenvalue

What is the difference between the spectral radius and dominant eigenvalue? If they are one and the same then why do both get mentioned, for instance here ...
-2
votes
1answer
80 views

How to show that if $A, B$, and $A + B$ are invertible matrices with the same size, then $A(A^{ −1} + B^{ −1} )B(A + B)^ {−1} = I$ [duplicate]

Show that if $A$, $B$, and $A + B$ are invertible matrices with the same size, then $$A(A^{-1} + B^{-1})B(A + B)^{-1} = I$$ What does the result in part $1$ tell you about the matrix $A^{-1} + ...
8
votes
3answers
129 views

Find the expansion for $\det(I+\epsilon A)$ where $\epsilon$ is small without using eigenvalue.

I'm taking a linear algebra course and the professor included the problem that prove $$ \rm{det}(I+\epsilon A) = 1 + \epsilon\,\rm{tr}\,A + o(\epsilon) $$ Since the professor hasn't covered the ...
2
votes
3answers
145 views

Prove that if $A$ is invertible then $AA^\top$ is positive definite [duplicate]

I need to prove that if $A$ is a square invertible matrix then $AA^\top$ ($A$ multiply $A$ transpose) is positive definite. I tried to prove that all the eigenvalues are positive. I know that ...
2
votes
0answers
115 views

Self-inverse matrices with integers with pairwise different absolut values.

Let A be a self-inverse matrix ($A=A^{-1}$) with integer values such that no two integers have the same absolut value. Let M be the maximum of the absolut values (maximum-norm) of A. Which M is the ...
6
votes
2answers
692 views

What can be said about a matrix which is both symmetric and orthogonal?

I tried to find matrices A, which are both orthogonal and symmetric, this means $A=A^{-1}=A^T$. I only found very special examples like I, -I or the matrix $$\begin{pmatrix} 0 &0& -1\\ ...
1
vote
1answer
32 views

Shears using Matrix Methods

Determine the equation of the image of the graph: $$y=(x-1)^3 -2$$ after a shear of factor $1$ away from the $y$-axis, relative to the line $y=1$.
2
votes
6answers
152 views

$A$ a $n\times n$ matrix with real entries such that $A^3 = I$ but $A \ne I$

Let $A$ a $n \times n$ matrix with real entries such that $A^3 = I$ but $A \ne I$. a) Give an example that satisfies this conditions. b) what are the eigenvalues ​​of $A$? Well for $a)$ i ...
0
votes
1answer
73 views

Properties of invertible matrices

Show that if $A$, $B$, and $A + B$ are invertible matrices with the same size, then $A(A^{-1}+B^{-1})B(A+B)^{-1}=I$ What does the result in the first part tell you about the matrix ...
0
votes
1answer
22 views

Forms - unitary group?

If $Id$ is the $n$ by $n$ identity matrix and $J$ is the $2n$ by $2n$ matrix with $Id$ in the upper right corner and $-Id$ in the lower left corner, then $Sp_{2n} = \{G\in Gl_{2n} : G^{tr}JG = J\}$ is ...
1
vote
1answer
33 views

How do I find matrix that satisfies following conditions?

How do I find matrix $A$ with integer entries given two $2\times 1 $ vectors $\vec{x}, \vec{a} $ such that $$\vec{x} = A \vec{a}$$
0
votes
0answers
59 views

Why is smallest singular value of a singular matrix zero?

In my book, it is stated that the smallest singular value ($\sigma_n$ ) of a singular matrix is zero. I don't understand what it is so, please someone explain the reason to me.
1
vote
0answers
86 views

Calculate Geodesic Path of $N\times N$ matrix on Riemannian manifold of fixed rank

If I have two matrices $A(0)$ at $t=0$ and $A(1)$ at $t=1$, they are $N\times N$ matrices, and they are on the Riemannian manifold of rank $K$. How to calculate the geodesic path $A(t)$? I haven't ...
3
votes
1answer
63 views

Skew-symmetric matrix and exp function $e^A$

Let $A_{nXn}(\mathbb{R})$ Skew-symmetric matrix $A=-A^t$ prove that $e^A(e^A)^t=I$ while: $e^A=\sum_{i=0}^{\infty} \frac{A^n}{n!}$ I tried this: $A=-A^t \Rightarrow A$ is Diagonalizable with ...
1
vote
1answer
49 views

Why is this finding inverse of a matrix by row operation not working?

the correct answer is $\begin{pmatrix} -5&3&-6\\-6&3&-7\\-2&1&-2 \end{pmatrix}$ So I think the mistake might be in the first two row operations but I see nothing?
3
votes
2answers
121 views

Determine if a particular matrix is diagonalizable

my teacher gave me this exercise: Determine if this matrix is diagonalizable $ \begin{pmatrix} 1 & 1&1&1\\ 1&2&3&4\\ 1&-1&2&-2\\ 0&0&1&-2 ...
0
votes
1answer
20 views

Transformation of matrix with variables

I have this matrix: $$\eqalign{\pmatrix{1&-1&-1&|&-2\cr 3&1&-1&|&b\cr a&8&2&|&7\cr} &\sim\pmatrix{1&-1&-1&|&-2\cr ...
2
votes
1answer
28 views

Acquiring $Df(\mathbf{x})$

Sorry for the probably easy and silly question, but I try to teach myself linear algebra and I am stucked at "the derivative as a matrix" part. I know how to differentiate partially and I know how ...
-1
votes
1answer
227 views

Reverse of matrix multiplication

If I have matrices A, B, and C so that C = A * B, how can I get A from B and C? This page ( http://mathworld.wolfram.com/MatrixInverse.html ) tells me that A sould be C*B^-1, but using python and ...
0
votes
2answers
63 views

How do I know if a matrix is irreducible?

My course at university mainly works with 3x3 matrices. We are asked to put them in reduced echelon form which is the easy part, however I come across many matrices that I cannot seem to reduce into ...
6
votes
2answers
58 views

Quotient group $\mathbb Z^n/\ \text{im}(A)$

Let $A$ be an $n \times n$ matrix with integer coefficients and nonzero determinant. Can we say something about $ \mathbb{Z}^n /\ \text{im}( \phi )$ (here $\phi : v \mapsto Av$ )? This problem ...
2
votes
2answers
103 views

Prove that $A^{t}A$ is positive definite

$A$ is an invertible matrix over $\mathbb{R}$ (nxn). Show that $A^{T}A$ is positive definite. I looked up for it and found this two relevent posts but still need help. positive definite and transpose ...
3
votes
2answers
63 views

Find all $3\times3$ square matrices which commute with any $3\times3$ upper triangular matrix.

I'm not sure how to proceed. Let us find all possible solutions for the matrix $A$ which commutes with any other matrix $X$. In other words: $$AX=XA$$ Stating the matrix multiplication explicitly ...
0
votes
1answer
103 views

Find a 4 × 4 matrix A =[a ij ] whose entries satisfy the stated condition.

Find a $4\times4$ matrix $A=[a_{ij}]$ whose entries satisfy the stated condition. (a) $a_{ij}=i+j$ (b) $a_{ij}=i^{j-1}$ (c) $a_{ij}= \begin{cases} 1,&\text{if }|i-j|>1\\ ...
0
votes
1answer
47 views

Show: $\phi: \mathbb{R}^3 \rightarrow \mathcal{su}(2)$, $h \mapsto h \cdot \sigma$ is an isometric isomorphism

I found this problem and need some help. It is given: $$ \sigma_1 = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} $$ $$ \sigma_2 = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix} $$ ...
0
votes
1answer
56 views

Operations with $\text{SL}_2(\mathbb{Z})$

We define $\Omega :=\left\lbrace z \in \mathbb{H}\colon -\frac{1}{2} \leq \operatorname{Re}z \leq \frac{1}{2} \wedge |z| \geq 1\right\rbrace$. I want to show that the following holds: $$ \forall \, ...
0
votes
1answer
98 views

Symmetric matrix over inner product space

I try really hard to prove this Question. let $A_{nXn}(\mathbb{R})$ Symmetric matrix $A=A^t$ let $\lambda$ be the greatest Eigenvalue of A. we will define over the field $\mathbb{R}$ with the ...
0
votes
1answer
24 views

Exponent of polynomials (of matrices)

$A$ is a matrix over $\mathbb R$ (reals). Prove that for every $f,g\in \mathbb R[x]$, $\displaystyle e^{f(A)}\times e^{g(A)} = e^{f(A)+g(A)}$ I tried using the sigma writing but got stuck (I ...
3
votes
1answer
41 views

Matrix $A$ with characteristic polynomial

Given: Matrix $A$ with characteristic polynomial $p(x) = (x+3)^2(x-1)(x-5)$ Also given: $\rho(A+2I) + \rho(A+3I) + \rho(A-5I) = 9$ (btw $\rho$ means rank of the matrix) Prove: $A$ is ...
0
votes
0answers
12 views

explicit expression for the solution of a lower-triangular Toeplitz linear system

I need to find an explicit expression for $c_k$ ($k=0,1,2,\cdots$) in terms of $A_m$ ($m=0,1,2,\cdots$) and $b_n$ ($n=0,1,2,\cdots$)from the following lower-triangular Toeplitz linear system of ...
0
votes
3answers
74 views

What do you mean by the subspace spanned by the matrices?

What do you mean by the subspace spanned by the matrices $\{1,A,A^2,\dots,A^n\}$ where A is and $n\times n$ real or complex matrix.
1
vote
2answers
74 views

Eigenvalue by inspection?

Can I guess the eigenvalues of a $3\times3$ matrix having all entries $1$? for e.g., consider the matrix $ \left( \begin{array}{ccc} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 ...
1
vote
3answers
114 views

Minimal polynomial of diagonalizable matrix

It's a if and only if sentence (have to prove both directions) If a matrix $A$ (over $\mathbb{C}$) is diagonalizable then its minimal polynomial's roots are all of algebraic multiplicity 1. Any idea ...
2
votes
3answers
66 views

How to check whether this matrix is diagonalizable or not.

Let $\rm A$ be a complex $3\times3$ matrix with $\rm A^3=-1$. Which of the following statements are correct: $\rm A$ has three distinct eigenvalues. $\rm A$ is diagonalizable over $\Bbb ...
1
vote
2answers
50 views

trace function ($2\times2$) with ordered bases as linear transformation

We got trace function as following: $$\operatorname{tr}\begin{pmatrix} a & b\\ c & d\\ \end{pmatrix}=a+d$$ So now have to write down $[\operatorname{tr}]_{S_1,S_2}$, ...
2
votes
1answer
779 views

2x2 symmetric matrix is a subspace of vector space.

Can you kindly check my proof of the problem and correct if possible. The following $S=\{A\in M_{2,2} | AA^T=A^TA\}$ is a subspace of $V=M_{2,2}$ all real $2\times2$ matrices. My proof: S ...
1
vote
1answer
89 views

How can I find $T^{-1}(x,y,z)$ (inverted matrix) of a linear operator $T:V_3 \to V_3$

How can I find $T^{-1}(x,y,z)$ (inverted matrix) of a linear operator $T:V_3 \to V_3$, which matrix relative to the basis: $A=\{ (1,0,0), (1,1,0), (1,1,1)\}$ is: $$T_A= \begin{bmatrix} 2 &0 ...
3
votes
6answers
1k views

How can I prove that a square matrix is invertible if it satisfies this polynomial equation?

For a 3x3 matrix $C$, it is given that $$C^3+I=3C^2-C$$ I am then required to prove that $C$ is invertible. I have attempted a proof, below, but I am not sure it is valid or if there is a better ...
2
votes
1answer
44 views

Inverse of partitioned matrix, checking result

$A$ is an $n\times n$ matrix, partitioned as $$A=\begin{bmatrix} A_{11} & A_{12}\\ A_{21} & A_{22} \end{bmatrix},$$ where $A_{11}$ has dimensions $k\times k$ and $A_{11}$ and $A_{22}$ are ...
0
votes
1answer
55 views

Find a matrix B that satisfies A'A+lambda I=B'B, where A and B are the same size

Let A and B be two (kxn) complex matrices, where $k<n$. How can I find a matrix ...
0
votes
1answer
26 views

Characteristic polynomial of triangular blocks matrix

Let A be a triangular blocks matrix (the blocks are: A1,...,Ak). Show that CA(t)=CA1(t)*...*CAk(t). Any help ? thanks ;) (edit: CA and CAj are the characteristic polynomials of the blocks)