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Questions tagged [linear-algebra]

For questions on linear algebra, including vector spaces, linear transformations, systems of linear equations, spanning sets, bases, dimensions and vector subspaces.

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394 votes
34 answers
143k views

If $AB = I$ then $BA = I$

If $A$ and $B$ are square matrices such that $AB = I$, where $I$ is the identity matrix, show that $BA = I$. I do not understand anything more than the following. Elementary row operations. Linear ...
Dilawar's user avatar
  • 6,165
74 votes
2 answers
12k views

Determinant of a rank $1$ update of a scalar matrix, or characteristic polynomial of a rank $1$ matrix

This question aims to create an "abstract duplicate" of numerous questions that ask about determinants of specific matrices (I may have missed a few): Characteristic polynomial of a matrix ...
Marc van Leeuwen's user avatar
42 votes
9 answers
13k views

Determinant of a matrix with diagonal entries $a$ and off-diagonal entries $b$ [duplicate]

I have the following $n\times n$ matrix: $$A=\begin{bmatrix} a & b & \ldots & b\\ b & a & \ldots & b\\ \vdots & \vdots & \ddots & \vdots\\ b & b & \ldots &...
M.B.M.'s user avatar
  • 5,426
21 votes
5 answers
6k views

reference for linear algebra books that teach reverse Hermite method for symmetric matrices

May 11, 2019. Evidently the original method should be attributed to Lagrange in 1759. I got confused, Hermite is much more recent. January 13, 2016: book that does this mentioned in a question today, ...
Will Jagy's user avatar
  • 140k
307 votes
5 answers
89k views

Norms Induced by Inner Products and the Parallelogram Law

Let $ V $ be a normed vector space (over $\mathbb{R}$, say, for simplicity) with norm $ \lVert\cdot\rVert$. It's not hard to show that if $\lVert \cdot \rVert = \sqrt{\langle \cdot, \cdot \rangle}$ ...
Hans Parshall's user avatar
825 votes
18 answers
173k views

What's an intuitive way to think about the determinant?

In my linear algebra class, we just talked about determinants. So far I’ve been understanding the material okay, but now I’m very confused. I get that when the determinant is zero, the matrix doesn’t ...
Jamie Banks's user avatar
  • 13.1k
62 votes
10 answers
21k views

A linear operator commuting with all such operators is a scalar multiple of the identity.

The question is from Axler's "Linear Algebra Done Right", which I'm using for self-study. We are given a linear operator $T$ over a finite dimensional vector space $V$. We have to show that $T$ is a ...
abeln's user avatar
  • 555
75 votes
13 answers
35k views

Do matrices $ AB $ and $ BA $ have the same minimal and characteristic polynomials?

Let $ A, B $ be two square matrices of order $n$. Do $ AB $ and $ BA $ have same minimal and characteristic polynomials? I have a proof only if $ A$ or $ B $ is invertible. Is it true for all cases?
Andy's user avatar
  • 2,276
138 votes
4 answers
107k views

Prove $\operatorname{rank}A^TA=\operatorname{rank}A$ for any $A\in M_{m \times n}$

How can I prove $\operatorname{rank}A^TA=\operatorname{rank}A$ for any $A\in M_{m \times n}$? This is an exercise in my textbook associated with orthogonal projections and Gram-Schmidt process, but I ...
jaynp's user avatar
  • 2,161
225 votes
13 answers
398k views

Inverse of the sum of matrices

I have two square matrices: $A$ and $B$. $A^{-1}$ is known and I want to calculate $(A+B)^{-1}$. Are there theorems that help with calculating the inverse of the sum of matrices? In general case $B^{-...
Tomek Tarczynski's user avatar
100 votes
8 answers
180k views

Union of two vector subspaces not a subspace?

I'm having a difficult time understanding this statement. Can someone please explain with a concrete example?
NSjonas's user avatar
  • 1,167
11 votes
1 answer
13k views

Kernels and reduced row echelon form - explanation

The following text is written in my textbook and I don't really understand it: If $A = (a_{ij}) \in$ Mat$(m x N, F)$ is a matrix in reduced row echelon form with $r$ nonzero rows and pivots in the ...
UnclearTextbook's user avatar
73 votes
4 answers
103k views

Is the product of symmetric positive semidefinite matrices positive definite?

I see on Wikipedia that the product of two commuting symmetric positive definite matrices is also positive definite. Does the same result hold for the product of two positive semidefinite matrices? ...
nullUser's user avatar
  • 28k
70 votes
9 answers
40k views

How to prove and interpret $\operatorname{rank}(AB) \leq \operatorname{min}(\operatorname{rank}(A), \operatorname{rank}(B))$?

Let $A$ and $B$ be two matrices which can be multiplied. Then $$\operatorname{rank}(AB) \leq \operatorname{min}(\operatorname{rank}(A), \operatorname{rank}(B)).$$ I proved $\operatorname{rank}(AB) \...
user avatar
59 votes
12 answers
41k views

What is a good book to study linear algebra? [duplicate]

I'm looking for a book to learn Algebra. The programme is the following. The units marked with a $\star$ are the ones I'm most interested in (in the sense I know nothing about) and those with a $\circ$...
266 votes
7 answers
358k views

Eigenvectors of real symmetric matrices are orthogonal

Can someone point me to a paper, or show here, why symmetric matrices have orthogonal eigenvectors? In particular, I'd like to see proof that for a symmetric matrix $A$ there exists decomposition $A = ...
Phonon's user avatar
  • 4,038
68 votes
6 answers
12k views

Similar matrices and field extensions

Given a field $F$ and a subfield $K$ of $F$. Let $A$, $B$ be $n\times n$ matrices such that all the entries of $A$ and $B$ are in $K$. Is it true that if $A$ is similar to $B$ in $F^{n\times n}$ then ...
Melesia's user avatar
  • 681
132 votes
1 answer
103k views

Prove that simultaneously diagonalizable matrices commute

Two $n\times n$ matrices $A, B$ are said to be simultaneously diagonalizable if there is a nonsingular matrix $S$ such that both $S^{-1}AS$ and $S^{-1}BS$ are diagonal matrices. a) Show that ...
diimension's user avatar
  • 3,430
108 votes
8 answers
69k views

Understanding of the theorem that all norms are equivalent in finite dimensional vector spaces

The following is a well-known result in functional analysis: If the vector space $X$ is finite dimensional, all norms are equivalent. Here is the standard proof in one textbook. First, pick a ...
user avatar
48 votes
4 answers
19k views

How can we prove Sylvester's determinant identity?

Sylvester's determinant identity states that if $A$ and $B$ are matrices of sizes $m\times n$ and $n\times m$, then $$ \det(I_m+AB) = \det(I_n+BA)$$ where $I_m$ and $I_n$ denote the $m \times m$ ...
Bruce George's user avatar
  • 1,970
32 votes
5 answers
16k views

The characteristic and minimal polynomial of a companion matrix

The companion matrix of a monic polynomial $f \in \mathbb F\left[x\right]$ in $1$ variable $x$ over a field $\mathbb F$ plays an important role in understanding the structure of finite dimensional $\...
DBr's user avatar
  • 4,820
96 votes
4 answers
96k views

Are the eigenvalues of $AB$ equal to the eigenvalues of $BA$?

First of all, am I being crazy in thinking that if $\lambda$ is an eigenvalue of $AB$, where $A$ and $B$ are both $N \times N$ matrices (not necessarily invertible), then $\lambda$ is also an ...
dantswain's user avatar
  • 1,125
87 votes
12 answers
218k views

How to show that $\det(AB) =\det(A) \det(B)$?

Given two square matrices $A$ and $B$, how do you show that $$\det(AB) = \det(A) \det(B)$$ where $\det(\cdot)$ is the determinant of the matrix?
Learner's user avatar
  • 2,706
71 votes
2 answers
57k views

Order of general- and special linear groups over finite fields.

Let $\mathbb{F}_3$ be the field with three elements. Let $n\geq 1$. How many elements do the following groups have? $\text{GL}_n(\mathbb{F}_3)$ $\text{SL}_n(\mathbb{F}_3)$ Here GL is the general ...
user avatar
192 votes
21 answers
386k views

Calculate Rotation Matrix to align Vector $A$ to Vector $B$ in $3D$?

I have one triangle in $3D$ space that I am tracking in a simulation. Between time steps I have the previous normal of the triangle and the current normal of the triangle along with both the current ...
user1084113's user avatar
  • 2,099
154 votes
16 answers
213k views

Where to start learning Linear Algebra? [closed]

I'm starting a very long quest to learn about math, so that I can program games. I'm mostly a corporate developer, and it's somewhat boring and non exciting. When I began my career, I chose it because ...
46 votes
5 answers
19k views

Given a matrix, is there always another matrix which commutes with it?

Given a matrix $A$ over a field $F$, does there always exist a matrix $B$ such that $AB = BA$? (except the trivial case and the polynomial ring?)
Strin's user avatar
  • 1,539
31 votes
3 answers
10k views

If a field $F$ is such that $\left|F\right|>n-1$ why is $V$ a vector space over $F$ not equal to the union of $n$ proper subspaces of $V$

If $U_1$, $U_2,\ldots,U_n$ are proper subspaces of a vector space $V$ over a field $F$, and $|F|\gt n-1$, why is $V$ not equal to the union of the subspaces $U_1$, $U_2,\ldots,U_n$?
Freeman's user avatar
  • 5,439
362 votes
11 answers
238k views

What is the importance of eigenvalues/eigenvectors?

What is the importance of eigenvalues/eigenvectors?
Ryan's user avatar
  • 5,519
74 votes
5 answers
25k views

Matrix is conjugate to its own transpose

Mariano mentioned somewhere that everyone should prove once in their life that every matrix is conjugate to its transpose. I spent quite a bit of time on it now, and still could not prove it. At the ...
George's user avatar
  • 1,897
57 votes
2 answers
24k views

Why is the ring of matrices over a field simple?

Denote by $M_{n \times n}(k)$ the ring of $n$ by $n$ matrices with coefficients in the field $k$. Then why does this ring not contain any two-sided ideal? Thanks for any clarification, and this is ...
awllower's user avatar
  • 16.6k
178 votes
7 answers
58k views

Is there a quick proof as to why the vector space of $\mathbb{R}$ over $\mathbb{Q}$ is infinite-dimensional?

It would seem that one way of proving this would be to show the existence of non-algebraic numbers. Is there a simpler way to show this?
Elchanan Solomon's user avatar
186 votes
24 answers
73k views

Looking for an intuitive explanation why the row rank is equal to the column rank for a matrix

I am looking for an intuitive explanation as to why/how row rank of a matrix = column rank. I've read the proof on Wikipedia and I understand the proof, but I don't "get it". Can someone ...
hari_sree's user avatar
  • 2,121
28 votes
3 answers
14k views

The Center of $\operatorname{GL}(n,k)$

The given question: Let $k$ be a field and $n \in \mathbb{N}$. Show that the centre of $\operatorname{GL}(n, k)$ is $\lbrace\lambda I\mid λ ∈ k^∗\rbrace$. I have spent a while trying to prove this ...
Ben's user avatar
  • 1,440
27 votes
2 answers
24k views

The range of $T^*$ is the orthogonal complement of $\ker(T)$

How can I prove that, if $V$ is a finite-dimensional vector space with inner product and $T$ a linear operator in $V$, then the range of $T^*$ is the orthogonal complement of the null space of $T$? I ...
user62182's user avatar
  • 370
51 votes
4 answers
26k views

When are minimal and characteristic polynomials the same?

Assume that we are working over a complex space $W$ of dimension $n$. When would an operator on this space have the same characteristic and minimal polynomial? I think the easy case is when the ...
rigot's user avatar
  • 513
230 votes
8 answers
251k views

Proof that the trace of a matrix is the sum of its eigenvalues

I have looked extensively for a proof on the internet but all of them were too obscure. I would appreciate if someone could lay out a simple proof for this important result. Thank you.
JohnK's user avatar
  • 6,484
49 votes
3 answers
18k views

Diagonalizable transformation restricted to an invariant subspace is diagonalizable

Suppose $V$ is a vector space over $\mathbb{C}$, and $A$ is a linear transformation on $V$ which is diagonalizable. I.e. there is a basis of $V$ consisting of eigenvectors of $A$. If $W\subseteq V$ is ...
NGY's user avatar
  • 1,027
41 votes
4 answers
11k views

Traces of all positive powers of a matrix are zero implies it is nilpotent

Let $A$ be an $n\times n$ complex nilpotent matrix. Then we know that because all eigenvalues of $A$ must be $0$, it follows that $\text{tr}(A^n)=0$ for all positive integers $n$. What I would like ...
yoshi's user avatar
  • 421
39 votes
7 answers
27k views

Determinant of a block lower triangular matrix

I'm trying to prove the following: Let $A$ be a $k\times k$ matrix, let $D$ have size $n\times n$, and $C$ have size $n\times k$. Then, $$\det\left(\begin{array}{cc} A&0\\ C&D \end{array}\...
Buddy Holly's user avatar
  • 1,189
37 votes
3 answers
10k views

Intersection of kernels and linear dependence of functionals

I am trying to prove the following. I have seen it alluded to in other places of the internet (this site included) but without proof. Let $L,L_1\ldots L_n$ be linear functionals on a vector space $X$....
201p's user avatar
  • 807
9 votes
3 answers
13k views

Finding Intersection of an ellipse with another ellipse when both are rotated

Equation of first ellipse=> $$\dfrac {((x-xFirstEllipseCenterPoint)\cdot \cos(A)+(y-yFirstEllipseCenterPoint)\cdot \sin(A))^2}{(a_1^2)}+\dfrac{((x-xFirstEllipseCenterPoint)\cdot \sin(A)-(y-...
andikat dennis's user avatar
113 votes
3 answers
16k views

Why are infinitely dimensional vector spaces not isomorphic to their duals?

Assuming the axiom of choice, set $\mathbb F$ to be some field (we can assume it has characteristics $0$). I was told, by more than one person, that if $\kappa$ is an infinite cardinal then the ...
Asaf Karagila's user avatar
  • 396k
76 votes
3 answers
51k views

Eigenvalues of the rank one matrix $uv^T$

Suppose $A=uv^T$ where $u$ and $v$ are non-zero column vectors in ${\mathbb R}^n$, $n\geq 3$. $\lambda=0$ is an eigenvalue of $A$ since $A$ is not of full rank. $\lambda=v^Tu$ is also an eigenvalue of ...
user avatar
18 votes
8 answers
11k views

How to calculate the determinant of all-ones matrix minus the identity? [duplicate]

How do I calculate the determinant of the following $n\times n$ matrices $$\begin {bmatrix} 0 & 1 & \ldots & 1 \\ 1 & 0 & \ldots & 1 \\ \vdots & \vdots & \ddots & \...
Mohan's user avatar
  • 15k
158 votes
8 answers
246k views

Show that the determinant of $A$ is equal to the product of its eigenvalues

Show that the determinant of a matrix $A$ is equal to the product of its eigenvalues $\lambda_i$. So I'm having a tough time figuring this one out. I know that I have to work with the characteristic ...
onimoni's user avatar
  • 6,466
146 votes
10 answers
229k views

When is matrix multiplication commutative?

I know that matrix multiplication in general is not commutative. So, in general: $A, B \in \mathbb{R}^{n \times n}: A \cdot B \neq B \cdot A$ But for some matrices, this equations holds, e.g. A = ...
Martin Thoma's user avatar
  • 9,881
69 votes
2 answers
47k views

Does non-symmetric positive definite matrix have positive eigenvalues?

I found out that there exist positive definite matrices that are non-symmetric, and I know that symmetric positive definite matrices have positive eigenvalues. Does this hold for non-symmetric ...
user avatar
25 votes
5 answers
25k views

How to compute the determinant of a tridiagonal Toeplitz matrix?

How to show that the determinant of the following $(n\times n)$ matrix $$\begin{pmatrix} 5 & 2 & 0 & 0 & 0 & \cdots & 0 \\ 2 & 5 & 2 & 0 & 0 & \cdots & ...
user46450's user avatar
  • 321
129 votes
8 answers
150k views

How to prove that eigenvectors from different eigenvalues are linearly independent [duplicate]

How can I prove that if I have $n$ eigenvectors from different eigenvalues, they are all linearly independent?
Corey L.'s user avatar
  • 1,309

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