In linear algebra, the transpose of a matrix is an operator which flips a matrix over its diagonal;
that is, it switches the row and column indices of the matrix A by producing another matrix, often denoted by AT (among other notations).The transpose of a matrix was introduced in 1858 by the British mathematician Arthur Cayley.
i-th column of ##cof~A## =
$$
\begin{bmatrix}
(-1)^{I+1} det~A_{1i} \\
(-1)^{I+2} det ~A_{2i}\\
\vdots \\
(-1)^{I+n} det ~A_{ni}\\
\end{bmatrix}$$
Therefore, the I-th row of ##(cof~A)^t## = ##\big[ (-1)^{I+1} det~A_{1i}, (-1)^{I+2} det ~A_{2i}, \cdots, (-1)^{I+n} det ~A_{ni} \big]##
The I-th...
One way to express a function of a matrix A is by a power series (a Taylor expansion). It is not too difficult to show that two functions f(A) and g(A) with such a power series representation must commute, i.e. f(A)g(A) = g(A)f(A). But matrices typically do not commute with their own transpose...
While the prefix of the thread is Python, this could be easily generalised to any language.
It is absolutely not the first time I am working with an array, but definitely the first time I am facing the task of defining the transpose of a non-square matrix. I have worked so much with arrays in...
Homework Statement
Suppose T:V→U is linear and u ∈ U. Prove that u ∈ I am T or that there exists ##\phi## ∈ V* such that TT(##\phi##) = 0 and ##\phi##(u)=1.
Homework Equations
N/A
The Attempt at a Solution
Let ##\phi## ∈ Ker Tt, then Tt(##\phi##)(v)=##\phi##(T(v))=0 ∀T(v) ∈ I am T. So...
Homework Statement
If ##A## is an ##n \times n## matrix, show that the eigenvalues of ##T(A) = A^{t}## are ##\lambda = \pm 1##
Homework EquationsThe Attempt at a Solution
First I assume that a matrix ##M## is an eigenvector of ##T##. So ##T(M) = \lambda M## for some ##\lambda \in \mathbb{R}##...
Let ##\Lambda## be a Lorentz transformation. The matrix representing the Lorentz transformation is written as ##\Lambda^\mu{}_\nu##, the first index referring to the rows and the second index referring to columns.
The defining relation (necessary and sufficient) for Lorentz transforms is...
Homework Statement
Show that ##A## and ##A^T## have the same eigenvalues.
Homework EquationsThe Attempt at a Solution
If they have the same eigenvalues, then ##Ax = \lambda x## iff ##A^T x = \lambda x##. In other words, we have to show that ##|A - \lambda I| = 0## iff ##|A^T - \lambda I| =...
Hi PF!
When proving ##\left(AB\right)^T = B^T A^T## I was thinking of writing ##\left(AB\right)_{ij} = A_{ik} B_{kj} = D_{ij}##. Then ##\left(D\right)^T_{ij} = D_{ji} = A_{jk} B_{ki} = A^TB^T## but clearly this is incorrect. Can someone tell me where my mistake is made?
Thanks!
I have a problem of proving an identity about determinants. For ##A\in M_{m\times n}(\mathbb{R}),## a matrix with ##m## rows and ##n## columns, prove the following identity.
$$|\det(A^tA)|=\sum_{1\le j_1\le ... \le j_n \le m} (det(A_{j_1...j_n}))^2$$
where ##A_{j_1...j_n}## is the matrix whose...
Hi! As the title says, what is the derivative of a matrix transpose?
I am attempting to take the derivative of \dot{q} and \dot{p} with respect to p and q (on each one).
Any advice?
Homework Statement
Let A be an n x n matrix, and let v, w ∈ ℂn.
Prove that Av ⋅ w = v ⋅ A†w
Homework Equations
† = conjugate transpose
⋅ = dot product
* = conjugate
T = transpose
(AB)-1 = B-1A-1
(AB)-1 = BTAT
(AB)* = A*B*
A† = (AT)*
Definitions of Unitary and Hermitian Matrices
Complex Mod...
Alternate title: Is the textbook contradicting itself?
imgur link: http://i.imgur.com/3sTVgwr.jpg
But
imgur link: http://i.imgur.com/33Ufncb.jpg
So...it would appear that transposing has the property of linearity, but no matrix can achieve it...is transposing a linear transformation? The...
Homework Statement
Let L be a mapping such that L(A) = A^t, the transpose mapping. Find a matrix representing L with respect to the standard basis [1,1,1,1]
Homework EquationsThe Attempt at a Solution
So should I end up getting a 4x4 matrix here? I got 1,0,0,0 for the first column, 0,0,1,0 for...
Define the following
$$Z=
\begin{pmatrix}
0 & A \\
B^T & T
\end{pmatrix}$$
where we define $A$ and $B$ as $r \times m $ matrices and $T$ is an $m \times m$ matrix with nonzero distinct indeterminates at the diagonal, that is, $T_{i,i} = t_i$.
What is the meaning of $B^T$ ?
Homework Statement
Show that if a square matrix A satisfies
A3 + 4A2 -2A + 7I = 0
Mod note: It took me a little while to realize that the last term on the left is 7I, seven times the identity matrix. The italicized I character without serifs appeared to me to be the slash character /.
then so...
B=PRn-Q(Rn-1)/R-1
Mod note: This thread is closed. @Rodo, this appears to be homework that is misplaced, with no effort shown. You are welcome to repost in the Homework & Coursework section, but you need to use the homework template and show what you have tried.
I am told to compute C^T .. what is this implying? I'm guessing maybe the transpose? Is this correct? Also should I post matrix related questions here or in the pre-calculus forum? This is a discrete mathematics class I am using these things in by the way.
Hello,
While studying dual vectors in general relativity, it was written as we all know that dual vectors (under Lorentz Transformation) transform as follows:
\tilde{u}_{a} = \Lambda^{b}_{a}μ_{b}
where \Lambda^{b}_{a}= η_{ac}L^{c}_{d}η^{db}
I was wondering if one can prove the latter...
What does ATx=0 means?
Does this notation means if A = [3,2;1,2;4,4], then, AT = [3,1,4;2,2,4]
and ATx [x1;x2;x3] = 0?
The nullspace of the transposed of the matrix A?
Homework Statement
Let ##n## be a positive integer and let ##V = P_n## be the space of polynomials over ##R##. Let D be the differentiation operator on ##V## . Find a basis for the null space of the transpose operator ##D^t: V^*\to V^*##.
Homework Equations
Let ##T:V\to W## be a linear...
My textbook (regarding continuum mechanics) has the following identity that is supposed to be true for all tensors:
a\cdotTb = b\cdotTTa
But I don't get the same result for both sides when I work it out.
For each side, I'm doing the dot product last. For example, I compute Tb first and...
Hi,
The following equations are from linear regression model notes but there is an aspect of the matrix algebra I do not get.
I have, \mathbf{y} and \tilde{\beta} are a mx1 vectors, and \mathbf{X} is a mxn matrix.
I understand the equation...
I'm using a book that has a loot of errors (luckly most of them are easy to recognize, like a = instead of a ≠ or viceversa, but some are way more serious), and I'm not sure if it's a new error or a thing I don't understand.
Either I didn't understood all the steps of the proof or the correct...
Starting out a Lin Alg class - my prof wrote this on the board.
(ABC-1Dt)t = DC-1BtAt
On the right hand side, I get why D is D, why A and B are now both transpose, but why is C still inverse? I know the rule (D-1)t = (Dt)-1, but I do not see how the heck it applies here or what would make the...
Hi,
I am new to Math so I am trying to get some intuition.
Let's say I have a matrix A of n x n and a vector B of n x 1 what is the difference between A x B and A' x B?
Thanks
Hi
What is the angle between a vector (e.g. a row vector A) and it's transpose (a column vector) ? I know what transpose means mathematically but what is the intuition?
Thanks guys
Homework Statement
Find (X * Y-1)T - (Y * X-1)T
When X = [3 5]
.....[1 2]
and Y = [3 4]
...[2 3]
Homework Equations
Inverse= 1/ad-bc [d -b]
......[-c a]
The Attempt at a Solution
I got:
[9 -6 ]
[14 -9]
But the answer is:
[-3 -2]
[6 3]I did the problem twice and got the same answer so I...
Homework Statement
Let A and B be two square matrices of order n such that AB = A and BA = B. Then, what is the value of [(A + B)t]2?
Homework Equations
The Attempt at a Solution
[(A + B)t]2 = AtAt + AtBt + BtAt + BtBt.
I tried to use the fact that AB = A and BA = B to keep...
Show that a matrix and its transpose have the same eigenvalues.
I must show that det(A-λI)=det(A^t-λI)
Since det(A)=det(A^t)
→det(A-λI)=det((A-λI)^t)=det(A^t-λI^t)=det(A^t-λI)
Thus, A and A^t have the same eigenvalues.
Is the above enough to prove that a matrix and its transpose have the...
Homework Statement
let transpose of A be noted by A`
Show that if the matrix product AB is permitted, then so is the product B`A`, where
B`A`=(AB)`
Homework Equations
C_{ij}=ƩA_{ik} B_{kj} where summing from k=1 to m
A`_{ij} = A_{ji}The Attempt at a Solution
It wants me to use the...
So I'm given a matrix A that is already in RREF and I'm supposed to find the null space of its transpose.
So I transpose it. Do I RREF the transpose of it? Because if I transpose a matrix that's already in RREF, it's no longer in RREF. But if I RREF the transpose, it gives me a matrix with 2...
I'm trying to find a way to prove that the determinant of the transpose of an endomorphism is the determinant of the original linear map (i.e. det(A) = det(Aᵀ) in matrix language) using Dieudonne's definition of the determinant expressed in terms of an alternating bilinear form but am having...
Homework Statement
here is the answer:
The Attempt at a Solution
I can't figure out how the matrix listed above in the answer is supposed to add up to -1. that's the only way that a and b can equal each other, that is, if they both add up to -1.
Homework Statement
I don't see how you multiply a matrix by its transpose. If a matrix is 3 x 2 then its transpose is 2 x 3. I thought you couldn't multiply matrices unless they have the same rows and columns.
In Sakurai's Modern Quantum Mechanics, he develops the Dirac notation of bras and kets. In one part, he states (page 17):
<B|X|A>
= (<A|X^|B>)*
= <A|X^|B>*
where X^ denotes the Hermitian adjoint (the conjugate transpose) of the operator X.
My question is, since a bra is the conjugate...
To find the least squares polynomial of degree 2 to
approximate points (X,Y) given in the table
X_____________Y
1_____________36
1.9_____________-49...
Homework Statement
Let A be an m x n matrix with rank(A) = m < n. As far as the eigenvalues of A^{T}A is concerned we can say that...
Homework Equations
The Attempt at a Solution
If eigenvalues exist, then
A^{T}Ax = λx where x ≠ 0.
The only thing I think I can show is that...
1Homework Statement
Prove that the transpose of a tensor is a tensor.
Homework Equations
Definition of the transpose: a\bulletTb = b\bulletT^Ta where a and be are arbitrary vectors
The Attempt at a Solution
This isn't homework per se, I'm 60 yo and studing continuum mechanics...
Homework Statement
We are looking for the matrix A
Homework Equations
(A^transpose)^transpose=A
The Attempt at a Solution
i would start with finding the transpose of the matrix.
-5 0
-8 -7
Why is every matrix (complex) similar to its transpose?
I am looking at a typical jordan block and I see that the transpose of the nilpotent part is again nilpotent and actually similar to the nilpotent part. I can see that the scalar part of the jordan block does not change under...
Homework Statement
Show that if A is orthogonal, then AT is orthogonal.
Homework Equations
AAT = I
The Attempt at a Solution
I would go about this by letting A be an orthogonal matrix with a, b, c, d, e, f, g, h, i , j as its entries (I don't know how to draw that here)...but...