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Questions tagged [matrices]

For any topic related to matrices. This includes: systems of linear equations, eigenvalues and eigenvectors (diagonalization, triangularization), determinant, trace, characteristic polynomial, adjugate and adjoint, transpose, Jordan normal form, matrix algorithms (e.g. LU, Gauss elimination, SVD, QR), invariant factors, quadratic forms, etc. For questions specifically concerning matrix equations, use the (matrix-equations) tag.

A matrix is a rectangular array of elements, usually numbers or variables, arranged in rows and columns. A matrix with $m$ rows and $n$ columns has $m \times n$ elements and is called an $m$ by $n$ matrix. Matrices are a part of .

Matrices can be added and subtracted. Furthermore, if they have compatible shapes, they can be multiplied. More precisely, given two matrices $A$ and $B$, the matrix $AB$ is defined when the number of columns of $A$ is equal to the number of rows of $B$. In particular, given a natural number $n$, any two matrices $A$ and $B$ with $n$ columns and $n$ rows can be multiplied in both ways (that is, both $AB$ and $BA$ exist).

For questions specifically concerning matrix equations, use the tag.

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Show that $|\det(A_n)|=n^{n/2}$

For k $\ge2$ we recursively define $A_{2^k}$ as $\begin{bmatrix} A_{2^{k-1}} & A_{2^{k-1}} \\ A_{2^{k-1}} & -A_{2^{k-1}} \end{bmatrix}$ and $A_1=[1]$ The problem is to show that $|\det(A_n)|=n^{n/2}$ My attempt: we do an induction on…
John Cataldo
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Derivatives of a the Matrix diagonal function

If A is a not diagonal but symetric matrix and diag() is a function such that returns the diagonal, i.e. diag(A) is a matrix of zeros except on the diagonal. Im interested in the derivative of vec(diag(A)) with respect to x where A is a function of…
Johan
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Higher powers of matrix

$$ A = \begin{bmatrix} 1 & 1 & 13 \\ 5 & 2 & 6 \\ -2 & -1 & -3 \\ \end{bmatrix} $$ Find $A^{14}+3A-2I$. One way is to find $A^2$, then $A^4$, then $A^8$, then $A^{14}$. Another way is using eigen values and diagonal matrix…
San
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how do you recognize a positive (semi)definite matrix?

I understand that the definition of (semi-)definiteness of matrix $A$ is $$\forall z_{\neq0}\in\mathbb R^k: z^TAz>0$$ I also know that this doesn't mean that all elements of a negative definite matrix $A$ are negative (in fact, they may all be…
user56834
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Rank of matrix as a difference of ranks

If $X$ is an $n \times p$ matrix of rank $r$ and $C = AX$ for some $q \times n$ matrix $A$ with rank$(A) = q$, how do I show that rank $(X(I-C^{-}C))=$ rank$(X)-$ rank$(C)$? I can show that rank $(X(I-C^{-}C))\geq $ rank$(X)-$ rank$(C)$, but how do…
Ken Dunn
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Why is Matrix Addition defined as element by element addition?

Matrix multiplication is defined as row by column multiplication. It represents linear transformation. Why isn't matrix addition defined in a similar way: row by column addition. Given to matrices A and B (of the same size). What transformation, if…
Hass
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Why is this determinant zero ? (block matrix)

I have two non-singular matrices $P_1$ and $P_2$ such that their sum $P_1+P_2$ is also non-singular. The calculations I need to do lead me to the following block matrix: $$\begin{pmatrix} (P_1+P_2)^{-1} & (P_1+P_2)^{-1}-P_2^{-1} \\…
M. Boyet
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Diagonally dominant matrix with matrix similarity

Applying similarity transform to a matrix $A$ gives: $$M=P^{-1}AP$$ $M$ and $A$ have same eigenvalues. What is the way to to find $P$ such that $M$ is diagonally dominant case of $A$? $M$ is diagonally dominant if $$|{m_{ii}}| \ge \sum\limits_{j \ne…
kotoll
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rotation matrix for particle track

Struggling with some equations here. I've got a particle I need to track in an absolute reference frame, but each step I move/rotate it relative to its own reference frame. I need to track it's absolute position while moving it relatively. I'm…
gallamine
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Matrix Multiplication $1 \times n$ and $n \times 1$ Case

I have seen a matrix multiplication with programming and other means such as the convention of dot product where $[3][2]=6$. However, I am unsure if it is technically correct to have it $[3][2]=[6]$. If you have a matrix $A$ which is $ 1 \times n$ …
W. G.
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Full rank vs short rank matrix

I am given the definition: "A matrix A is of full rank if and only if the vector $d$ for which $Ad=0$ is $d=0$." I don't understand: if we have the matrix $$\begin{pmatrix}1&2&3\\ 4&5&6\\ 13&19&88\end{pmatrix}$$ It is not of full rank, but what…
metrix
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How to calculate the negative half power of a matrix

I have a square matrix called A. How can I find $A ^ {-1/2}$. Should I compute $a_{ij} ^ {-1/2}$ for all of its elements? Thanks
Taban
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Prove or disprove: For $2\times 2$ matrices $A$ and $B$, if $(AB)^2=0$, then $(BA)^2=0$.

My goal is to prove or disprove the following claim: For $2\times 2$ matrices $A$ and $B$, if $(AB)^2=0$, then $(BA)^2=0$. My thoughts on the question: I know that $AB=O$ does not imply that $BA=O$, so my first impression was that it is false. I…
Taladris
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What is a generic matrix

What is a generic matrix? I try to google this but even can not find a definition. And what is a generic nilpotent matrix? Does generic has some canonical meaning?
user198206
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Moore Penrose Inverse for symmetric matrix

Is there any proof that a MPI of symetric matrix is symmetric matrix? Basically I need that Moore-Penroses invers of positive semidefinite matrix is positive semidefinite. I can show that x^T(A+)x >=0. But A+ also need to be symmetric for positive…
Glass12
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