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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.

55954 questions
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Must basis for column space be consist columns?

Lets say we have the following Matrix $$\left[ \begin{matrix} 2 & 0 & 0\\ 0 & 2 & 0\\ 0 & 0 & 0 \\ \end{matrix} \right]$$ Obviously a basis for this would be…
Sahar Rabinoviz
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Recurrence relation for a matrix

Question: If $A=\left[\begin{array}{ccc} 1 & 0 & 0\\ 1 & 0 & 1\\ 0 & 1 & 0\end{array}\right]$ then show that $A^n=A^{n-2}+A^2-I$, $n\geq3$. Hence find $A^{50}$. Where do I begin? I solved upto $A^8=4A^2-I$ and $A^9=4A^2+A-I$ with similar pattern…
Nitin
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Which topic(s) does this matrix come from? What is the name of this matrix?

Which topic(s) does this matrix come from? What is the name of this matrix? $$\pmatrix{1&1&1\\1&\omega&\omega^2\\1&\omega^2&\omega}$$
BooKInquiry
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Complex matrix operations question

If we have $4$ Real matrices $A,B,C,D$, is it possible to show that $(A+Bi)(C+Di) = E+Fi$ in $3$ nxn matrix multiplications?
Pat
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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?
anonymous
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Is there ever a case where a mapping is not preserved/closed under scalar multiplication.

Is there ever a case where a mapping is not preserved/closed under scalar multiplication? Every problem I have encountered has always been closed under scalar multiplication. However, closed under addition is easier to see if a mapping is linear or…
user3170251
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Need help to understand one line in the proof

Lemma. If the matrices $\mathbf{X}$ ($n\times p$ design matrix of full column rank ) and $\mathbf{L}_{2}$ satisfy $\mathbf{L}_{2}^{\prime}\mathbf{X}=\mathbf{0}$ and $\Omega$ is positive definite then $$ \Omega - \Omega…
MYaseen208
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The number of symmetric matrices of order 5 with each element either 0 or 1

Question is to find The number of symmetric matrices of order 5 with each element either 0 or 1 . What i am trying is If i take matrix of order 2 $$A=\left[\matrix{ A & B \\ B & C \ }\right]$$ Here i am having $2.2.2$ cases in total .Assuming A…
godonichia
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matrix convexity

How to determine the convexity of the following functions: $X^p$, in which $p$ is a real number and $X$ is $n \times n$ symmetric positive definite matrix. $e^X$ in which $X$ is a $n \times n$ symmetric matrix and $n \geq 2$ The convexity…
mining
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Prove multiplication of column matrix and skew symmetric matrix is null matrix

I was reading this proof from our text book. I didnt get one step. Q. If A is a skew symmetric matrix and X is a column matrix, then show X'AX is a null matrix. Proof Since A is skew symmetric, A'=-A Let A be square matrix of order n, X of nx1 and…
Mahesha999
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Change of basis matrix verification

Let $B$ and $C$ be two bases. To find the change of basis matrix $\phi_{B,C}$, I compute $\phi_{SB,B}$ and $\phi_{SB,C}$. Create the new matrix $T=[\phi_{SB,B}|\phi_{SB,C}]$. Reducing it to reduced row echelon form, should yield $T=[I|\phi_{B,C}$,…
Andrew
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prove value of trace of a matrix

Suppose that $X$ is an $m \times n$ matrix and that the matrix $X^TX$ is invertible. $H = X(X^TX)^{-1}X^T$ where $X^T$ is transpose of $X$; $(X^TX)^{-1}$ is inverse of $X^TX$ we are asked to show that $\text{tr}(H) = n$. By considering $H^2v$,…
Samuel Song
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prove the identity of the kronecker product of four matrix

Suppose that $H_i$($i=1,2,3,4$) are four arbitrary matrices, the Kronecker product relation below $$ (H_1\otimes H_2)\otimes(H_3\otimes H_4)=H_1\otimes (H_2\otimes H_3)\otimes H_4 $$ holds due to the fact that $(H_1\otimes H_2)\otimes H_3=H_1\otimes…
Roger209
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Build a rotation matrix that rotates 30 degrees along the axis (1,1,1)?

Why does the following image equal what it equals? Why does x,y,z equal that? 1/sqrt(3),1/sqrt(3),1/sqrt(3)
terry
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Inversion of n x n matrix

A matrix F is given: $$ F = [e^{i\frac{2\pi kl}{n}}]_{k,l=0}^{n-1} $$ Find $$ F^{-1} $$ I know Gaussian method for inverting matrices but I suppose it doesn't apply to matrices with not given exact n value. Could you tell me what are the methods…
wisniak
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