# 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.

37,271 questions
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### Range space of matrices over $\mathbb{Z}$

Let A and B be $m \times n$ matrices over $\mathbb{Z}$ such that $B=PAQ$ for some invertible matrices P and Q. Then can we tell that Range space of A is same as that of the range space of B when A ...
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### $A$ is a $n\times n$ matrix and there exists a unique matrix $B$ such that $AB=I_n$. Prove that $BA=I_n$ and $B=A^{-1}$. [duplicate]

$A$ is a $n\times n$ matrix and there exists a unique matrix $B$ such that $AB=I_n$. Prove that $BA=I_n$ and $B=A^{-1}$. I have no idea how $BA=I_n$ and $AB=I_n$, one implies another. Please help me ...
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### What does it mean if, for a real input $x, x^T x$ is a singular matrix? [closed]

I was trying to find a least squares solution using the normal equation. I created an 'input' vector $x$ with a fixed coefficient times $i$ plus a random number. As far as I know, for a $x^T * x$ to ...
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### Finding Eigenvalues of a 3x3 Matrix INVOLVING LAMBDA

Need some help with determinants involving eigen's. I understand the steps used in the process below, but I don't understand how my teacher knew that he had to do those steps to get a nice 0 row with ...
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### Let $I \neq A ∈ M_{n×n}(\mathbb{R})$ be an involutory matrix. Show that the Jordan canonical form of $A$ is a diagonal matrix.

Let $I \neq A ∈ M_{n×n}(\mathbb{R})$ be an involutory matrix. Show that the Jordan canonical form of $A$ is a diagonal matrix. I'm not sure how to do this, any solutions/hints are greatly appreciated....
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### Computing the variance of a matrix [closed]

If $X$ and $Y$ are independent random variables and we let $x=\begin{bmatrix}X+2\\3X\\3Y-1\end{bmatrix}$ How do you define a variance-like characteristic of a random vector variable? Unfortunately, ...
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### SVD proof to If $A$ is of full rank, then $A^{*}A$ is of full rank

Provided $A$ is a full rank matrix $\in\mathbb{C^{m\times n}}$, then $A^{*}A$ is of full rank. Suppose $m\gt n$. There is a solution to this problem: solution link, and the top solution makes sense ...
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### Column vector of simultaneous equaations' solution

Struggling with some basics of Linear Algebra. Please help. Let's restrict the discussion to 2D space & consider the following simultaneous equations: $2x + 3y = 8, x + 2y = 5$ I understand ...
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### Question on how a matrix is calculated from an example [closed]

I have the following laplacian matrix given to me in a textbook. In the textbook, the matrix calculation is always done from the 3 x 3 matrix (the methods I learnt makes me cut the matrix further ...
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### $LU$ Decomposition of antidiagonal matrix

I cannot find the $LU$ decomposition of anti-diagonal matrix $$\begin{bmatrix} 0 &0 &0 &1 \\ 0 &0 &2 &0 \\ 0 &3 &0 &0 \\ 4 &0 &0 &0 \end{bmatrix}.$$ ...
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### Why do the columns of the inverse of a matrix (defined as a linear operator) form an orthogonal basis in an inner product space?

Let V be a vector space over C and W be an inner product space over C with inner product <., .> and T:V --> W be a linear transformation. Find an orthogonal basis for V = R^3 with the inner product ...
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