Articles of matrices

Nilpotent matrices with same minimal polynomial and nullity

From Hoffman and Kunze. Let $N_1$ and $N_2$ be 6 X 6 nilpotent matrices over the field F. Suppose that $N_1$ and $N_2$ have the same minimal polynomial and the same nullity. Prove that $N_1$ and $N_2$ are similar. Show that this is not true for 7 X 7 nilpotent matrices. Right so nilpotent matrices […]

Eigenvalue alteration counter-proof

If $A$ is a square $n\times n$ matrix, with $\lambda_1,\ldots,\lambda_n$ being the eigenvalues of $A$, $v_1$ being the eigenvector associated with eigenvalue $\lambda_1$, and $d$ the column vector of dimension $n$, then can anyone provide a counterexample to show the following statement is false: the eigenvalues of $A+v_1d^T$ are equal to $\lambda_1+d^Tv_1,\lambda_2,\ldots,\lambda_n$ ? Any help […]

Let $A:\mathbb R_2\rightarrow \mathbb R_2$ is a linear transformation defined as $(A(p))(x)=p'(x+1)$. Find matrix of $(A(p))(x)$.

Let $A:\mathbb R_2[x]\rightarrow \mathbb R_2[x]$ is a linear transformation defined as $(A(p))(x)=p'(x+1)$ where $\mathbb R_2[x]$ is the space of polynomials of the second order. Find all $a,b,c\in\mathbb R$ such that the matrix $ \begin{bmatrix} a & 1 & 0 \\ b & 0 & 1 \\ c & 0 & 0 \\ \end{bmatrix}$ is the […]

How to find $A=UDU^H$ in this case

I am given a matrix $A$. I find out it is normal. And I compute $\det(A-\lambda)=0$ and find that not all $\lambda_i$ are different, i.e., the eigenvalues are not distinct. Thus, I am not sure if the eigenvectors are L.I. Now, I want find a decomposition $$A=UDU^H$$ but how do I do it? And under […]

Strassen Multiplication?

How are the values of the 7 new matrices derived? I’m referring to the values that reduce matrix multiplication to 7 multiplications per level: $M_1 = \left(A_{1,1} + A_{2,2}\right)\left(B_{1,1} + B_{2,2}\right)$ … $M_7 = \left(A_{1,2} – A_{2,2}\right)\left(B_{2,1} + B_{2,2}\right)$ Could someone show how these values were arrived at?

How to solve the matrix equation $ABA^{-1}=C$ with $\operatorname{Tr}(A)=a$

I have the following matrix equation: $$ABA^{-1}=C$$ with $B$ and $C$ given and $A$ unknown. The constraint on $A$ is $\operatorname{Tr}(A)=a$ with $a\in\mathbb{R}$. The matrices are $N\times N$.

Difficulties understanding these statements about change of basis

I understood more or less what a change of basis matrix is and how I can use it to pass to one coordinate system to another. Basically, a change of basis matrix is a matrix whose columns are the entries of the vectors in the new basis. For example, if we have the basis $B […]

Special matrices with determinant 0

Define a quadratic matrix A with n rows and n columns by filling it with consecutive primes, starting with some prime p. The object is, to find the least starting prime p, such that A has determinant 0 , if there is one. For n=3 , PARI yields [2213 2221 2237] [2239 2243 2251] [2267 […]

Name of this matrix product?

Suppose $A$ and $B$ have columns as follows, $$A = \begin{bmatrix} a_1 & a_2 & \dots & a_n \end{bmatrix},$$ $$B = \begin{bmatrix} b_1 & b_2 & \dots & b_n \end{bmatrix}.$$ Is there any name for the following matrix “product”, or simple way of expressing in terms of standard matrix operations: $$A \times B := \begin{bmatrix} […]

What's the fastest way to take powers of a square matrix?

So I know that you can use the Strassen Algorithm to multiply two matrices in seven operations, but what about multiplying two matrices that are exactly the same. Is there a faster way to go about doing that (ie. by reducing the number of multiplications per iteration to something less than 7) ?