# Questions tagged [diagonalization]

For questions about matrix diagonalization, that is, writing a matrix, a bilinear form or an operator into a "basis" making this one diagonal. This tag is **NOT** for diagonalization arguments from logic and set theory.

109 questions
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### Simultaneous diagonalization

Let $V$ be a vector space of finite dimension and let $T,S$ linear diagonalizable transformations from $V$ to itself. I need to prove that if $TS=ST$ every eigenspace $V_\lambda$ of $S$ is $T$-...
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### Eigenvalues of the rank one matrix $uv^T$

Suppose $A=uv^T$ where $u$ and $v$ are non-zero column vectors in ${\mathbb R}^n$, $n\geq 3$. $\lambda=0$ is an eigenvalue of $A$ since $A$ is not of full rank. $\lambda=v^Tu$ is also an eigenvalue of ...
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### Square root of Positive Definite Matrix

Let $A$ be an $n\times n$ positive definite matrix. Show that there exists a unique positive definite matrix $B$ such that $B^2=A$. I do know the existence. But what about the uniqueness? Would you ...
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### Finding a 2x2 Matrix raised to the power of 1000

Let $A= \pmatrix{1&4\\ 3&2}$. Find $A^{1000}$. Does this problem have to do with eigenvalues or is there another formula that is specific to 2x2 matrices?
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### Diagonalization of an infinite matrix

Let A be an infinite matrix with all its first column elements equal to 1 and the rest of them equal to 0. A=\begin{pmatrix} 1 & 0 & 0 & 0 & \cdots\\ 1 & 0 & 0 & 0 &\...
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### New proof about normal matrix is diagonalizable.

I try to prove normal matrix is diagonalizable. I found that $A^*A$ is hermitian matrix. I know that hermitian matrix is diagonalizable. I can not go more. I want to prove statement use only this ...
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### If $A$ has two eigenvalues $\lambda _1, \lambda_2$ and $\dim (E_{\lambda_1})=n-1$, then $A$ is diagonalizable

Suppose that $A \in M_{n\times n}(\Bbb F)$ has two distinct eigenvalues $\lambda_{1}$ and $\lambda_{2}$ and that $\dim (E_{\lambda_1})=n-1$ show that $A$ is diagnolizable.
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### Is the matrix diagonalizable for all values of t?

For t∈R, let $A_t = \left( \begin{array}{ccc} t & 1 & 1 \\ 1 & t & 1 \\ 1 & 1 & t \end{array} \right)$. Find the Eigenvalues and Eigenvectors. Is $A_t$ diagonalizable for all ...
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### Where M is a matrix calculate a formula for M^n

Let $$M = \begin{bmatrix} -7 & 8 \\ -8 & -7 \end{bmatrix}.$$ Find formulas for the entries of $M^n$ where $n$ is a positive integer. (Your formulas should not contain complex numbers.) Your ...
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### Diagonalize the matrix A or explain why it can't be diagonalized

Diagonalize the matrix or explain why it cant be diagonalized $A=\begin{pmatrix}1 & 2 & 4 \\3 & 5 & 2 \\2 & 6 & 1\end{pmatrix}$ Hint: One eigenvalue is $λ=9$ So, i began the ...
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Let A = $\begin{bmatrix}1 & 1 & 4\\0 & 3 & -4\\0&0&-1\end{bmatrix}$. Is the matrix A diagonalizable? If so find a matrix P that diagonalizes A. Can you write A as a linear ...
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### Proving that if $A$ is a $8\times 8$ matrix over $\mathbb{R}$ and $A^3=A$, then $A$ is diagonalizable.

If $A$ is a $8\times 8$ matrix over $\mathbb{R}$ and $A^3=A$ then prove that $A$ is diagonalizable. I have got that the minimal polynomial of $A$ may be $x^3-x$ or $x$ or $x(x+1)$ or $x(x-1)$ in the ...