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Q34E

Expert-verifiedFound in: Page 337

Book edition
5th

Author(s)
Otto Bretscher

Pages
442 pages

ISBN
9780321796974

**suppose a certain ****${\mathbf{4}}{\mathbf{\times}}{\mathbf{4}}$ matrix A has two distinct real Eigenvalues. what could the algebraic multiplicities of These eigenvalues be? Give an example for each possible Case and sketch the characteristic polynomial.**

**A is a $\mathbf{4}\mathbf{\times}\mathbf{4}$ matrix its characteristic polynomial is of degree 4, which means it is** .

$\left(\lambda -{\lambda}_{1}\right)\left(\lambda -{\lambda}_{2}\right)\left(\lambda -{\lambda}_{3}\right)\left(\lambda -{\lambda}_{4}\right)$

role="math" localid="1659586683331" $A=\left\{\begin{array}{cccc}17& 0& 0& 0\\ 0& 46& 0& 0\\ 0& 0& 46& 0\\ 0& 0& 0& 17\end{array}\right\}$

**A polynomial is a mathematical expression made up of indeterminates and coefficients that only involves the operations of addition, subtraction, multiplication, and non-negative integer exponentiation of variables.**

$4\times 4$Matrix A has two distinct real Eigenvalues:

If A is a $4\times 4$ matrix, its characteristic polynomial is of degree , which means it is $\left(\lambda -{\lambda}_{1}\right)\left(\lambda -{\lambda}_{2}\right)\left(\lambda -{\lambda}_{3}\right)\left(\lambda -{\lambda}_{4}\right)$.

If we want two distinct eigenvalues, then we have two cases:

The two eigenvalues are of algebraic multiplicities 1 and 3 respectively.

Example is:** **

$A=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 2& 0& 0\\ 0& 0& 2& 0\\ 0& 0& 0& 2\end{array}\right]$

The eigenvalue is a number that indicates how much variance exists in the data in that direction; in the example above, the eigenvalue is a number that indicates how spread out the data is on the line.

Both eigenvalues are of algebraic multiplicity 2

Example is:

$A=\left\{\begin{array}{cccc}17& 0& 0& 0\\ 0& 46& 0& 0\\ 0& 0& 46& 0\\ 0& 0& 0& 17\end{array}\right\}$

Hence,

_{}$A=\left\{\begin{array}{cccc}17& 0& 0& 0\\ 0& 46& 0& 0\\ 0& 0& 46& 0\\ 0& 0& 0& 17\end{array}\right\}$

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