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Found in: Page 337

### Linear Algebra With Applications

Book edition 5th
Author(s) Otto Bretscher
Pages 442 pages
ISBN 9780321796974

# suppose a certain ${\mathbf{4}}{\mathbf{×}}{\mathbf{4}}$ 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 $\mathbf{4}\mathbf{×}\mathbf{4}$ 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" $A=\left\{\begin{array}{cccc}17& 0& 0& 0\\ 0& 46& 0& 0\\ 0& 0& 46& 0\\ 0& 0& 0& 17\end{array}\right\}$

See the step by step solution

## Step 1: definition of polynomial

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.

$4×4$Matrix A has two distinct real Eigenvalues:

If A is a $4×4$ 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:

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

## Step 2: definition of eigenvalues

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,

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