suppose a certain 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 matrix its characteristic polynomial is of degree 4, which means it is .
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.
Matrix A has two distinct real Eigenvalues:
If A is a matrix, its characteristic polynomial is of degree , which means it is .
If we want two distinct eigenvalues, then we have two cases:
The two eigenvalues are of algebraic multiplicities 1 and 3 respectively.
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
In all parts of this problem, let V be the linear space of all 2 × 2 matrices for which is an eigenvector.
(a) Find a basis of V and thus determine the dimension of V.
(b) Consider the linear transformation T (A) = A from V to . Find a basis of the image of T and a basis of the kernel of T. Determine the rank of T .
(c) Consider the linear transformation L(A) = A from V to . Find a basis of the image of L and a basis of the kernel of L. Determine the rank of L.
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