just calculus one

1. Government Engineering
college,Bhavnagar
• presented by : paresh parmar
(140210107040)

2. DIAGONALIZATION
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3. Problem reduction
• A square matrix M is called diagonalizable if
we can find an invertible matrix, say P, such
that the product P–1 M P is a diagonal matrix.
• A diagonalizable matrix can be raised to a high
power easily.
– Suppose that P–1 M P = D, D diagonal.
– M = P D P–1.
– Mn = (P D P–1) (P D P–1) (P D P–1) … (P D P–1)
= P Dn P–1.
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4. Example of diagonalizable matrix
• Let
• A is diagonalizable because we can find a
matrix
such that
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5. Now we know how fast it
converges to 0.2
• The matrix can be diagonalized
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6. Convergence to equilibrium
• The trajectory of the unemployment rate
– the initial point is set to 0.1
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7. EIGENVECTOR AND EIGENVALUE
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8. How to diagonalize?
• How to determine whether a matrix M is
diagonalizable?
• How to find a matrix P which diagonalizes a
matrix M?
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9. From diagonalization to
eigenvector
• By definition a matrix M is diagonalizable if
P–1 M P = D
for some invertible matrix P, and diagonal
matrix D.
or equivalently,
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10. The columns of P are special
• Suppose that
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11. Definition
• Given a square matrix A, a non-zero vector v is
called an eigenvector of A, if we an find a real
number (which may be zero), such that
• This number is called an eigenvalue of A,
corresponding to the eigenvector v.
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Matrix-vector product Scalar product of a vector

12. Important notes
• If v is an eigenvector of A with eigenvalue ,
then any non-zero scalar multiple of v also
satisfies the definition of eigenvector.
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k 0

13. Geometric meaning
• A linear transformation L(x,y) given by: L(x,y) = (x+2y, 3x-4y)
• If the input is x=1, y=2 for example,
the output is x = 5, y = -5.
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x x + 2y
y 3x – 4y

14. Invariant direction
• An Eigenvector points at a direction which is invariant under the linear
transformation induced by the matrix.
• The eigenvalue is interpreted as the magnification factor.
• L(x,y) = (x+2y, 3x-4y)
• If input is (2,1), output is magnified by a factor of 2, i.e., the eigenvalue is 2.
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15. Another invariant direction
• L(x,y) = (x+2y, 3x-4y)
• If input is (-1/3,1), output is (5/3,-5). The length is increased by a factor of 5, and
the direction is reversed. The corresponding eigenvalue is -5.
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16. Eigenvalue and eigenvector of
First eigenvalue = 2, with eigenvector
where k is any nonzero real number.
Second eigenvalue = -5, with eigenvector
where k is any nonzero real number.
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17. Summary
• Motivation: want to solve recurrence
relations.
• Formulation using matrix multiplication
• Need to raise a matrix to an arbitrary power
• Raising a matrix to some power can be easily
done if the matrix is diagonalizable.
• Diagonalization can be done by eigenvalue
and eigenvector.
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