Quadratic form - Its lecture note PDF

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Rank, Signature & Index of the Quadratic form Let 𝑞 = 𝑋 𝑇 𝐴𝑋 be a quadratic form in the matrix form

i).Rank: The number of non-zero Eigen values of the matrix 𝐴 is called rank of the quadratic form.

ii).Signature: Signature of a quadratic form is defined as the triplet (𝑛0 , 𝑛+ , 𝑛− )

𝑛0 − 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑍𝑒𝑟𝑜 𝐸𝑖𝑔𝑒𝑛 𝑣𝑎𝑙𝑢𝑒𝑠

𝑛+ − 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒 𝐸𝑖𝑔𝑒𝑛 𝑣𝑎𝑙𝑢𝑒𝑠

𝑛− − 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑛𝑒𝑔𝑎𝑡𝑖𝑣𝑒 𝐸𝑖𝑔𝑒𝑛 𝑣𝑎𝑙𝑢𝑒𝑠

iii).Index: The Index of the quadratic form is the number of positive Eigen values of A.

Nature of a Quadratic form

A quadratic form 𝑞 = 𝑋 𝑇 𝐴𝑋 is said to be

i). Positive definite – Positive Eigen values ii). Negative definite – Negative Eigen values iii). Positive semi definite – Positive Eigen values and at least one is zero iv). Negative semi definite – Negative Eigen values and at least one is zero v). Indefinite – Positive as well as negative Eigen values Questions Find the nature, rank, signature & index of the following quadratic forms 1) 3𝑥 2 + 5𝑦 2 + 3𝑧 2 − 2𝑥𝑦 + 2𝑥𝑧 − 2𝑦𝑧 Soln:

3 −1 1 The matrix associated with the quadratic form is, 𝐴 = [−1 5 −1] 1 −1 3

Characteristic equation is

.𝜆3 − 𝑡𝑟(𝐴)𝜆2 + (𝐴11 + 𝐴22 + 𝐴33 )𝜆 − |𝐴| = 0 .𝜆3 − 11𝜆2 + 36𝜆 − 36 = 0

..𝜆 = 2, 3, 6

All the Eigen values are positive. So the quadratic form is positive definite.

Rank = Number of non-zero Eigen values = 3 Signature = (𝑛0 , 𝑛+ , 𝑛− ) = (0, 3, 0)

Index = Number of positive Eigen values = 3

2) 𝑥1 2 + 2𝑥2 2 + 𝑥3 2 − 2𝑥1 𝑥2 + 2𝑥2 𝑥3 Soln:

1 The matrix associated with the quadratic form is, 𝐴 = [−1 0

Characteristic equation is

−1 2 1

0 1] 1

.𝜆3 − 𝑡𝑟(𝐴)𝜆2 + (𝐴11 + 𝐴22 + 𝐴33 )𝜆 − |𝐴| = 0 .𝜆3 − 4𝜆2 + 3𝜆 = 0

𝜆(𝜆2 − 4𝜆 + 3) = 0 ..𝜆 = 0, 1,3

Here 2 Eigen values are positive and 1 Eigen value is zero. So the quadratic form is positive semi definite. Rank = Number of non-zero Eigen values = 2 Signature = (𝑛0 , 𝑛+ , 𝑛− ) = (1, 2, 0)

Index = Number of positive Eigen values = 2

3) 𝑥1 2 + 4𝑥2 2 + 𝑥3 2 − 4𝑥1 𝑥2 + 2𝑥1 𝑥3 − 4𝑥2 𝑥3 Soln:

1 −2 The matrix associated with the quadratic form is, 𝐴 = [−2 4 1 −2

Characteristic equation is

.𝜆3 − 𝑡𝑟(𝐴)𝜆2 + (𝐴11 + 𝐴22 + 𝐴33 )𝜆 − |𝐴| = 0 .𝜆3 − 6𝜆2 = 0

1 −2] 1

𝜆2 (𝜆 − 6) = 0 ..𝜆 = 0, 0,6

Here 1 Eigen value is positive and 2 Eigen values are zero. So the quadratic form is positive semi definite. Rank = Number of non-zero Eigen values = 1 Signature = (𝑛0 , 𝑛+ , 𝑛− ) = (2,1, 0)

Index = Number of positive Eigen values = 1

4) −3𝑥 2 − 3𝑦 2 − 3𝑧 2 − 2𝑥𝑦 − 2𝑥𝑧 + 2𝑦𝑧 Soln:

−3 −1 The matrix associated with the quadratic form is, 𝐴 = [ −1 −3 −1 1

Characteristic equation is

.𝜆3 − 𝑡𝑟(𝐴)𝜆2 + (𝐴11 + 𝐴22 + 𝐴33 )𝜆 − |𝐴| = 0 .𝜆3 + 9𝜆2 + 24𝜆 + 16 = 0 ..𝜆 = −1, −4, −4

All the Eigen values are negative. So the quadratic form is negative definite.

Rank = Number of non-zero Eigen values = 3 Signature = (𝑛0 , 𝑛+ , 𝑛− ) = (0,0, 3)

Index = Number of positive Eigen values = 0

5) 2𝑥1 𝑥2 + 2𝑥2 𝑥3 + 2𝑥1 𝑥3 Soln:

0 1 The matrix associated with the quadratic form is, 𝐴 = [ 1 0 1 1 Characteristic equation is

.𝜆3 − 𝑡𝑟(𝐴)𝜆2 + (𝐴11 + 𝐴22 + 𝐴33 )𝜆 − |𝐴| = 0 .𝜆3 − 3𝜆2 − 2 = 0

1 1] 0

−1 1] −3

..𝜆 = −1, −1,2

Here 2 Eigen values are negative & 1 Eigen value is positive. So the quadratic form is indefinite.

Rank = Number of non-zero Eigen values = 3 Signature = (𝑛0 , 𝑛+ , 𝑛− ) = (0,1, 2)

Index = Number of positive Eigen values = 1...