This set of Linear Algebra Problems focuses on “Canonical Form or Sum of Squares Form”.
1. Which of the following is not a condition for a given real nonsingular quadratic form, Q = XTAX, to be a negative definite quadratic form?
a) The number of positive square terms in the quadratic form is equal to zero
b) The rank of the matrix A is equal to the number of variables in the quadratic form (index)
c) All the eigen values of A are negative
d) The rank and index are equal
View Answer
Explanation: The quadratic form is said to be negative definitive if the rank is equal to index and the number of square terms is equal to zero or all the eigen values of the matrix are negative.
2. Signature of a quadratic form is the difference between the positive and negative terms in the canonical form.
a) True
b) False
View Answer
Explanation: Signature of a quadratic form is defined as ‘the difference between the number of positive and negative square terms in the canonical form.’
3. Determine the nature of the given matrix.
\(\begin{bmatrix}2 & 0 & 0 \\ 1 & 2 & 1 \\ 0 & 0 & 1\end{bmatrix} \)
a) Indefinite
b) Positive definite
c) Negative definite
d) Positive semi-definite
View Answer
Explanation: To find the nature of the matrix, we find the eigenvalues, |A- λI|=0
\(\begin{vmatrix}2-λ & 0 & 0 \\1 & 2-λ & 1 \\ 0 & 0 & 1-λ \end{vmatrix} = 0 \)
(2- λ) ((2- λ)(1- λ)) = 0
λ = 1, 2, 2
Since all the eigenvalues are positive, the condition for positive definite quadratic form is satisfied.
4. What is the signature of the quadratic form, \(Q = 7x_1^2+ 2x_2^2- 3x_3^2+x_{31}+ x_{23}=0\)?
a) 2
b) -2
c) 1
d) -1
View Answer
Explanation: We know that signature is the difference between the number of positive and negative square terms of a quadratic form. Therefore, we have signature = 2 – 1 = 1.
5. What is the index of the quadratic form, \(3x_1^2+ x_3^2+8x_{13}+ 9x_{21}+ 2x_{23}=0\)?
a) 2
b) 4
c) 3
d) 1
View Answer
Explanation: The index of a quadratic form is the number of positive square terms.
Hence, from the given form, we have index = 2.
6. If A is a matrix, such that, Ak = 0, for positive integer k, then, A is known as Nilpotent matrix.
a) True
b) False
View Answer
Explanation: For a matrix A, Ak+1 = A, where k is a positive integer is known as periodic matrix.
Whereas, if A2 = A, i.e. k=1, then it is known as idempotent matrix and if Ak = 0, then it is known as Nilpotent matrix.
7. Reduce the quadratic form to canonical form, \(3x_1^2+ 2x_2^2+8x_{12}+8x_{23}+8x_{31}=0\).
a) \(\begin{bmatrix}3 & 4 & 4 \\4 & 0 & 4 \\ 4 & 4 & 2\end{bmatrix} \)
b) \(\begin{bmatrix}3 & 4 & 4 \\4 & 2 & 0 \\ 4 & 4 & 0\end{bmatrix} \)
c) \(\begin{bmatrix}3 & 4 & 4 \\4 & 2 & 4 \\ 4 & 4 & 0\end{bmatrix} \)
d) \(\begin{bmatrix}3 & 4 & 0 \\4 & 2 & 4 \\ 4 & 4 & 0\end{bmatrix} \)
View Answer
Explanation: Given quadratic form is, \(3x_1^2+ 2x_2^2+8x_{12}+8x_{23}+8x_{31}=0\)
General form of the matrix can be written as, \(\begin{bmatrix}x_{11} & x_{12} & x_{13} \\ x_{21} & x_{22} & x_{23} \\ x_{31} & x_{32} & x_{33}\end{bmatrix} \)
Hence, the matrix form can be obtained by,
Placing the square term coefficients in the diagonal of the matrix such that, \(x_{ii}= x_i^2,\)
\(\begin{bmatrix}3 & x_{12} & x_{13} \\ x_{21} & 2 & x_{23} \\ x_{31} & x_{32} & 0\end{bmatrix} \)
Dividing the coefficients of terms xij between xij and xji positions, for example, the coefficient of x12 is 8, hence the term x12 =x21= 8/2 = 4.
Therefore, the matrix form of the given quadratic equation is,
\(\begin{bmatrix}3 & 4 & 4 \\ 4 & 2 & 4 \\ 4 & 4 & 0\end{bmatrix} \)
8. Write the expression for spur of a matrix for a 3×3 matrix whose entries are in the form of aij.
a) a11+ a12+a13
b) a11+ a21+a31
c) a12+ a22+a32
d) a11+ a22+a33
View Answer
Explanation: The spur of a matrix is nothing but trace of the matrix. It is defined as, ‘the sum of the diagonal elements of the matrix’.
9. The Canonical form is also known as ‘sum of squares’ form.
a) False
b) True
View Answer
Explanation: The canonical form is also known as sum of squares form since after reducing to canonical form, we get the terms as sum of squares.
10. Which among the following is not a type of quadratic form?
a) Positive Semi-definite
b) Negative definite
c) Partial definite
d) Indefinite
View Answer
Explanation: Quadratic forms can be classified based on the nature of the eigen values of the matrix into 5 types:
i. Positive definite
ii. Negative definite
iii. Positive Semi-definite
iv. Negative Semi-definite
v. Indefinite
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