#csirnet #mathematics #linearalgebra FEB 2022 | Signature of Quadratic Form

Hello friends, this is Mahesh Yavelekar and you are watching Next Gen Maths. If you have not subscribed my channel, then please subscribe to the channel and press the bell icon. In this video, we are going to discuss the problem of CSI 8-10222, problem of Linear Algebra. The question is V equal to set A belongs to 3 cross 3 matrix over real numbers with the condition A transpose plus A belongs to R dot I, where I is the identity matrix. Consider the quadratic form defined as, quadratic form, square of trace of A minus trace of A square. What is the signature of this quadratic form? There are again 4 options, 1st option plus plus plus plus, 2nd option plus 0 0 0, 3rd option plus minus minus minus and 4th option minus minus minus 0. Solution? To solve this problem, we have to construct quadratic form of A, but that A is given in V. So, what is V? V is a set of 3 cross 3 matrix over real numbers with the condition A transpose plus A belongs to R dot I. So, first of all, R dot I is a set K times identity matrix, where K belongs to R. Therefore, K 0 0, 0 K 0, 0 0 K, K belongs to R are the elements of R dot I. Now we are given that A transpose plus A belongs to R dot I. Therefore, A transpose plus A equal to K 0 0, 0 K 0, 0 0 K, for some K belongs to R. Consider matrix A with a generalized entry ABCBEFGHI. This A belongs to V. Therefore, satisfy the condition A transpose plus A belongs to R dot I. A transpose is equal to ADGBEHCFI. A plus A transpose equal to 2A, B plus D, C plus G, B plus D, 2E, F plus H, C plus G, F plus H and 2I. Now, A plus A transpose belongs to R dot I. Therefore, A plus A transpose equal to K 0 0, 0 K 0, 0 0 K, for some K belongs to real number. So, if we compare A plus A transpose with the scalar matrix K 0 0, 0 K 0, 0 0 K, we have 2A equal to K, so A equal to K by 2, 2A equal to K, so A equal to K by 2 and 2I equal to K, so I equal to K by 2. Then, we have B plus D equal to 0, so D equal to minus B, C plus G equal to 0, therefore G equal to minus C and F plus H equal to 0, therefore H equal to minus F. Now, substitute all these values in matrix A, so A equal to K by 2, B, C minus B, K by 2, F, G equal to minus C, so minus C minus F, K by 2. This is matrix A. We want to calculate quadratic form of A, which is square of trace of A minus trace of A square. So, we need trace of A, so trace of matrix A is sum of its diagonal, so K by 2 plus K by 2 plus K by 2, which will give the 3K by 2. Now, we need trace of A square, first we will find A square, A square equal to A into A, for the trace we required only diagonal values, so we are calculating only in diagonal values. So, first diagonal value is multiplication of first row, first column, which will give the K square by 4 minus B square minus C square. Second diagonal value is a multiplication of second row, second column, which will give K square by 4 minus B square minus F square and third diagonal value is a multiplication of third row and third column, which will give K square by 4 minus C square minus F square. Trace of A square is a sum of diagonal values of A square, so if we take the sum of diagonal values of A square, we have 3K square by 4 minus 2B square minus 2C square minus 2F square. Quadratic form is a square of trace of A minus trace of A square, so trace of A is 3K and if we take the square, so 3K whole square minus trace of A square is minus 3K square by 4 minus 2B square minus 2C square minus 2F square. Now, if we simplify, then we have 9K square by 4 minus 3K square by 4, which will give 6K square by 4 plus 2B square plus 2C square plus 2F square. Therefore, quadratic form is 6 by 4 equal to 3 by 2 K square plus 2B square plus 2C square plus 2F square. Now matrix associated with this quadratic form is M equal to 3 by 2 0 0 0, 0 2 0 0 0, 0 0 2 0 0 0 0 2, because we have only square entries, other entries are 0 and square entry only give the diagonal values. M is the diagonal matrix, therefore the eigenvalues of M are the diagonal values 3 by 2, 2, 2, 2. All the diagonal values of M are positive, therefore the signature of matrix M is plus plus plus plus, therefore signature of quadratic form is plus plus plus plus, which is option number 1 and option number 1 is the correct answer. Thanks for watching, for more videos like, share, subscribe, press the bell icon of next video. Bye.