Hello Students! I'm Dr Gajendra Purohit and you're watching my youtube channel where I upload videos on Engg mathematics & BSc If you're preparing for any competitive exam where higher mathematics is asked then my youtube channel will be very helpful Click on the i tab to watch to watch my previous videos I've uploaded videos on eigen values & eigen vectors Today i'll be discussing more concepts on eigen values & eigen vectors This is the modified content that'll help you a lot This matrix formed has determinant in the form of an equation which is called characteristic equation on solving the characteristic equation we get the value of lambda i.e. the eigen value let's look at an example when we multiply the lambda inside we get lambda in the places of 1 so we directly subtract lambda inside the matrix The determinant will be this is the characteristic polynomial and when we equate it to 0 it becomes characteristic polynomial now we solve and the values of lambda will be the Eigen values For the concepts of homogeneous equation click on i tab homogeneous equation are always consistent which means that the solution is either trivial (zero) or non trivial (non-zero) so this is a matrix, we subtract lambda in the diagonal values using the determinant we get the characteristic polynomial we equate the polynomial to 0 and calculate the values of lambda These values are eigen values If you form the equation incorrectly then the eigen values will be incorrect and eigen vectors will be incorrect So there is a short trick to check, click on i tab to watch the video Using short trick I've explained how to calculate the eigen values & characteristic polynomial click on the i tab to watch the video on solving the equation we get the value of X which is the corresponding eigen vector this is my first class on eigen values & vectors in which i'll be discussing the theory, concepts & fundamentals In the next class i'll be solving a lot of examples & questions we calculate the determinant this is the characteristic polynomial for 22 matrix click on the i tab to watch for 33 matrix to check if the eigen values are correct or not, the sum of the eigen values (roots) should be equal to the sum of the diagonal elements of the matrix now let's learn to calculate the eigen vectors let's see how to calculate the eigen vectors for lambda = 1 we put lambda = 1 in place of lambda In a matrix which has both its eigen values as non zero and its rank is 2 let's see for an example this has rank 2 and non zero As we put 1 in it this becomes zero and another non zero so the rank gets reduced by 1, the number of non zero will be equal to the rank whenever the rank is 1 this type of equation is homogeneous equation and the rank is less than 1 so the solution will be non zero solution or infinite solution and this infinite solution is the eigen vector we take x1 = k so x2 will be equal to -3k this infinite solution will be equal to eigen vectors these values are the eigen values corresponding to lambda this here is the infinite solution when 3 eigen values are different for eg 1, 2 and 3 the eigen vectors will be linearly independent in the next video I'll be explaining it with an example if in a case two eigen values are same, the corresponding eigen vectors will be linearly independent or linearly dependent which falls under another topic that is diagonalization which we'll discuss in next class eigen values are also known as characteristic vales, eigen roots, characteristic roots or latent roots the set of eigen values is called the spectrum of matrix A and the largest eigen value is called the spectral radius of A With these two concepts we get another topic that is diagonalization In the next class we'll discuss it with an example for eg the eigen values of a matrix is 1, 2 and 3 these values are repeating only once so the algebraic multiplicity of these values will be 1 now if we have 1, 1, 3 now as 1 is repeating twice the algebraic multiplicity of 1 will be 2 and of 3 will be 1 when we write this in equation form we get the same form so this is the algebraic multiplicity that is power Geometric multiplicity is related to eigen vectors, the number of linearly independent eigen vectors the algebraic multiplicity is greater than or equal to the geometric multiplicity there is a 3*3 matrix with eigen values as lambda 1, lambda 2 & lambda 3 the determinant will be equal to the product of its eigen values the trace of the matrix is the sum of the diagonal elements so the trace is equal to the sum of eigen values for example so the trace =7 will be equal to the sum of the eigen values Question 1 the determinant will be Lambda is 1,6 the sum of the eigen values is 7 and so is the sum of diagonal elements of the given matrix the product of the eigen values is 6 and so is the determinant of the given matrix In the next class i'll be solving a lot of questions & examples on the same In the next class i'll be solving a lot of questions & examples on the same This is my playlists on CSIR NET general aptitude This is my playlist on real algebra This is my new channel that I'm working on This is my old channel on Higher Mathematics