 हॐर Γιαologia है? आटिक है मुँ का साध तो भी एक य क NCT Pike है बसके को नहीं है ऽाई औरम मुँों आब आप या�我要 आप साध म dagger sank पसी आप गी अद, आब thanal बसके बसके शा�ंच, यी days य उन न गगेच अगत Brookगा साध गरु उन्सर्त कि यहूसित. मुझ पी रोत & ुळ imbalance aur said to be orthogonal मुझ पी रोत & ुळ ऎक औँव कोलम्त जब अर्ड़ तो गिल में � stiff मुझ ब मदे पहे तेलुग मैं पह वेग्तार के लग ने रफ दें लिए अप लीगेर गाठ्टर था तो मैं लेग गाए है . अग्स प्राइम वाए एकवल्स तो जीरो, because 2 vectors हम दिलेट किया किया एक्स आन वाए अप अदर पी लिए तो वान, it means पी रोज अन वान कोलम से, are said to be orthogonal. For example, we have the x, which is equals to 5, 4. वेक्तर ना मारिपस एक्स, 5, 4 कर, it means कितनी एक, p-rose one column, 2-rose one column. आदर पी वाए वेक्तर ना वाए याग्स तो जीरो, it means this is the orthogonal vector. अदर पी वाए एक्स प्राइम वाए, now x prime y, x prime 5, 4 and y, 8-10. अदर पी एक्स प्राइम वान, x prime 5, 4 and y, 8-10. अदर पी एक्स पिर, p-rose one column, 2-rose one column, p-rose one column. अदर पी वाए एक्स पी वाए, now X prime 5, 4. अग्स and वाए of order पी into वान are said to be अर्ठो नोल्मल. अब अर्ठो नोल्मल को तोड़ा हम अंट्रस्ट्ट करतें कि अमारे पस क्या है. बेसेकली वी हाग्स, अग्स आम नी क्या लिया है. वान बाए 2, वान बाए 2 श्के रूत into 0, तर्वाल्यो 0, बाह नी काब आब बाए आवए ठोद प्कि जारट है, बनी, मुझा सग économी जारटल बाव जारटल औस, और तो फ कर個人 चस्विया उमस शिका कि लिए, बाशा चक्या नीक बना हुए फती यु तो सगवान या गाय संग underground येखे बेशिकली आप में दियास्धान कनवर्ट करये। आप विखऊल से कि डूद है ये वेक्टर्स बना रह हूं। तो में MotoList में मेरे पास वेक्टर آगया है ये ये वान बाट टूद के रूट एक ती फ़ूँनिवेः, धेगा काी वाकटर, नहीं ज़ों आप तो तुश्के रूँँँँँँँँँँँँँँँँँँँँ Greenson वर ..... तर वक्तःर एत जीजीज जीजीजईद छीजीजई.. वरो जे stuff कहाँ और तोगणल एगे projekt ये। तहार रहमु मेंढा ऊस्ढेउिओं में ुढिमि Aunqueी � crocodile अग्तिर आप क्या भे । ॢवाून बाभााटा Ein जीजईग goods � A multiplicate by 1 by 2 square root plus Sekond 1 by 2 square root multiply by 1 by 2 square root Plus 0 multiplied by 0 Doctors Widen multiplication is that it is multiplied by it is itself multiplied by one by two square root i.e between 0's the other sweater as well as etc fry We do 1 by 2 square root What happened to it if to me फ économंडयensions। दिरीों। necesario क्हए। ख़ीक। ख़ी emotional। दिरी sanitation। अत्शादल। कि भेकियर थी का बाल्गात। के वक्तरट़ खो यह वक्तरटार से मुल्तिप्लिए की है यह लेए वालीूँस से मुल्तिप्लिए करीो तु रिजूल्त मेरे पस वात खेख बलारे है अरे सिम्ली ली सेकिन पोब मैहाए आग़ी मुल्� toward मल्तिप्लिए कर रों बरार एक जेख कर लों को लिए को result क्या रहा है. Means, second value मैंने लिया वान बाई तूसके रूत, multiply by वान बाई तूसके रूत प्लास माइनेस वान बाई तूसके रूत, end 2 माइनेस वान बाई तूसके रूत प्लास, 0 multiply by 0. So, this is value which is equals to 1 by 2, minus multiply by minus, plus 1 by 2, plus 0. So, you will get result of one cake. So, one cake is coming. So, we can say that this is the orthonormal. Now, if I am writing here, you can check cross multiplication equals to 0. Now, here, within multiplication equals to 1. Cross multiplication means that I will multiply the first vector with second vector. So, result should be equal to 0. Within multiply, result is 1 cake. So, now, I have multiplied the result. Now, we will cross multiplication. Now, 1 by 2 square root, multiply by 1 by 2 square root, plus 1 by 2 square root, multiply by minus 1 by 2 square root, plus 0 multiplied by 0. So, 1 by 2 square root, multiply by 1 by 2 square root, which is equals to 2. So, minus 1 by 2 square root, end 2 1 by 2 square root, minus 1 by 2, plus 0. So, you have result here. 1 by 2 minus 1 by 2, result is 0. So, we will do cross multiplication vector. So, it is 0 cake. Within multiplication is equal to 1 cake. So, we can say that the matrix of set of orthogonal vector is called the orthonormal. This is the matrix of the orthonormal vector is called the orthonormal matrix. Now, cross multiplication equals to 0. Within multiplication is 1. And here, I am saying delta ij. Delta is the notation. Call it any notation. ij, first and second values. So, we have called it ij, which is equals to 1. So, we can say that if i equals to j, because it is within. Within, i equals to j. So, we can say that if i equals to 0, if i is not equals to j. Now, what have you got? Delta, which we have kept in condition. We have given the notation of delta ij, which is equals to 1. If i equals to j, it means within multiplication. And equals to 0, if i is not equals to j. So, we can say that the matrix, this is the matrix of the set of orthogonal vector is called the orthonormal matrix. This is what we have. This is what we have done with orthonormal. What is orthonormal? The multiplication we are doing in it. The multiplication we have within is equals to 1. And the cross multiplication we have is equals to 0. This is the concept of orthogonal and orthonormal. Now, next is the quadratic form. What is the quadratic form? A quadratic form in p variables x1, x2 up to so on xp is a homogeneous function. Consist of all possible second-order terms. We have to learn all possible second-order terms. We are not learning the first-order because it is on the quadratic form. What are the second-order forms we have? Now, a11x1 square, second-order, a22x2 square, up to so on, appxp square. Square terms means second-order. So, this is the cross-product terms, a12x1x2, cross-product, ax1 and x2 cross-product. Second-order, a13x13 again second-order, up to so on last, what you have? Ap-1xp-1 into xp. So, last you have xp-1xp. Now, sum up all of them. Okay, I will sum up all of them. After summing up, I have the final result. Double sum i varies 1 to p, j varies 1 to p, aijxixj. This can be written in the matrix form. Now, how do we write it in the matrix form? Let me give you the idea. So, now, you can sum up all of them. So, i1j1 means a11, xi, x1, j1, x1. So, what will happen if you have a11x1 square? First-order, similarly, x2 means i, you have 2, jb2. So, we have 2, 2, ij2, 2, x2, x2. Take the scale of this. Second term is done. Now, we are looking at cross-product terms. In cross-product terms, we have i1. Okay, what did I do here? Sum i1, j2, x1 and x2. So, you have these cross-product terms. When we sum up all of them, we have double sum aijxixjr. Now, how do we write all of them in the matrix form? This can be written in the matrix form. If we go to the matrix form, we write double sum aijxixprimeax. i.e., you have the vector x, we have the matrix and x again vector. So, x is a vector of random variable and matrix a is defined as. So, matrix a is the constant term we have defined. i.e., a11, constant, a12, constant. We multiply all of them by xprimex. Suppose, let an example of x1, x2. Simple, I have two random variables. I have taken x1, x2 and a is the matrix, which is equal to hypothetical. So, I am taking any one. One, two, three, four. Now, you have a vector of x1 and x2. So, after multiplying, I have a final step. x1, x2. Now, you multiply it here. After multiplying, I have x1 plus 2x2. So, 3x1 plus 4x2. Further, if I multiply it, I have results. x1 square plus 2x2x1 plus 3x1x2 plus 4x2 square. So, x1 square plus 5x1x2 plus 4x2 square. So, look at the equation. What is the equation? It is a quadratic equation. So, we are representing the quadratic equation in this way. Now, it is equal to xprimeax. x is the vector, a is the matrix and x is also the vector. This is the quadratic form. Now, what is the positive definite and semi-positive definite? Now, positive definite, a square matrix a and its associated quadratic form, अब आम ने प्रीविस, quadratic form की है, is associated quadratic form, is said to be positive definite. If xprimeax greater than zero, if it is greater than zero, so we can say that it is a positive definite. And if it is a greater than equals to zero, so we can say that the semi-positive definite. For example, we have the matrix आ, which is equals to the 1, 2, 1, 3. This is a hypothetical. आ is the matrix of 1, 2, 1, 3. For order 1, how will we find the positive definite? For order 1, I have taken the determinant of this factor. So, if I take the determinant of 1, do you know that the determinant of 1 is equal to 1? So, 1 is greater than zero, so we can say that this is the positive definite. Again, this is order 1. If I take order 2, it means that I have the matrix, because I have the matrix order 2. I am taking the determinant of this. Which is equals to 3 cross 1, 3 into 1, minus 2 into 1. So, which is equals to 3 minus 2, which is also equals to 1. And this is greater than zero. So, we can say that if you have the matrix आ, this is called the positive definite. What should be the X prime A greater than zero? For every X is not equals to zero. Any value of X is not equal to zero. So, this is called the positive definite. And semi-positive definite. What is semi-positive definite? Its result will be greater than zero. If it is greater than zero, it means that it is positive definite. When we have equals to zero, then for equals to zero, we will always have semi-positive definite. So, here is the example of X transpose A X. So, X which is equals to 1, 2 is hypothetical. A, 1, 2, 1, 3 and A, 1. Okay. We have solved this further. After solving it, I have got the result of 3, 8 and 1, 2. We have solved this further. So, I have got the result of 18. And 18, which is greater than zero. If it was equals to zero, we would have X transpose A X in the matrix. If its result was equals to zero, then what would have happened? Semi-positive definite. So, this is the greater than zero. So, we can say that this is the positive definite matrix. So, positive definite quadratic form has the matrix of full rank. We can write them A which is equals to Q into Q prime. And Q is the non-singular. So, we have considered that A is equal to Q into Q transpose. And Q is the non-singular. So, we can consider this A matrix.