PRIMARY DECOMPOSITION FORM | Lec-16 | CSIR NET MATHEMATICS | IFAS

Primary Decomposition Theorem in Linear Algebra explained with clear concepts and exam-focused approach. Learn minimal polynomial factorization, block diagonal form, diagonalizability, triangularizability, Jordan form and PYQ tricks for CSIR NET, GATE and SET. 🧭 Get Personalized Preparation Guidance: https://ifasonline.com/csir-net/mathe... 🔥Join India's Best Coaching for CSIR NET and GATE Exams: https://ifasonline.com/csir-net/mathe... 📞 Need Guidance? Call/WhatsApp: 9172266888 ━━━━━━━━━━━━━━━━━━━━━━━━━ ⏳ Lecture Timeline: [00:30] Introduction to Primary Decomposition Theorem [01:37] Statement of PDT [05:46] Minimal Polynomial Factorization [09:03] Direct Sum Decomposition into T-Invariant Subspaces [11:11] Primary Components and Generalized Eigenspaces [13:15] Over Algebraically Closed Field: Generalized Eigenspaces [17:02] Diagonalizability and Triangularizability Conditions [19:10] Diagonalizable iff Direct Sum of Eigenspaces [22:54] Triangularizable iff Characteristic Polynomial Splits [23:38] Jordan Block Count Formulas [24:21] Restriction on Generalized Eigenspace: λI + N [25:33] Example: 3x3 Matrix with Minimal Polynomial [28:29] Question 1: 6x6 Complex Matrix, Find Jordan Form [30:39] Question 2: Impossible Number of Jordan Blocks [33:49] Question 3: Maximum Independent Eigenvectors for 5x5 [38:42] Question 4: 3x3 Matrix with Characteristic Polynomial [42:02] Question 5: Direct Sum Decomposition with p²q [44:22] Conclusion and Course Promotion [45:09] End Of Lecture ━━━━━━━━━━━━━━━━━━━━━━━━━ 📖 Topics Covered in This Lecture: Primary Decomposition Theorem, minimal polynomial, irreducible factors, block diagonal matrix, T-invariant subspaces, generalized eigenspaces, diagonalizability, triangularizability, Jordan blocks, algebraic and geometric multiplicity. ━━━━━━━━━━━━━━━━━━━━━━━━━ 📚 Lecture Summary: In this session, IFAS Expert Faculty focuses on Primary Decomposition Theorem in Linear Algebra for CSIR NET, GATE and SET. Covers statement, factorization of minimal polynomial into monic irreducibles, direct sum decomposition into T-invariant primary components, block diagonal representation, generalized eigenspaces over algebraically closed fields. Key results: diagonalizable iff minimal polynomial has distinct linear factors, triangularizable iff characteristic polynomial splits, Jordan block count formulas, and restriction on generalized eigenspace is λI + N with N nilpotent. Solved PYQs on Jordan form, impossible block counts, maximum independent eigenvectors, diagonalizability over C and R, and direct sum decomposition. Learn shortcuts and exam-focused problem-solving. ━━━━━━━━━━━━━━━━━━━━━━━━━ 🎓 This lecture is highly beneficial for students preparing for: ✅ CSIR NET Mathematical Science ✅ GATE Mathematics ✅ SET Mathematics ✅ TIFR Mathematics ✅ PhD Entrance Exams ━━━━━━━━━━━━━━━━━━━━━━━━━ 🏆 Why IFAS is India’s No. 1 Choice for CSIR NET & GATE Mathematics? (Our Results Speak): ✅ CSIR NET Dec 2025: 4000+ Selections (AIR 3, 8, 9.....) ✅ CSIR NET June 2025: 4200+ Selections (AIR 1, 2, 6, 10...) ✅ CSIR NET Dec 2024: 4000+ Selections (AIR 5, 6, 7, 8, 9...) ✅ CSIR NET June 2024: 2000+ Selections (AIR 3, 10, 13...) ✅ CSIR NET Dec 2023: 1200+ Selections (AIR 2, 4, 5...) 📄 View IFAS Result Sheet: 👉 https://drive.google.com/file/d/1THh6... ━━━━━━━━━━━━━━━━━━━━━━━━━ 🔗 Important Links: 🌍 Visit our Website: https://ifasonline.com 📞 Contact for Queries: 9172266888 ━━━━━━━━━━━━━━━━━━━━━━━━━ 🔗 India's Best Online Classes and Books: 🔥CSIR NET Mathematical Science Live & Recorded Course: https://ifasonline.com/csir-net/mathe... 🎯 GATE Mathematics Recorded Course: https://ifasonline.com/gate/mathemati... 📘CSIR NET Mathematical Science Books: https://ifasonline.com/csir-net/mathe... ━━━━━━━━━━━━━━━━━━━━━━━━━ 📲 Download IFAS App: Android: https://play.google.com/store/apps/de... IOS: https://apps.apple.com/us/app/ifas/id... ━━━━━━━━━━━━━━━━━━━━━━━━━ Stay Updated Via Other Social Media: ✔Telegram: https://t.me/NET_GATE_SET_Maths_Discu... ✔Instagram:   / csir_net_mathematical_science   #csirnetmathematicalscience #gatemathematics #csirnetmaths #csirnetdec2026 #csirnetlecture #ifasmathematics