JORDAN CANONICAL FORM | Lec - 15 | CSIR NET MATHEMATICS | IFAS

Jordan Canonical Form in Linear Algebra explained with clear concepts and exam-focused approach. Learn Jordan blocks, characteristic and minimal polynomial, geometric multiplicity, diagonalizability 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: [00:00] Introduction [01:37] Course Promotion [03:04] JCF Fundamentals [03:17] What is JCF [04:00] Block Diagonal Format [04:46] Jordan Blocks [06:13] Existence of JCF [07:19] Minimal Polynomial Form [07:45] Number of Jordan Blocks [08:05] Max Block Order [09:50] Diagonalizable Condition [10:36] Triangularizable Condition [11:36] Question 1: JCF from Rank Conditions [12:01] Matrix A 7x7 Complex [14:38] GM from Rank [17:09] Block Sizes [18:08] Answer: Option B [18:15] Question 2: Nilpotent Matrix Block Size [18:26] Nullity Conditions [22:52] Index 3, Minimal Polynomial [23:31] JCF Construction [26:00] Number of 2x2 Blocks [26:19] Answer: 0 [28:38] Question 3: JCF from Trace and Determinant [29:10] Minimal Polynomial Given [32:55] Check Block Sizes [34:02] Contradiction with Trace [35:28] No Such Matrix Exists [36:44] Question 4: Dimension of Space Generated by A [37:04] Algebra Generated by A [39:03] Dimension Equals Degree of Minimal Polynomial [39:22] Minimal Polynomial from JCF [40:39] Answer: 3 [40:49] Homework [42:12] End Of Lecture ━━━━━━━━━━━━━━━━━━━━━━━━━ 📖 Topics Covered in This Lecture: Jordan Canonical Form, Jordan blocks, block diagonal form, characteristic polynomial, minimal polynomial, algebraic multiplicity, geometric multiplicity, diagonalizability, triangularizability, rank-nullity, PYQ solutions. ━━━━━━━━━━━━━━━━━━━━━━━━━ 📚 Lecture Summary: In this session, IFAS Expert Faculty focuses on Jordan Canonical Form in Linear Algebra for CSIR NET, GATE and SET. Covers definition of JCF, Jordan blocks, existence over splitting fields, uniqueness up to order, relation between minimal polynomial powers, block sizes, algebraic and geometric multiplicity. Key results: number of Jordan blocks for λ equals geometric multiplicity, maximum block order equals power in minimal polynomial, diagonalizable iff all powers are 1. Solved PYQs on JCF from rank conditions, nilpotent matrix block counts, non-existence of matrix with given trace/determinant, and dimension of polynomial space in A. ━━━━━━━━━━━━━━━━━━━━━━━━━ 🎓 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