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Bài giảng Matlab - Phần I: Basis and languge

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ꢀꢁꢂꢃꢁꢄꢅꢆꢄꢇꢈꢉꢈꢊꢋꢋꢌꢄꢄꢄꢍꢎꢄꢏꢅꢐꢄꢑꢒꢒꢓꢄꢂꢆꢏꢁꢔꢕꢆꢖꢄꢗꢅꢆꢖꢄꢒꢆꢅꢐꢖꢔꢘꢄꢏꢅꢐꢄꢙꢂꢆꢄꢁꢔꢃꢅꢚꢄꢕꢁꢄꢂꢚꢂꢏꢛꢄ  
Matlab  
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part I  
Basis and languge  
PhÇn 1. NhËp m«n vÒ ng«n ng÷ Matlab  
Ch−¬ng 1. C¸c kh¸i niÖm c¬ b¶n.  
1.1 Khëi ®éng matlab (trong windows 9x).  
1.2 C¸c phÝm chuyªn dông vµ c¸c lÖnh th«ng dông hÖ thèng.  
1.3 BiÕn vµ thao t¸c biÕn  
1.4 Sè phøc  
1.5 Lµm viÖc ë chÕ ®é héi tho¹i (command window).  
1.6 Lµm viÖc ë chÕ ®é so¹n ch−¬ng tr×nh (edit window).  
1.7 S¬ l−îc vÒ ®å ho¹.  
1.8 C¸c hµm ©m thanh.  
Ch−¬ng 2. Ma trËn vµ c¸c phÐp to¸n ma trËn  
2.1 V« h−íng, vector vµ ma trËn.  
2.2 C¸c ma trËn ®Æc biÖt.  
2.3 C¸c phÐp to¸n v« h−íng  
2.4 C¸c phÐp to¸n ma trËn  
Ch−¬ng 3. LËp tr×nh trong Matlab  
3.1 C¸c phÇn tö c¬ b¶n  
3.2 Hµm vµ t¹o hµm  
3.3 C©u lÖnh CÊu tróc  
Ch−¬ng 4. §å ho¹ 2 chiÒu  
4.1 C¸c phÐp biÕn ®æi ®å ho¹.  
4.2 PhÐp biÕn ®æi Affine trong mÆt ph¼ng  
4.3. C¸c hµm chuÈn ®å ho¹ 2 chiÒu  
Ch−¬ng 5. §å ho¹ kh«ng gian 3D  
5.1. Hµm ®ång møc (contour).  
5.2. Chia l−íi.  
5.3. §å ho¹ 3D.  
5.4. MÆt l−íi ®å ho¹ 3D.  
5.5. §å ho¹ bÒ mÆt.  
5.6. §iÓm nh×n vµ phÐp phèi c¶nh.  
5.7. MÆt c¾t (slice).  
5.8. MÇu s¾c vµ c¸c hÖ mµu  
Page 1 of 62  
part II  
matlab symbolic  
part III  
numerical method  
Page 2 of 62  
Basic concepts  
ꢜꢂꢄꢁꢃꢝꢆꢄꢞꢜꢂꢁꢃꢕꢟꢠꢡꢄ ꢄꢄ  
ꢢꢣꢙꢄꢁꢤꢄꢞꢢꢒꢙꢁꢅꢃꢠꢡꢄ ꢄ  
ꢢꢥꢄꢔꢦꢧꢆꢖꢄꢞꢀꢙꢂꢗꢂꢃꢠꢡꢄ ꢄ  
ꢑꢕꢨꢐꢄꢑꢔꢩꢕꢞꢙꢒꢗꢗꢠꢡꢄ  
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Page 3 of 62  
PhÇn I  
NhËp m«n vÒ ng«n ng÷ Matlab  
Giíi thiÖu  
ꢪꢫꢂꢁꢗꢂꢬꢄꢗꢭꢄꢓꢔꢮꢆꢄꢜꢯꢜꢄꢙꢰꢂꢄꢙꢥꢆꢖꢄꢁꢏꢄꢫꢂꢁꢔꢄꢱꢅꢃꢑꢀꢄꢙꢲꢆꢖꢄꢔꢅꢭꢄꢳꢣꢙꢛꢄ  
ꢪꢴꢥꢆꢖꢄꢵꢶꢆꢖꢄꢡꢄꢫꢂꢁꢗꢂꢬꢄꢗꢭꢄꢙꢔꢦꢤꢆꢖꢄꢁꢃꢷꢆꢔꢄꢓꢔꢮꢆꢄꢜꢯꢜꢄꢁꢃꢸꢄꢖꢕꢹꢓꢄꢙꢔꢅꢄꢢꢕꢺꢙꢄꢁꢻꢆꢔꢄꢁꢅꢼꢆꢄꢢꢭꢄꢔꢕꢨꢆꢄꢁꢔꢽꢄꢄ  
ꢙꢼꢙꢄꢬꢭꢕꢄꢁꢅꢼꢆꢄꢑꢾꢄꢁꢔꢐꢝꢁꢛꢄꢫꢂꢁꢗꢂꢬꢄꢿꢦꢸꢙꢄꢿꢕꢯꢐꢄꢑꢔꢕꢨꢆꢄꢬꣀꢕꢄꢁꢝꢓꢄꢔꢸꢓꢄꢙꢼꢙꢄꢗꢺꢆꢔꢘꢄꢁꢼꢙꢄꢿꢲꢆꢖꢄꣁꢐꢂꢄꢬꢭꢆꢄ  
ꢓꢔꢻꢜꢄꢁꢃꣂꢆꢄꢙꣃꢂꢄꢀ꣄ꢄꢿꢕꢯꢐꢄꢑꢔꢕꢨꢆꢄꢞꢴꢅꢜꢜꢂꢆꢵꢄꢱꢕꢆꢵꢅꢚꢠꢛꢄꣅ꣆ꢄꢙ꣇ꢆꢖꢄꢙꢔꢅꢄꢓꢔꢣꢓꢄꢜꢲꢁꢄꢑꢔ꣈ꢄꢆ꣉ꢆꢖꢄ  
ꢗꢝꢓꢄꢁꢃꢷꢆꢔꢄꢢꢧꢕꢄꢁꢔꢥꢆꢖꢄꢵꢽꢙꢔꢄꢁꢝꢓꢄꢔꢸꢓꢄꢗꢺꢆꢔꢪꢗꢝꢓꢄꢁꢃꢷꢆꢔꢄscrip fileꢛꢄ  
ꢪꢫꢂꢁꢗꢂꢬꢄꢗꢭꢄꢢꢕ꣊ꢁꢄꢁ꣋ꢁꢄꢙꢰꢂꢄꢁ꣌ꢄꢫꢂꢁꢃꢕꢟꢄ꣍ꢂꢬꢅꢃꢂꢁꢅꢃꢏꢛꢄ  
ꢪ꣎꣏ꢙꢄꢿꢕꢨꢜꢄꢙꢰꢂꢄꢆꢖꢥꢆꢄꢆꢖ꣐ꢄꢫꢂꢁꢗꢂꢬꢡꢄ  
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꣑꣒ꣃꢄꢗ꣓ꢄꢙꢼꢙꢄꢜꢂꢄꢁꢃꢝꢆꢄꢁꢔ꣔ꢙꢘꢄꢓꢔ꣕ꢙꢘꢄꢙꢼꢙꢄꢟ꣖ꢐꢄꢑꢻꢄꢁ꣔ꢛꢄ  
꣑꣗꣊ꢁꢄꢔꢸꢓꢄꢓꢔꢣꢓꢄꢁꢻꢆꢔꢄꢗꢅꢖꢕꢙꢄꢢꢧꢕꢄꢀꢩꢄꢔ꣘ꢙꢛꢛꢄ  
꣑ꢴꢼꢙꢄꢔꢭꢜꢄꢿ꣙ꢄꢔꢅ꣚ꢄꢜ꣚ꢆꢔꢄꢢꢭꢄꢵ꣛ꢄꢵꢭꢆꢖꢄꢑ꣊ꢁꢄꢔꢸꢓꢄꢢꢧꢕꢄꣁꢐꢼꢄꢁꢃꢷꢆꢔꢄꢁꢻꢆꢔꢄꢁꢅꢼꢆꢛꢄ  
꣑꣜꣝ꢄꢁꢃꢸꢄꢙꢼꢙꢄꢙꢥꢆꢖꢄꢙꢶꢄꢜ꣚ꢆꢔꢄꢙꢔꢅꢄꢙꢼꢙꢄ꣕ꢆꢖꢄꢵꢶꢆꢖꢄꢙꢶꢄꢁꢔꢨꢄ  
ꢪꢴꢔ꣊ꢄꢿꢲꢄꢗꢭꢜꢄꢢꢕꢺꢙꢡꢄ  
꣑꣜ꢲꢕꢄꢁꢔꢅ꣚ꢕꢄꢆꢖꢦ꣞ꢕꢪꢜꢼꢏꢄꢞꢐꢀꢒꢃꢪꢜꢂꢙꢔꢕꢆꢒꢄꢂꢁꢄꢙꢅꢜꢜꢂꢆꢵꢄꢚꢕꢆꢵꢅꢚꢠꢡꢄꢴꢔꢅꢄꢓꢔꢣꢓꢄꢆꢔꢝꢓꢄꢗꢺꢆꢔꢄ  
ꢁ꣌ꢄꢬꢭꢆꢄꢓꢔꢻꢜꢄꢢꢭꢄꢁꢔꢐꢄꢿꢦꢸꢙꢄꢙꢼꢙꢄꢑ꣊ꢁꢄꣁꢐ꣈ꢄꢀꣃꢄꢗ꣓ꢗꢺꢆꢔꢄꢁ꣌ꢄꢜꢼꢏꢄꢁꢻꢆꢔꢛꢄ  
꣑ꢴꢔ꣊ꢄꢿꢲꢄꢗꢝꢓꢄꢁꢃꢷꢆꢔꢡꢄꢴꢼꢙꢄꢗꢺꢆꢔꢄꢿꢦꢸꢙꢄꢙꢔꢐ꣟ꢆꢄꢬꢽꢄꢁꢃꣂꢆꢄꢜꢲꢁꢄ ꢎꢕꢗꢒꢄꢙꢔꢦꢤꢆꢖꢄꢁꢃꢷꢆꢔꢄꢢꢭꢄꢙꢔꢅꢄ  
ꢓꢔꢣꢓꢄꢙꢔ꣚ꢏꢄꢙꢔꢦꢤꢆꢖꢄꢁꢃꢷꢆꢔꢄꢁ꣌ꢄꢙꢼꢙꢄꢎꢕꢗꢒꢄꢗꢺꢆꢔꢄꢆꢭꢏꢛꢄ  
ꢪꢴ꣆ꢄꢆꢔꢕꢯꢐꢄꢓꢔꢕꣂꢆꢄꢬ꣈ꢆꢄꢑꢔꢼꢙꢄꢆꢔꢂꢐꢄꢞꢢꢒꢃꢀꢕꢅꢆꢀꢠꢘꢄꢙꢔꢅꢄꢓꢔꢣꢓꢄꢙꢔ꣚ꢏꢄꢁꢃꣂꢆꢄꢙꢼꢙꢄꢜꢼꢏꢄꢁꢻꢆꢔꢄꢙ꣆ꢄꢔꢺꢄꢿꢕꢯꢐꢄ  
ꢔꢭꢆꢔꢄꢑꢔꢼꢙꢄꢆꢔꢂꢐꢄꢆꢔꢦꢄ꣠꣡ꢳꢘꢄ꣢ꣅꢍ꣒ꢘꢄ꣣꣤꣤꣍꣥ꢘꢄꢱꢍꣅ꣠꣡ꢱꢳꢘꢄ꣍ꢍꣅ꣢꣒ꢛꢛꢛꢄ  
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ꢪ꣦ꢧꢕꢄꢫꢂꢁꢗꢂꢬꢘꢄꢙꢼꢙꢄꢢ꣧ꢆꢄꢿꢯꢄꢙꢮꢆꢄꢖꢕ꣈ꢕꢄꣁꢐꢏ꣊ꢁꢄꢙꢰꢂꢄꢬ꣚ꢆꢄꢀ꣨ꢄꢿꢦꢸꢙꢄꢓꢔ꣖ꢆꢄꢁꢻꢙꢔꢄꢢꢭꢄꢟꣃꢄꢗ꣓ꢄꢁꢔꢒꢅꢄꢇꢄꢬꢦꢧꢙꢄ  
ꢆꢔꢦꢄꢀꢂꢐ꣩ꢌ꣪ꢡꢄ  
B−íc 1: §Æt vÊn ®Ò.  
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꣫ꢭꢕꢄꢁꢅꢼꢆꢄꢿ꣏ꢁꢄꢃꢂꢄꢙꢮꢆꢄꢿꢦꢸꢙꢄꢓꢔ꣖ꢆꢄꢁꢻꢙꢔꢘꢄꢬꢕꢨꢐꢄꢵꢕ꣛ꢆꢄꢜꢲꢁꢄꢙꢼꢙꢔꢄꢃ꣬ꢄꢃꢭꢆꢖꢄꢢꢭꢄꢙꢶꢄ  
ꢁꢔꢨꢛꢄ꣎꣖ꢏꢄꢗꢭꢄꢬꢦꢧꢙꢄꢜꣀꢄꢿꢮꢐꢄꢃ꣧ꢁꢄꣁꢐꢂꢆꢄꢁꢃ꣘ꢆꢖꢘꢄꢆ꣆ꢄꣁꢐꢏ꣊ꢁꢄꢿꢽꢆꢔꢄꢁꢅꢭꢆꢄꢬꢲꢄꢔꢦꢧꢆꢖꢄ  
ꢖꢕ꣈ꢕꢄꣁꢐꢏ꣊ꢁꢄꢁꢕ꣊ꢓꢄꢁꢔꢒꢅꢄꢙꢰꢂꢄꢬꢭꢕꢄꢁꢅꢼꢆꢄꢿ꣏ꢁꢄꢃꢂꢛꢄ  
B−íc 2: M« t¶ c¸c gi¸ trÞ d÷ liÖu vµo/ra.  
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꣦ꢕꢺꢙꢄꢜꢥꢄꢁ꣈ꢄꢙꢼꢙꢄꢁꢔꢥꢆꢖꢄꢁꢕꢆꢄꢙꢮꢆꢄꢖꢕ꣈ꢕꢄꢿꢼꢓꢄꢙ꣆ꢄꢗꢕꣂꢆꢄꣁꢐꢂꢆꢄꢁꢃ꣔ꢙꢄꢁꢕ꣊ꢓꢄꢿ꣊ꢆꢄꢙꢼꢙꢄ  
ꢁꢔꢂꢜꢄꢀꢩꢄꢿꢦꢸꢙꢄꢀꣃꢄꢵꢶꢆꢖꢄꢁꢃꢅꢆꢖꢄꣁꢐꢼꢄꢁꢃꢷꢆꢔꢄꢁꢻꢆꢔꢄꢁꢅꢼꢆꢘꢄꢬꣀꢕꢄꢢꢝꢏꢄꢬꢦꢧꢙꢄꢆꢭꢏꢄꢙꢮꢆꢄ  
ꢿꢦꢸꢙꢄꢁꢕ꣊ꢆꢄꢔꢭꢆꢔꢄꢙ꣟ꢆꢄꢁꢃ꣘ꢆꢖꢛꢄ꣭ꢃꢅꢆꢖꢄꢆꢔꢕꢯꢐꢄꢁꢃꢦ꣞ꢆꢖꢄꢔꢸꢓꢘꢄꢀꢤꢄꢿ꣙ꢄꢑꢔꢩꢕꢄꢿꢦꢸꢙꢄꢀꣃꢄ  
ꢵꢶꢆꢖꢄꢿꢨꢄꢟꢼꢙꢄꢿꢽꢆꢔꢄꢢꢽꢄꢁꢃꢻꢄꢗꢐ꣙ꢆꢖꢄꢢꢭꢅꢈꢃꢂꢘꢄꢁꢐꢏꢄꢆꢔꢕꣂꢆꢄꢿꢥꢕꢄꢑꢔꢕꢄꢙꢔꢹꢆꢖꢄꢙꢔ꣮ꢄꢗꢭꢄꢔꢲꢓꢄ  
ꢿꢒꢆꢄꢢꢷꢄꢑꢔꢥꢆꢖꢄꢁꢔꢨꢄꢟꢼꢙꢄꢄꢿꢽꢆꢔꢄꢿꢦꢸꢙꢄꢗꢐ꣙ꢆꢖꢄꢃꢂꢄꢁ꣚ꢕꢄꢜꢲꢁꢄꢿꢕꢨꢜꢄꢆꢭꢅꢄꢿ꣆ꢄꢁꢃꢅꢆꢖꢄꢙꢼꢙꢄ  
ꢬꢦꢧꢙꢛꢄꢫ꣏ꢙꢄꢵꢮꢐꢄ ꢢꢝꢏꢘꢄꢁꢂꢄꢢ꣯ꢆꢄꢙꢔ꣮ꢄꢃꢂꢄꢿꢦꢸꢙꢄꢆꢔ꣐ꢆꢖꢄꢁꢔꢥꢆꢖꢄꢁꢕꢆꢄꢿꢨꢄꢁꢻꢆꢔꢄꢁꢅꢼꢆꢄ  
ꢗꢐ꣙ꢆꢖꢄꢃꢂꢛꢄ  
B−íc 3: C¸c tÝnh to¸n b»ng tay víi c¸c tËp d÷ liÖu ®Çu vµo ®¬n gi¶n  
ꢄ
꣎꣖ꢏꢄꢗꢭꢄꢬꢦꢧꢙꢄꢿꢨꢄꢆꢔ꣰ꢜꢄꢁꢷꢜꢄꢑꢕ꣊ꢜꢄꢆꢔ꣐ꢆꢖꢄꢖꢕ꣈ꢕꢄꢓꢔꢼꢓꢄꢙꢶꢄꢁꢔꢨꢘꢄꢬ꣚ꢆꢄꢑꢔꢥꢆꢖꢄꢆꣂꢆꢄ  
ꢬ꣱ꢄꣁꢐꢂꢄꢑꢨꢄꢙ꣈ꢄꢿꢩꢕꢄꢢꢧꢕꢄꢙꢼꢙꢄꢬꢭꢕꢄꢁꢅꢼꢆꢄꢿꢤꢆꢄꢖꢕ꣈ꢆꢛꢄꣅ꣊ꢐꢄꢁꢃꢅꢆꢖꢄꢬꢦꢧꢙꢄꢆꢭꢏꢄꢬ꣚ꢆꢄ  
ꢙꢔꢦꢂꢄꢗ꣧ꢏꢄꢿꢦꢸꢙꢄꢵ꣐ꢄꢗꢕꢺꢐꢄꢔꢂꢏꢄꢙꢔꢦꢂꢄꢁꢻꢆꢔꢄꢿꢦꢸꢙꢄꢿꢮꢐꢄꢃꢂꢄꢁꢔꢷꢄꢙ꣆ꢄꢁꢔꢨꢄꢙꢔꢐꢏꢨꢆꢄꢀꢂꢆꢖꢄ  
ꢬꢦꢧꢙꢄꢑ꣊ꢄꢁꢕ꣊ꢓꢛꢄ  
B−íc 4: ChuyÓn bµi to¸n sang gi¶i ph¸p b»ng Matlab.  
ꢄ
ëꢄꢬꢦꢧꢙꢄꢆꢭꢏꢄꢬ꣚ꢆꢄꢀ꣨ꢄꢀꣃꢄꢵꢶꢆꢖꢄꢙꢼꢙꢄꢔꢭꢜꢄꢁꢅꢼꢆꢘꢄꢙ꣇ꢆꢖꢄꢗꢭꢄꢙꢼꢙꢄꢗꢺꢆꢔꢄꢿꢨꢄꢜꢥꢄꢁ꣈ꢄꢬꢭꢕꢄ  
Page 4 of 62  
ꢁꢅꢼꢆꢄꢁꢔꢒꢅꢄꢫꢂꢁꢗꢂꢬꢄ  
B−íc 5: KiÓm tra.  
ꢄ
꣎꣖ꢏꢄ ꢗꢭꢄ ꢬꢦꢧꢙꢄ ꢙꢐꢩꢕꢄ ꢙꣲꢆꢖꢄ ꢁꢃꢅꢆꢖꢄ ꢁꢕ꣊ꢆꢄ ꢁꢃꢷꢆꢔꢄ ꢖꢕ꣈ꢕꢄ ꢬꢭꢕꢄ ꢁꢅꢼꢆꢛꢄ ꣫ꢭꢕꢄ ꢁꢅꢼꢆꢄ ꢿꢦꢸꢙꢄ  
ꢑꢕꢨꢜꢄꢁꢃꢂꢄꢬ꣰ꢆꢖꢄꢙꢼꢙꢄꢵ꣐ꢄꢗꢕꢺꢐꢄꢿꢮꢐꢄꢢꢭꢅꢛꢄꢫꢂꢁꢗꢂꢬꢄꢁꢔ꣔ꢙꢄꢔꢕꢺꢆꢄꢬꢭꢕꢄꢁꢅꢼꢆꢄꢢꢭꢄꢙꢔꢅꢄ  
ꢬ꣚ꢆꢄꢑ꣊ꢁꢄꣁꢐ꣈ꢄꣀꢄꢿꢮꢐꢄꢃꢂꢛꢄ  
꣭ꢃꢅꢆꢖꢄꢁꢃꢦ꣞ꢆꢖꢄꢔꢸꢓꢄꢑꢔꢥꢆꢖꢄꢙ꣆ꢄꢑ꣊ꢁꢄꣁꢐ꣈ꢄꢔꢅ꣏ꢙꢄꢑ꣊ꢁꢄꣁꢐ꣈ꢄꢀꢂꢕꢄꢁꢔꢷꢄꢿꢕꢯꢐꢄꢿ꣆ꢄꢙ꣆ꢄꢆꢖꢔꣳꢂꢄꢗꢭꢄ  
ꢫꢂꢁꢗꢂꢬꢄꢙꢔꢦꢂꢄꢁꢔ꣔ꢙꢄꢔꢕꢺꢆꢄꢿꢦꢸꢙꢄꢬꢭꢕꢄꢁꢅꢼꢆꢘꢄꢬ꣚ꢆꢄꢙꢮꢆꢄꢑꢕꢨꢜꢄꢁꢃꢂꢄꢗ꣚ꢕꢄꢙ꣈ꢄꢁꢻꢆꢔꢄꢁꢅꢼꢆꢄꢬ꣰ꢆꢖꢄ  
ꢁꢂꢏꢄꢢꢭꢄꢁꢔꢂꢅꢄꢁꢼꢙꢄꢬ꣰ꢆꢖꢄꢫꢂꢁꢗꢂꢬꢛꢄ  
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Page 5 of 62  
Ch−¬ng 1  
C¸c kh¸i niÖm c¬ b¶n  
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1.1Khëi ®éng matlab (trong windows 9x).  
ꢪ꣗ꢔꢧꢕꢄꢿꢲꢆꢖꢄꢜꢂꢁꢗꢂꢬꢡꢄ꣗ꢻꢙꢔꢄꢙꢔꢐꢲꢁꢄꢢꢭꢅꢄꢆꢹꢁꢄStart ꢀꢄꢙꢔ꣘ꢆꢄProgram ꢀ ꢙꢔ꣘ꢆ Matlab ꢀ  
ꢙꢔ꣘ꢆ Matlab 5.3 ꢔꢅ꣏ꢙꢄꢑꢻꢙꢔꢄꢄꢙꢔꢐꢲꢁꢄꢢꢭꢅꢄbiÓu t−îng cña Matlabꢄꢁꢃꣂꢆꢄ꣠ꢒꢀꢑꢁꢅꢓꢛꢄ  
ꢪꣴꢕꢂꢅꢄꢵꢕꢺꢆꢄꢙꢰꢂꢄꢫꢂꢁꢗꢂꢬꢄꢖ꣙ꢜꢄꢡꢄ  
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ꢄ
꣑ꢫꢲꢁꢄꢙꣃꢂꢄꢀ꣄ꢄꢗꢺꢆꢔꢄꢄ  
꣑ꢫꢲꢁꢄꢙꣃꢂꢄꢀ꣄ꢄꢀꢅ꣚ꢆꢄꢁꢔ꣈ꢅꢄꢄ  
꣑ꣅꢔꢕꢯꢐꢄꢙꣃꢂꢄꢀꢩꢄꢿ꣙ꢄꢔꢅ꣚ꢄ  
(command window)  
(edit window)  
(figures)  
ꢪ꣦ꢕꢺꢙꢄꢆꢖ꣋ꢁꢄꢙꢔꢦꢤꢆꢖꢄ ꢁꢃꢷꢆꢔꢄ ꢿꢂꢆꢖꢄ ꢁꢔ꣔ꢙꢄ ꢔꢕꢺꢆꢄꢑꢔꢥꢆꢖꢄꢿꢹꢆꢖꢄ ꢁꢔꢒꢅꢄ ꢏꣂꢐꢄ ꢙꢮꢐꢄꢁꢔꢥꢆꢖꢄꣁꢐꢂꢄ ꢓꢔꢻꢜꢄ  
ꢆ꣆ꢆꢖꢄCtrl+Cꢄꢛꢄ  
ꢪ꣎ꢨꢄꢁꢔꢅꢼꢁꢄꢃꢂꢄꢑꢔ꣱ꢕꢄꢜꢥꢕꢄꢁꢃꢦ꣞ꢆꢖꢄꢗꢭꢜꢄꢢꢕꢺꢙꢄꢫꢂꢁꢗꢂꢬꢘꢄꢙꢔꢹꢆꢖꢄꢁꢂꢄꢙ꣆ꢄꢁꢔꢨꢄꢀꣃꢄꢵꢶꢆꢖꢄꢙꢼꢙꢄꢗꢺꢆꢔꢄꢙꢰꢂꢄ  
ꢫꢂꢁꢗꢂꢬꢄꢗꢭꢄꢡꢄꢄ» quitꢄꢄꢄꢔꢅ꣏ꢙꢄꢄꢄ» exitꢄꢄ  
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1.2C¸c phÝm chuyªn dông vµ c¸c lÖnh th«ng dông hÖ thèng.  
ꢴꢼꢙꢄꢓꢔꢻꢜꢄꢗꢺꢆꢔꢄꢁꢔꢥꢆꢖꢄꢵꢶꢆꢖꢡꢄ  
ꣴ꣘ꢕꢄꢗ꣚ꢕꢄꢗꢺꢆꢔꢄꢢ꣌ꢂꢄꢁꢔ꣔ꢙꢄꢔꢕꢺꢆꢄꢁꢃꢦꢧꢙꢄꢿ꣆ꢛꢄ  
ꣴ꣘ꢕꢄꢗ꣚ꢕꢄꢗꢺꢆꢔꢄꢿꣵꢄꢿꢼꢆꢔꢄꢢꢭꢅꢄꢁꢃꢦꢧꢙꢄꢿ꣆ꢛꢄ  
ꢴꢔꢐꢏꢨꢆꢄꢙꢅꢆꢄꢁꢃ꣱ꢄꢀꢂꢆꢖꢄꢓꢔ꣈ꢕꢄꢌꢄꢑꢻꢄꢁ꣔ꢛꢄ  
ꢴꢔꢐꢏꢨꢆꢄꢙꢅꢆꢄꢁꢃ꣱ꢄꢀꢂꢆꢖꢄꢁꢃꢼꢕꢄꢌꢄꢑꢻꢄꢁ꣔ꢛꢄ  
ꢴꢔꢐꢏꢨꢆꢄꢙꢅꢆꢄꢁꢃ꣱ꢄꢀꢂꢆꢖꢄꢁꢃꢼꢕꢄꢌꢄꢁ꣌ꢛꢄ  
ꢴꢔꢐꢏꢨꢆꢄꢙꢅꢆꢄꢁꢃ꣱ꢄꢀꢂꢆꢖꢄꢓꢔ꣈ꢕꢄꢌꢄꢁ꣌ꢛꢄ  
ꢴꢔꢐꢏꢨꢆꢄꢙꢅꢆꢄꢁꢃ꣱ꢄꢢꢯꢄꢿꢮꢐꢄꢵꣶꢆꢖꢛꢄ  
꣒ꢅꢼꢄꢁ꣌ꢄꢙꢅꢆꢄꢁꢃ꣱ꢄꢿ꣊ꢆꢄꢙꢐꢩꢕꢄꢵꣶꢆꢖꢛꢄ  
꣒ꢅꢼꢄꢑꢻꢄꢁ꣔ꢄꢁꢃꢦꢧꢙꢄꢙꢅꢆꢄꢁꢃ꣱ꢛꢄ  
↑ hoÆc Ctrl-p  
↓ hoÆc Ctrl-n  
→ hoÆc Ctrl-f  
← hoÆc Ctrl-b  
Ctrl-l hoÆc Ctrl-←  
Ctrl-r hoÆc Ctrl-→  
Ctr-a hoÆc Home  
Ctrl-k  
Back Space  
Del  
꣒ꢅꢼꢄꢑꢻꢄꢁ꣔ꢄꢁ꣚ꢕꢄꢢꢽꢄꢁꢃꢻꢄꢙꢅꢆꢄꢁꢃ꣱ꢛꢄ  
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ꢴꢼꢙꢄꢗꢺꢆꢔꢄꢔꢺꢄꢁꢔꢩꢆꢖꢡꢄ  
clc  
clf  
꣒ꢅꢼꢄꢁꢅꢭꢆꢄꢬꢲꢄꢙꣃꢂꢄꢀ꣄ꢄꢵꣶꢆꢖꢄꢗꢺꢆꢔꢛꢄ  
꣒ꢅꢼꢄꢙꣃꢂꢄꢀ꣄ꢄꢿ꣙ꢄꢔꢅ꣚ꢄ  
computer  
ctrl-c  
demo  
exit,quit  
help  
꣍ꢺꢆꢔꢄꢕꢆꢄꢵꣶꢆꢖꢄꢑꢻꢄꢁ꣔ꢄꢙꢔꢅꢄꢬꢕ꣊ꢁꢄꢗꢅ꣚ꢕꢄꢜꢼꢏꢄꢁꢻꢆꢔꢛꢄ  
꣠꣌ꢆꢖꢄꢙꢔꢦꢤꢆꢖꢄꢁꢃꢷꢆꢔꢄꢑꢔꢕꢄꢃꢤꢕꢄꢢꢭꢅꢄꢁꢷꢆꢔꢄꢁꢃ꣚ꢆꢖꢄꢗ꣏ꢓꢄꢢꢥꢄꢔ꣚ꢆꢛꢄ  
꣍ꢺꢆꢔꢄꢙꢔꢅꢄꢟꢒꢜꢄꢙꢼꢙꢄꢙꢔꢦꢤꢆꢖꢄꢁꢃꢷꢆꢔꢄꢜ꣯ꢐꢛꢄ  
꣭ꢔꢅꢼꢁꢄꢑꢔ꣱ꢕꢄꢜꢥꢕꢄꢁꢃꢦ꣞ꢆꢖꢄꢫꢂꢁꢗꢂꢬꢛꢄ  
꣍ꢺꢆꢔꢄꢟꢒꢜꢄꢁꢃꢸꢄꢖꢕꢹꢓꢄꢢꢯꢄꢆꢖꢥꢆꢄꢆꢖ꣐ꢘꢄꢔꢭꢜꢘ ꢄ  
ꣅꢔꢝꢓꢄꢵ꣐ꢄꢗꢕꢺꢐꢄꢁ꣌ꢄꢬꢭꢆꢄꢓꢔꢻꢜꢄ  
input  
load  
꣭꣈ꢕꢄꢙꢼꢙꢄꢬꢕ꣊ꢆꢄꢗꢦꢐꢄꢁꢃꢅꢆꢖꢄꢌꢄꢎꢕꢗꢒꢄꢿꢦꢂꢄꢢꢭꢅꢄꢢꣲꢆꢖꢄꢗꢭꢜꢄꢢꢕꢺꢙꢛꢄ  
꣍ꢺꢆꢔꢄꢁ꣚ꢜꢄꢁꢔ꣞ꢕꢄꢵ꣌ꢆꢖꢄꢙꢔꢦꢤꢆꢖꢄꢁꢃꢷꢆꢔꢄꢿ꣊ꢆꢄꢓꢔꢻꢜꢄꢬ꣧ꢁꢄꢑꢷꢄꢿꢦꢸꢙꢄꢆꢔ꣧ꢆꢛꢄ  
꣍ꢦꢐꢄꢖꢕ꣐ꢄꢙꢼꢙꢄꢬꢕ꣊ꢆꢄꢢꢭꢅꢄꢎꢕꢗꢒꢄꢫꢂꢁꢗꢂꢬꢛꢜꢂꢁꢄ  
pause  
save  
version  
ver  
what  
ꢴꢔꢅꢄꢬꢕ꣊ꢁꢄꢓꢔꢕꣂꢆꢄꢬ꣈ꢆꢄꢔꢕꢺꢆꢄꢁ꣚ꢕꢄꢙꢰꢂꢄꢫꢂꢁꢗꢂꢬꢛꢄ  
ꢴꢔꢅꢄꢬꢕ꣊ꢁꢄꢓꢔꢕꣂꢆꢄꢬ꣈ꢆꢄꢔꢕꢺꢆꢄꢁ꣚ꢕꢄꢙꢰꢂꢄꢙꢼꢙꢄꢁꢅꢅꢗꢬꢅꢟꢄꢿꣵꢄꢙ꣆ꢛꢄ  
ꢴꢔꢅꢄꢬꢕ꣊ꢁꢄꢵꢂꢆꢔꢄꢀꢼꢙꢔꢄꢫꢂꢁꢗꢂꢬꢪꢎꢕꢗꢒꢄꢁꢔꢦꢄꢜꢶꢙꢄꢔꢕꢺꢆꢄꢁ꣚ꢕꢛꢄ  
Page 6 of 62  
which  
ꢴꢔꢅꢄꢬꢕ꣊ꢁꢄꢿꢦ꣞ꢆꢖꢄꢵ꣯ꢆꢄꢙꢰꢂꢄꣷꢎꢕꢗꢒ꣸ꢆꢂꢜꢒꣷꢛꢄ  
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1.3BiÕn vµ thao t¸c biÕn.  
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1.3.1 BiÕn trong Matlab.  
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꣭ꣂꢆꢄꢬꢕ꣊ꢆꢄꢁꢃꢅꢆꢖꢄꢫꢂꢁꢗꢂꢬꢄꢞꢁ꣌ꢄꢇꢛ꣹ꢄꢁꢃꣀꢄꢿꢕꢠꢄꢙ꣆ꢄꢁꢔꢨꢄꢵꢭꢕꢄꢁꢧꢕꢄ꣹ꢌꢄꢑꢻꢄꢁ꣔ꢄꢬꢂꢅꢄꢖ꣙ꢜꢄꢙꢼꢙꢄꢙꢔ꣐ꢄꢙꢼꢕꢄ  
꣣ꢪ꣺ꢄꢔꢂꢏꢄꢂꢪꣻꢄꢙꣲꢆꢖꢄꢙꢼꢙꢄꢙꢔ꣐ꢄꢀꢩꢄꢢꢭꢄꢵ꣧ꢐꢄꢖ꣚ꢙꢔꢄꢵꢦꢧꢕꢄꣷ꣸ꣷꢘꢄꢁꣂꢆꢄꢓꢔ꣈ꢕꢄꢿꢦꢸꢙꢄꢬ꣋ꢁꢄꢿꢮꢐꢄꢬ꣰ꢆꢖꢄꢙꢔ꣐ꢄꢙꢼꢕꢘꢄ  
ꢁꣂꢆꢄꢬꢕ꣊ꢆꢄꢑꢔꢥꢆꢖꢄꢿꢦꢸꢙꢄꢿ꣏ꢁꢄꢁꢃꣲꢆꢖꢄꢢꢧꢕꢄꢁꣂꢆꢄꢙꢼꢙꢄꢁ꣌ꢄꢑꢔꢅꢼꢛꢄ꣭ꣂꢆꢄꢙꢼꢙꢄꢔꢭꢜꢄꢿꣵꢄꢿꢦꢸꢙꢄꢿ꣏ꢁꢄꢙ꣇ꢆꢖꢄꢙ꣆ꢄ  
ꢁꢔꢨꢄꢿꢦꢸꢙꢄꢀꣃꢄꢵꢶꢆꢖꢄꢗꢭꢜꢄꢁꣂꢆꢄꢙꢰꢂꢄꢬꢕ꣊ꢆꢘꢄꢢꢭꢄꢔꢭꢜꢄꢆꢭꢏꢄꢀ꣨ꢄꢜ꣧ꢁꢄꢁꢼꢙꢄꢵꢶꢆꢖꢄꢙꢔꢅꢄꢿ꣊ꢆꢄꢑꢔꢕꢄꢬꢕ꣊ꢆꢄꢿ꣆ꢄ  
ꢿꢦꢸꢙꢄꢟꢅꢼꢛꢄ  
ꢄ
ꢴꢼꢙꢄꢔꢭꢜꢄꢁꢼꢙꢄꢿꢲꢆꢖꢄꢗꣂꢆꢄꢬꢕ꣊ꢆꢡꢄ  
clear all  
꣒ꢅꢼꢄꢁ꣧ꢁꢄꢙ꣈ꢄꢙꢼꢙꢄꢬꢕ꣊ꢆꢄꢔꢕꢺꢆꢄꢙ꣆ꢄꢁꢃꢅꢆꢖꢄꢢꣲꢆꢖꢄꢆꢔꢧꢛꢄ  
ꢴꢔ꣮ꢄꢟꢅꢼꢄꢙꢼꢙꢄꢬꢕ꣊ꢆꢄꢙ꣆ꢄꢁꣂꢆꢄꢙꢔ꣮ꢄꢿꢽꢆꢔꢛꢄ  
꣭ꢃ꣈ꢄꢢꢯꢄꢿꢽꢆꢔꢄꢆꢖꢔꣳꢂꢄꢙꢰꢂꢄꣷꢆꢂꢜꢒꣷꢡꢄ  
ꢋꢄꢆ꣊ꢐꢄꣷꢆꢂꢜꢒꣷꢄꢑꢔꢥꢆꢖꢄꢁ꣙ꢆꢄꢁ꣚ꢕꢛꢄ  
clear name2,name2,...  
exist('name')  
ꢌꢄꢆ꣊ꢐꢄꣷꢆꢂꢜꢒꣷꢄꢗꢭꢄꢬꢕ꣊ꢆꢛꢄ  
ꢊꢄꢆ꣊ꢐꢄꣷꢆꢂꢜꢒꣷꢄꢗꢭꢄꢁꣂꢆꢄꢜꢪꢎꢕꢗꢒꢛꢄ  
꣹ꢄꢆ꣊ꢐꢄꣷꢆꢂꢜꢒꣷꢄꢗꢭꢄꢄꢫ꣥꣒ꢪꢎꢕꢗꢒꢛꢄ  
꣼ꢄꢆ꣊ꢐꢄꣷꢆꢂꢜꢒꣷꢄꢗꢭꢄꢫ꣠꣍ꢪꢎꢕꢗꢒꢛꢄ  
ꢇꢄꢆ꣊ꢐꢄꣷꢆꢂꢜꢒꣷꢄꢔꢭꢜꢄꢬꢐꢕꢗꢁꢪꢕꢆꢛꢄ  
ꣽꢄꢆ꣊ꢐꢄꣷꢆꢂꢜꢒꣷꢄꢗꢭꢄ꣤ꢪꢎꢕꢗꢒꢛꢄ  
ꣾꢄꢆ꣊ꢐꢄꣷꢆꢂꢜꢒꣷꢄꢗꢭꢄꢁꣂꢆꢄꢁꢔꢦꢄꢜꢶꢙꢛꢄ  
pack  
꣍ꢺꢆꢔꢄꢁꢔ꣔ꢙꢄꢔꢕꢺꢆꢄꢀ꣋ꢓꢄꢟ꣊ꢓꢄꢗ꣚ꢕꢄꢢꣲꢆꢖꢄꢆꢔꢧꢄꢙꢔꢅꢄꢙꢼꢙꢄ  
ꢬꢕ꣊ꢆꢛꢄ꣗ꢔꢕꢄꢬꢲꢄꢆꢔꢧꢄꢜꢼꢏꢄꢁꢻꢆꢔꢄꢿꢮꢏꢘꢄꢗꢺꢆꢔꢄꢓꢂꢙꢑꢄꢙꢔꢅꢄ  
ꢓꢔꢣꢓꢄꢁ꣚ꢅꢄꢃꢂꢄꢁꢔꣂꢜꢄꢢꣲꢆꢖꢄꢆꢔꢧꢄꢙꢔꢅꢄꢬꢕ꣊ꢆꢄꢜꢭꢄ  
ꢑꢔꢥꢆꢖꢄꢓꢔ꣈ꢕꢄꢟꢅꢼꢄꢙꢼꢙꢄꢬꢕ꣊ꢆꢄꢿꣵꢄꢁ꣙ꢆꢄꢁ꣚ꢕꢛꢄ  
꣜ꢕꢨꢆꢄꢁꢔꢽꢄꢁꣂꢆꢄꢙꢼꢙꢄꢬꢕ꣊ꢆꢛꢄ  
who  
whos  
꣜ꢕꢨꢆꢄꢁꢔꢽꢄꢁꣂꢆꢄꢢꢭꢄꢑꢕꢨꢐꢄꢄꢙꢼꢙꢄꢬꢕ꣊ꢆꢛꢄ  
꣜ꢕꢨꢆꢄꢁꢔꢽꢄꢙꢼꢙꢄꢬꢕ꣊ꢆꢄꢙꢶꢙꢄꢬꢲꢛꢄ  
꣜ꢕꢨꢆꢄꢁꢔꢽꢄꢙꢼꢙꢄꢬꢕ꣊ꢆꢄꢁꢃꢅꢆꢖꢄꢁꣂꢓꢄꢎꢕꢗꢒꢆꢂꢜꢒꢛꢜꢂꢁꢛꢄ  
who global  
who -file 'filename'  
ꢄ
1.3.2 KÝch th−íc cña biÕn.  
꣎ꢲꢄꢗꢧꢆꢘꢄꢔꢂꢏꢄꢙꢔꢕꢯꢐꢄꢵꢭꢕꢄꢙꢰꢂꢄꢬꢕ꣊ꢆꢄꢢꢒꢙꢁꢅꢃꢄꢙ꣇ꢆꢖꢄꢆꢔꢦꢄꢜꢂꢄꢁꢃꢝꢆꢄꢿꢦꢸꢙꢄꢟꢼꢙꢄꣁꢐꢂꢄꢙꢼꢙꢄꢔꢭꢜꢄ  
ꢄ
ꢙ꣆ꢄꢀꣿꢆꢄꢙꢰꢂꢄꢫꢂꢁꢗꢂꢬꢛꢄ  
ꢀꢕꣻꢒꢞ꣣ꢠꢄ  
꣭ꢃ꣈ꢄꢢꢯꢄꢢꢒꢙꢁꢅꢃꢄꢊꢄꢓꢔꢮꢆꢄꢁꣃꢘꢄꢓꢔꢮꢆꢄꢁꣃꢄꢿꢮꢐꢄꢙꢔꢅꢄꢬꢕ꣊ꢁꢄꢀꢩꢄꢔꢭꢆꢖꢄꢙꢰꢂꢄꢜꢂꢄ  
ꢁꢃꢝꢆꢘꢄꢄꢓꢔꢮꢆꢄꢁꣃꢄꢿꢮꢐꢄꢙꢔꢅꢄꢬꢕ꣊ꢁꢄꢀꢩꢄꢙꢲꢁꢄꢙꢰꢂꢄꢜꢂꢄꢁꢃꢝꢆꢄ  
꣩ꢜꢘꢆ꣪꤀ꢀꢕꣻꢒꢞ꣣ꢠꢄ ꣭ꢃ꣈ꢄꢖꢕꢼꢄꢁꢃꢽꢄꢿꢲꢄꢗꢧꢆꢄꢙꢰꢂꢄꢜꢂꢄꢁꢃꢝꢆꢘꢄꢜꢪꢀꢩꢄꢔꢭꢆꢖꢘꢄꢆꢪꢀꢩꢄꢙꢲꢁꢛꢄ  
ꢀꢕꣻꢒꢞ꣣ꢘꢓꢠꢄ  
꣭ꢃ꣈ꢄꢢꢯꢄꢀꢩꢄꢔꢭꢆꢖꢄꢆ꣊ꢐꢄꢓ꤀ꢌꢄꢄꢢꢭꢄꢀꢩꢄꢙꢲꢁꢄꢆ꣊ꢐꢄꢓ≥ꢊꢛꢄ  
꣭ꢃ꣈ꢄꢄꢢꢯꢄꢖꢕꢼꢄꢁꢃꢽꢄꢀꢩꢄꢗꢦꢸꢆꢖꢄꢓꢔꢮꢆꢄꢁꣃꢄꢙꢰꢂꢄꢟꢛꢄ  
꣭ꢃ꣈ꢄꢖꢕꢼꢄꢢꢯꢄꢁꢃꢽꢄꢜꢂꢟꢞꢜꢘꢆꢠꢄꢢꢧꢕꢄ꣩ꢜꢘꢆ꣪꤀ꢀꢕꣻꢒꢞ꣣ꢠꢛꢄ  
ꢗꢒꢆꢖꢁꢔꢞꢟꢠꢄ  
ꢗꢒꢆꢖꢁꢔꢞ꣣ꢠꢄ  
ꢄ
1.3.3 Mét sè biÕn ®−îc ®Þnh nghÜa tr−íc.  
ꢄ
ꢄ
ans  
eps  
BiÕn cã s½n ®−îc dïng l−u kÕt qu¶ phÐp tÝnh cuèi cïng.  
Tr¶ vÒ ®é chÝnh x¸c t−¬ng ®èi tÝnh to¸n cña m¸y x¸c ®Þnh kho¶ng tõ  
1.0 ®Õn sè ph¶y ®éng lín nhÊt tiÕp theo. EPS ®−îc sö dông lµ sai sè  
Page 7 of 62  
cho phÐp mÆc ®Þnh cña pinv, rank vµ c¸c hµm kh¸c.  
Tr¶ vÒ trÞ 3.1415926535897  
Cho biÕt gi¸ trÞ lín nh  
pi  
realmax  
Êt m¸y tÝnh cã thÓ tÝnh to¸n ®−îc, c¸c sè lín h¬n gi¸ trÞ nµy sÏ g©y  
trµn trªn.  
realmin  
Cho biÕt gi¸ trÞ nhá nhÊt m¸y tÝnh cã thÓ tÝnh to¸n ®−îc, c¸c sè lín  
bÐ gi¸ trÞ nµy sÏ g©y trµn d−íi.  
inf  
biÓu diÔn sè v« cïng ∝  
NaN  
biÓu diÔn kÕt qu¶ 0/0, ∝-∝  
1.4Sè phøc.  
1.4.1 C¸c phÐp to¸n ®èi víi sè phøc.  
ꢴꢔꢅꢄꢙꢼꢙꢄꢀꢩꢄꢓꢔ꣕ꢙꢄꣻꢌ꤀ꢂꢌ꣑ꢕ꤁ꢬꢌꢘꢄꢄꣻꢊ꤀ꢂꢊ꣑ꢕ꤁ꢬꢊꢘꢄꢢꢧꢕꢄꢂꢌꢘꢂꢊꢘꢬꢌꢘꢬꢊꢄꢗꢭꢄꢙꢼꢙꢄꢀꢩꢄꢁꢔ꣔ꢙꢛꢄ  
ꢄ
PhÐp to¸n  
z1 z2  
z1*z2  
KÕt qu¶  
(a1+a2) i(b1+b2)  
(a1*a2-b1*b2)+i(a1*b2+a2*b1)  
(a1 *a2 + b1 *b2 ) + i(a2 *b1 − a1b2 )  
z1  
z2  
a22 + b22  
a12 + b12  
a1-ib1  
|z1|  
z1*  
§é lín (magnitude)hay modul cña sè phøc.  
Sè phøc liªn hîp.  
1.4.2 Mét sè hµm ®Æc biÖt cña sè phøc.  
real(a+i*b)  
imag(a+i*b)  
conj(a+i*b)  
abs(a1+i*b1)  
Hµm cho gi¸ trÞ phÇn thùc a cña sè phøc.  
Hµm cho gi¸ trÞ phÇn ¶o b cña sè phøc.  
Hµm tr¶ vÒ sè phøc liªn hîp a-i*b.  
Tr¶ vÒ modul a12 + b12 cña sè phøc  
angle(a+i*b)  
TÝnh gãc cã gi¸ trÞ lµ atan2(b,a), gi¸ trÞ gãc ∈[−π,π]  
ꢄ
1.4.3 To¹ ®é biÓu diÔn sè phøc.  
꣭ꢂꢄꢙ꣆ꢄꢁꢔꢨꢄꢬꢕꢨꢐꢄꢵꢕ꣛ꢆꢄꢀꢩꢄꢓꢔ꣕ꢙꢄꢂ꣑ꢕ꤁ꢬꢄꢁꢃꣂꢆꢄꢔꢺꢄꢁꢃꢶꢙꢄꢁꢅ꣚ꢄꢿꢲꢛꢄ꣎ꢩꢕꢄꢢꢧꢕꢄꢔꢺꢄꢁꢃꢶꢙꢄꢁꢅ꣚ꢄꢿꢲꢄ  
ꢿꢯꢙꢼꢙꢄꢓꢔꢮꢆꢄꢁꢔ꣔ꢙꢄꢿꢦꢸꢙꢄꢬꢕꢨꢐꢄꢵꢕ꣛ꢆꢄꢁꢃꣂꢆꢄꢁꢃꢶꢙꢄꢟꢡꢄꢟ꤀ꢂꢘꢄꢓꢔꢮꢆꢄ꣈ꢅꢄꢿꢦꢸꢙꢄꢬꢕꢨꢐꢄꢵꢕ꣛ꢆꢄꢁꢃꣂꢆꢄꢁꢃꢶꢙꢄꢏꢡꢄ  
ꢄ
ꢏ꤀ꢬꢛꢄ꣎ꢩꢕꢄꢢꢧꢕꢄꢔꢺꢄꢁꢅ꣚ꢄꢿꢲꢄꢙ꣔ꢙꢘꢄꢀꢩꢄꢓꢔ꣕ꢙꢄꢿꢦꢸꢙꢄꢬꢕꢨꢐꢄꢵꢕ꣛ꢆꢄꢬꣀꢕꢄρꢘθꢛꢄ  
ꢄ
꣭ꢃꢅꢆꢖꢄꢿ꣆ꢡꢄ  
= a2 + b2  
y
ρ
,
(a,b)  
a
b
ꢄ
ꢄ
b
tan−1  
.
ρ
θ
=
a
x
θ
ꢄ
ꢄ
ꢆꢖꢦꢸꢙꢄꢗ꣚ꢕꢡꢄ  
a =  
ρ
cos(  
θ ),  
ꢄ
b =  
ρ
sin(θ  
).  
ꢄ
ꢄ
H×nh BiÓu diÔn sè phøc  
Page 8 of 62  
ꢄ ꣭ꢃꢅꢆꢖꢄꢔꢺꢄꢁꢅ꣚ꢄꢿꢲꢄꢙ꣔ꢙꢡꢄꢿꢲꢄꢗꢧꢆꢄꢞꢜꢂꢖꢆꢕꢁꢐꢵꢒꢠꢄꢢꢭꢄꢓꢔꢂꢄꢞꢓꢔꢂꢀꢒꢠꢄꢙꢰꢂꢄꢀꢩꢄꢓꢔ꣕ꢙꢄꢀ꣨ꢄꢿꢦꢸꢙꢄꢁꢻꢆꢔꢄ  
ꢁꢅꢼꢆꢄꢆꢔꢦꢄꢀꢂꢐꢡꢄ  
>>r=abs(z);  
>>theta=angle(z);  
ꢄ
ꢄ
꣫ꢕꢨꢐꢄꢵꢕ꣛ꢆꢄꢀꢩꢄꢓꢔ꣕ꢙꢄꢁꢔꢒꢅꢄꢿꢲꢄꢗꢧꢆꢄꢢꢭꢄꢓꢔꢂꢀꢒꢄꢆꢔꢦꢄꢀꢂꢐꢡꢄ  
>>z=r*axp(i*theta);  
꣭ꢃꢅꢆꢖꢄꢔꢺꢄꢁꢅ꣚ꢄꢿꢲꢄꢿꢯꢙꢼꢙꢘꢄꢓꢔꢮꢆꢄꢁꢔ꣔ꢙꢄꢞꢃꢒꢂꢗꢠꢄꢢꢭꢄꢓꢔꢮꢆꢄ꣈ꢅꢄꢞꢕꢜꢂꢖꢕꢆꢂꢃꢏꢠꢄꢀꢒꢄꢿꢦꢸꢙꢄꢁꢻꢆꢔꢄꢁꢅꢼꢆꢄ  
ꢆꢔꢦꢄꢀꢂꢐꢡꢄ  
>>a=real(z);  
>>b=imag(z);  
ꢄ
꣫ꢕꢨꢐꢄꢵꢕ꣛ꢆꢄꢀꢩꢄꢓꢔ꣕ꢙꢡꢄ  
>>z=a+i*b;  
1.5Lµm viÖc ë chÕ ®é héi tho¹i (command window).  
ꢄ
ꢪ꣭꣚ꢕꢄꢵ꣧ꢐꢄꢜ꣞ꢕꢄ ꢖ꣬ꢄꢵꣶꢆꢖꢄꢗꢺꢆꢔꢘꢄꢵꣶꢆꢖꢄꢗꢺꢆꢔꢄꢀ꣨ꢄꢿꢦꢸꢙꢄꢁꢔ꣔ꢙꢄꢔꢕꢺꢆꢄꢆꢖꢂꢏꢄꢗꢝꢓꢄꢁ꣕ꢙꢄꢀꢂꢐꢄꢑꢔꢕꢄꢆꢔ꣧ꢆꢄ  
»
ꢓꢔꢻꢜꢄEnter ↵ꢄꢛꢄ꣗꣊ꢁꢄꣁꢐ꣈ꢄꢀ꣨ꢄꢿꢦꢸꢙꢄꢿꢦꢂꢄꢃꢂꢄꢙꣃꢂꢄꢀ꣄ꢄꢗꢺꢆꢔꢄꢔꢅ꣏ꢙꢄꢙꣃꢂꢄꢀ꣄ꢄꢿ꣙ꢄꢔꢅ꣚ꢛꢄꣅ꣊ꢐꢄꢄꢙ꣆ꢄꢗ꣝ꢕꢘꢄ  
ꢗ꣝ꢕꢄꢀ꣨ꢄꢿꢦꢸꢙꢄꢖꢔꢕꢄꢆꢖꢂꢏꢄꢁ꣚ꢕꢄꢙꣃꢂꢄꢀ꣄ꢄꢗꢺꢆꢔꢛꢄ  
ꢪ꤂ꢐꢕꢄꢁ꣋ꢙꢄꢢꢕ꣊ꢁꢄꢵꣶꢆꢖꢄꢗꢺꢆꢔꢡꢄꢄ  
ꢄ
꣑꣭ꢃꣂꢆꢄꢌꢄꢵꣶꢆꢖꢄꢙ꣆ꢄꢁꢔꢨꢄꢢꢕ꣊ꢁꢄꢆꢔꢕꢯꢐꢄꢗꢺꢆꢔꢘꢄꢙꢼꢙꢄꢗꢺꢆꢔꢄꢿꢦꢸꢙꢄꢓꢔ꣖ꢆꢄꢙꢼꢙꢔꢄꢬꣀꢕꢄꢵ꣧ꢐꢄꣷꢘꣷꢄꢔꢅ꣏ꢙꢄꣷ꤃ꣷꢛꢄ  
ꣅ꣊ꢐꢄꢑ꣊ꢁꢄꣁꢐ꣈ꢄꢗꢺꢆꢔꢄꢁꢃ꣈ꢄꢢꢯꢄꢖꢕꢼꢄꢁꢃꢽꢄꢀꢩꢄꢢꢭꢄꢑ꣊ꢁꢄꢁꢔꢹꢙꢄꢬꣀꢕꢄꢵ꣧ꢐꢄꣷꢘꣷꢄꢑ꣊ꢁꢄꣁꢐ꣈ꢄꢀ꣨ꢄꢿꢦꢸꢙꢄꢕꢆꢄꢁꢃꣂꢆꢄꢙꣃꢂꢄꢀ꣄ꢄ  
ꢗꢺꢆꢔꢘꢄꢙꣶꢆꢄꢆ꣊ꢐꢄꢑ꣊ꢁꢄꢁꢔꢹꢙꢄꢬꣀꢕꢄꢵ꣧ꢐꢄꣷ꤃ꣷꢄꢑ꣊ꢁꢄꣁꢐ꣈ꢄꢄꢀ꣨ꢄꢑꢔꢥꢆꢖꢄꢿꢦꢸꢙꢄꢕꢆꢄꢃꢂꢛꢄ꣦ꢧꢕꢄꢙꢼꢙꢄꢗꢺꢆꢔꢄꢑꢔꢥꢆꢖꢄꢁꢃ꣈ꢄ  
ꢢꢯꢄꢬꢕ꣊ꢆꢄꢆꢭꢅꢘꢄꢵ꣧ꢐꢄꢓꢔ꣖ꢆꢄꢙꢼꢙꢔꢄꢗꢺꢆꢔꢄꣷꢘꣷꢄꢢꢭꢄꣷ꤃ꣷꢄꢗꢭꢄꢆꢔꢦꢄꢆꢔꢂꢐꢛꢄ  
ꢄ ꣑꣗ꢔꢕꢄꢗꢺꢆꢔꢄꣁꢐꢼꢄꢵꢭꢕꢘꢄꢔꢅ꣏ꢙꢄꢜꢐꢩꢆꢄꢢꢕ꣊ꢁꢄꢁꢃꣂꢆꢄꢆꢔꢕꢯꢐꢄꢵꣶꢆꢖꢄꢗꢺꢆꢔꢄꢵꣲꢆꢖꢄꢵ꣧ꢐꢄꢛꢛꢛꢄꢿꢨꢄꢜꢂꢁꢗꢂꢬꢄ  
ꢆꢔꢝꢆꢄꢵꣶꢆꢖꢄꢁꢕꢯꢓꢄꢁꢔꢒꢅꢛꢄ  
ꢪꢴꢔ꣮ꢄꢆꣂꢆꢄꢀꣃꢄꢵꢶꢆꢖꢄꢙꢔ꣊ꢄꢿꢲꢄꢔꢲꢕꢄꢁꢔꢅ꣚ꢕꢄꢑꢔꢕꢄꢖꢕ꣈ꢕꢄꢬꢭꢕꢄꢁꢅꢼꢆꢄꢙ꣆ꢄꢙ꣧ꢐꢄꢁꢃꢹꢙꢄꢿꢤꢆꢄꢖꢕ꣈ꢆꢛꢄꢴꣶꢆꢄꢢꢧꢕꢄꢬꢭꢕꢄ  
ꢁꢅꢼꢆꢄꢙ꣆ꢄꢙ꣧ꢐꢄꢁꢃꢹꢙꢄꢓꢔ꣕ꢙꢄꢁ꣚ꢓꢄꢆꣂꢆꢄꢀꣃꢄꢵꢶꢆꢖꢄꢙꢔ꣊ꢄꢿꢲꢄꢗꢝꢓꢄꢁꢃꢷꢆꢔꢛꢄ  
VÝ dôꢄꢡꢄꣴꢕ꣈ꢕꢄꢓꢔꢦꢤꢆꢖꢄꢁꢃꢷꢆꢔꢄ꣹ꢟꢊꢪꢊꢟꢪꢌ꤀ꢋ꤃ꢄꢄꢄꢢꢭꢄꢂꢟꢊ꣑ꢬꢟ꣑ꢙ꤀ꢋ꤃ꢄ  
» eq=[3 -2 -1];  
» n0=roots(eq)  
» syms a b c x;  
» eq=a*x^2+b*x+c;  
» n0=solve(eq,x)  
n0 =  
n0 =  
1.0000  
-0.3333  
ꢄ
[ 1/2/a*(-b+(b^2-4*a*c)^(1/2))]  
[ 1/2/a*(-b-(b^2-4*a*c)^(1/2))]  
ꢄ
1.6Lµm vÖc ë chÕ ®é viÕt ch−¬ng tr×nh.  
ꢄ
ꢪ꣭꣚ꢕꢄꢙꣃꢂꢄꢀ꣄ꢄꢗꢺꢆꢔꢄꢞꢴꢅꢜꢜꢂꢆꢵꢄꢱꢕꢆꢵꢅꢚꢠꢄꢙꢔ꣘ꢆꢄFile ꢜꢒꢆꢐꢄꢀ New ꢀ m-file ꢿꢨꢄꢢꢭꢅꢄꢙꢔ꣊ꢄ  
ꢿꢲꢄꢀꢅ꣚ꢆꢄꢁꢔ꣈ꢅꢄꢢꢕ꣊ꢁꢄꢙꢔꢦꢤꢆꢖꢄꢁꢃꢷꢆꢔꢛꢄ꣜ꢅ꣏ꢙꢄꢜꣀꢄꢜꢪꢎꢕꢗꢒꢄꢙ꣆ꢄꢀꣿꢆꢄꢿꢨꢄꢀꣃꢂꢘꢄꢙꢔ꣘ꢆꢄFile ꢜꢒꢆꢐ ꢀ  
Open ꢀ ꢙꢔ꣘ꢆ m-file ꢙꢮꢆꢄꢀꣃꢂꢘꢄ  
ꢄꢔꢅ꣏ꢙꢄꢢꢕ꣊ꢁꢄꢁ꣚ꢕꢄꢙꣃꢂꢄꢀ꣄ꢄꢗꢺꢆꢔꢄꢵꣶꢆꢖꢄꢗꢺꢆꢔꢄ» edit filename.  
ꢪꢴꢔ꣚ꢏꢄꢎꢕꢗꢒꢄꢙꢔꢦꢤꢆꢖꢄꢁꢃꢷꢆꢔꢄꢿꣵꢄꢀꢅ꣚ꢆꢄꢁꢔ꣈ꢅꢡꢄꢴ꣆ꢄ꣹ꢄꢙꢼꢙꢔꢛꢄ  
꣑ꢴꢔ꣚ꢏꢄꢆꢖꢂꢏꢄꢁ꣚ꢕꢄꢙꣃꢂꢄꢀ꣄ꢄꢀꢅ꣚ꢆꢄꢁꢔ꣈ꢅꢡꢄTool ꢜꢒꢆꢐꢄꢀ Runꢄꢘꢄꢙꢔ꣮ꢄꢵꣲꢆꢖꢄꢿꢦꢸꢙꢄꢑꢔꢕꢄꢎꢕꢗꢒꢄ  
ꢿꢦꢸꢙꢄꢗꢦꢐꢄꢞSave)ꢄꢢꢭꢅꢄꢁꢔꢦꢄꢜꢶꢙꢄꢔꢕꢺꢆꢄꢁ꣚ꢕꢄꢞꢙꢐꢃꢃꢒꢆꢁꢄꢓꢂꢁꢔꢠꢛꢄ  
ꢄ
Page 9 of 62  
ꢄ
ꢄ
꣑ꢴꢔ꣚ꢏꢄꢁꢃꢅꢆꢖꢄꢙꣃꢂꢄꢀ꣄ꢄꢗꢺꢆꢔꢡꢄ꤄꤄ꢒꢵꢕꢁꢄꢁꣂꢆ꣸ꢜ꣸ꢎꢕꢗꢒꢄꢄ↵ꢄꢛꢄ  
꣑ꢴꢔ꣚ꢏꢄꢁꢃꢅꢆꢖꢄꢙꣃꢂꢄꢀ꣄ꢄꢗꢺꢆꢔꢡꢄꢄFile ꢜꢒꢆꢐ ꢀRun Script... ꢢꢕ꣊ꢁꢄꢁꣂꢆꢄꢎꢕꢗꢒꢘꢄꢔꢅ꣏ꢙꢄꢙꢔ꣘ꢆꢄ  
꣫ꢃꢅꢚꢀꢒꢄꢿꢨꢄꢁꢷꢜꢄꢜꢪꢎꢕꢗꢒꢄꢙꢮꢆꢄꢙꢔ꣚ꢏꢛꢄ  
ꢄ
1.7S¬ l−îc vÒ ®å häa.  
1.7.1 C¸c lÖnh ®å ho¹ th«ng dông.  
ꢄ
꣗ꢔ꣈ꢄꢆ꣉ꢆꢖꢄꢿ꣙ꢄꢔꢅ꣚ꢄꢁꢃꢅꢆꢖꢄꢫꢂꢁꢗꢂꢬꢄꢃ꣧ꢁꢄꢓꢔꢅꢆꢖꢄꢓꢔꢹꢄꢙꢔꢅꢄꢓꢔꢣꢓꢄꢢ꣨ꢄꢔꢮꢐꢄꢔ꣊ꢁꢄꢙꢼꢙꢄꢵ꣚ꢆꢖꢄꢿ꣙ꢄ  
ꢁꢔꢽꢄꢢꢭꢄꢙꢼꢙꢄꢵ꣚ꢆꢖꢄꢬꢕꢨꢐꢄꢿ꣙ꢄꢙꢔꢅꢄꢙꢼꢙꢄꢁꢔꢥꢆꢖꢄꢁꢕꢆꢄꢁꢃꢅꢆꢖꢄꢜ꣏ꢁꢄꢓꢔ꤅ꢆꢖꢄꢞꢊ꣠ꢠꢄꢢꢭꢄꢁꢃꢅꢆꢖꢄꢑꢔꢥꢆꢖꢄꢖꢕꢂꢆꢄ  
ꢞ꣹꣠ꢠꢛꢄ  
ꢄ
ꢙꢼꢙꢄꢗꢺꢆꢔꢄꢿ꣙ꢄꢔꢅ꣚ꢄꢿꢤꢆꢄꢖꢕ꣈ꢆꢡꢄ  
plot(x1,y1,s1,x2,y2,s2,..) VÏ c¸c ®å thÞ trong mÆt ph¼ng to¹ ®é ®Òc¸c  
víi kiÓu ®−êng chØ ®Þnh trong s1,s2..  
plot(y)  
NÕu y lµ d·y sè thùc≡plot(i=1:length(y),y).  
nÕu y lµ sè phøc≡plot(real(y),imag(y)).  
VÏ c¸c ®å thÞ trong kh«ng gian to¹ ®é ®Òc¸c  
víi kiÓu ®−êng chØ ®Þnh trong s1,..  
§−a tiªu ®Ò ®å thÞ.  
plot3(x1,y1,z1,s1,..)  
title('tiªu ®Ò ®å thÞ')  
xlabel, ylabel, zlabel  
legend  
§−a vµo c¸c nh·n cho c¸c trôc x, y, z.  
LÖnh ®−a vµo chó thÝch ®−êng vÏ.  
LÖnh vÏ c¸c ®−êng dãng trªn ®å thÞ.  
grid  
ꢄ
꣦ꢻꢄꢵꢶꢡ  
Vi du ve ham 3D  
» t=linspace(0,2*pi,40);  
8
» plot3(sin(t),t.*cos(t),t);  
» title('Vi du ve ham 3D');  
» xlabel('x'),ylabel('axis y'),  
» zlabel('t'), grid on;  
6
4
2
0
10  
ꢄꢄꢄꢄꢄꢄꢄꢄꢄꢄꢄꢄꢄꢄꢄ  
1
5
0.5  
0
0
ꢄ
ꢄ
-0.5  
-5  
-1  
axis  
y
x
ꢄ
H×nh §å thÞ kh«ng gian 3D  
1.7.2 In Ên trªn cöa sæ ®å ho¹.  
꣦ꢕꢺꢙꢄꢕꢆꢄꢙꢼꢙꢄ꣈ꢆꢔꢄꢿ꣙ꢄꢔꢅ꣚ꢄꢙ꣆ꢄꢁꢔꢨꢄꢁꢔ꣔ꢙꢄ  
ꢔꢕꢺꢆꢄꢁꢔꢥꢆꢖꢄꣁꢐꢂꢄꢙꢼꢙꢄꢜꢒꢆꢐꢄꢗꢺꢆꢔꢄꢁꢃꣂꢆꢄꢙꣃꢂꢄꢀ꣄ꢄꢿ꣙ꢄꢔꢅ꣚ꢄꢔꢅ꣏ꢙꢄꢙꢼꢙꢄꢗꢺꢆꢔꢄꢙꢰꢂꢄꢫꢂꢁꢗꢂꢬꢛꢄ  
>>print;  
>>print filename  
%In cöa sæ ®å ho¹ hiÖn thêi ra m¸y in.  
% In cöa sæ ®å ho¹ hiÖn thêi ra tÖp filename.fig.  
ꢄ
1.7.3 Mét vµi vÝ dô minh ho¹.  
꣦꣨ꢄꢿ꣙ꢄꢁꢔꢽꢄꢙꢼꢙꢄꢔꢭꢜꢄꢏ꤀ꢀꢕꢆꢞꢟꢠꢘꢄꢏ꤀ꢟ꤁ꢙꢅꢀꢞꢟꢠꢘꢄꢏ꤀ꢗꢆꢞꢟꢊ꣑ꢌꢠꢄꢁꢃꢅꢆꢖꢄꢑꢔꢅ꣈ꢆꢖꢄꢪꢊ≤ꢟ≤ꢊꢛꢄ  
» x=linspace(-2,2);  
» y1=sin(x);y2=x.*cos(x);y3=log(x.^2+1);  
» plot(x,y1,x,y2,x,y3);  
» legend('sin(x)','xcos(x)','ln(x^2+1)');  
Page 10 of 62  
2
1.5  
1
sin(x)  
xcos(x)  
2
ln(x +1)  
0.5  
0
-0.5  
-1  
-2  
-1.5  
-1  
-0.5  
0
0.5  
1
1.5  
2
ꢄ
3
x = t − 3t +1;  
ꢄ
꣦꣨ꢄꢿꢦ꣞ꢆꢖꢄꢙꢅꢆꢖꢄꢁꢔꢂꢜꢄꢀꢩꢄꢙꢔꢅꢄꢬꣀꢕꢄꢓꢔꢦꢤꢆꢖꢄꢁꢃꢷꢆꢔꢡ  
ꢄ
3
y = t + 3t +1;  
» t=linspace(-2.5,2.5);x=t.^3-3*t+1;y=t.^3+3*t-1;  
» plot(x,y);grid on;axis tight;  
» xlabel('x'); ylabel('y');title('Do thi duong cong tham so');  
ꢄ
ꢄ
Do thi duong cong tham so  
ꢄ
20  
ꢄ
15  
ꢄ
10  
ꢄ
5
ꢄ
0
ꢄ
-5  
ꢄ
-10  
ꢄ
-15  
ꢄ
-20  
ꢄ
-6  
-4  
-2  
0
2
4
6
8
ꢄ
ꢄ
ꢄ
x
ꢄ
H×nh  
2
2
ꢄ
꣦꣨ꢄꢓꢔ꣄ꢄꢜꢭꢐꢄꢙꢰꢂꢄꢜ꣏ꢁꢄ z = f (x, y) = (x2 −3y2 )e−(x + y )  
,
ꢁꢃꢅꢆꢖꢄꢑꢔꢥꢆꢖꢄꢖꢕꢂꢆꢄꢊ꣠ꢛꢄ  
» [x,y]=meshgrid(linspace(-2,2));  
» z=(x.^2-3*y.^2).*exp(-(x.^2+y.^2));pcolor(x,y,z);  
» shading interp; colorbar;  
ꢄ
ꢄ
ꢄ
ꢄ
ꢄ
ꢄ
ꢄ
ꢄ
ꢄ
ꢄ
ꢄ
ꢄ
ꢄ
ꢄ
ꢄ
ꢄ
ꢄ
H×nh  
Page 11 of 62  
1.8C¸c hµm ©m thanh.  
ꢄ
ꢄ
ꢫꢂꢁꢗꢂꢬꢄꢙꢔꢅꢄꢓꢔꢣꢓꢄꢁ꣚ꢅꢄ꣖ꢜꢄꢁꢔꢂꢆꢔꢄꢁꢔꢥꢆꢖꢄꣁꢐꢂꢄꢙꢼꢙꢄꢢꢒꢙꢁꢅꢃꢄꢬꣀꢕꢄꢔꢭꢜꢄꢀꢅꢐꢆꢵꢞꢠꢛꢄ  
sound(v)  
꣭꣚ꢅꢄꢁꢻꢆꢄꢔꢕꢺꢐꢄꢁ꣌ꢄꢢꢒꢙꢁꢅꢃꢄꢢꢄꢃꢂꢄ꣖ꢜꢄꢁꢔꢂꢆꢔꢘꢄꢢꢒꢙꢁꢅꢃꢄꢢꢄꢓꢔ꣈ꢕꢄ  
ꢿꢦꢸꢙꢄꢙꢔꢐ꣟ꢆꢄꢔꢅꢼꢄꢪꢌ≤ꢏ≤ꢌꢘꢄꢙꢼꢙꢄꢖꢕꢼꢄꢁꢃꢽꢄꢆꢖꢅꢭꢕꢄꢑꢔꢅ꣈ꢆꢖꢄꢀ꣨ꢄ  
ꢬꢽꢄꢙ꣋ꢁꢘꢄꢁꢩꢙꢄ ꢿꢲꢄ ꢜ꣯ꢐꢄ ꢜ꣏ꢙꢄꢿꢽꢆꢔꢄꢗꢭꢄꢉꢌ꤆ꢊ꣜ꣻꢛꢄ ꢫꢐꢩꢆꢄꢙ꣆ꢄ  
꣖ꢜꢄꢁꢔꢂꢆꢔꢄꢀꢁꢒꢃꢒꢅꢄꢢꢄꢗꢭꢄꢢꢒꢙꢁꢅꢃꢄꢊꢄꢙꢲꢁꢛꢄ  
sound(v,fs)  
sound(y,fs,bits)  
꣭ꢔ꣔ꢙꢄꢔꢕꢺꢆꢄꢆꢔꢦꢄꢔꢭꢜꢄꢀꢅꢐꢆꢵꢞꢢꢠꢄꢢꢧꢕꢄꢎꢄꢗꢭꢄꢵ꣈ꢕꢄꢁꢮꢆꢄ꣜ꣻꢛꢄ  
꣭꣚ꢅꢄ꣖ꢜꢄꢁꢔꢂꢆꢔꢄꢢꢧꢕꢄꢁꢩꢙꢄꢿꢲꢄꢜ꣯ꢐꢄꢬꢕꢁꢀꢈꢜ꣯ꢐꢄꢆ꣊ꢐꢄꢙ꣆ꢄꢑꢔ꣈ꢄ  
ꢆ꣉ꢆꢖꢛꢄꢄ꣜ꢮꢐꢄꢔ꣊ꢁꢄꢙꢼꢙꢄꢁꢔꢕ꣊ꢁꢄꢬꢽꢄꢙꢔꢅꢄꢓꢔꢣꢓꢄꢬꢕꢁꢀ꤀ꢉꢄꢔꢂꢏꢄꢌꣽꢛꢄ  
꣭ꢦꢤꢆꢖꢄꢁ꣔ꢄꢀꢅꢐꢆꢵꢞꢠꢄꢆꢔꢦꢆꢖꢄꢀ꣨ꢄꢁ꣚ꢅꢄ꣖ꢜꢄꢁꢔꢂꢜꢔꢄꢢꢧꢕꢄꢖꢕꢼꢄꢁꢃꢽꢄ  
ꢀꢩꢄꢬ꣧ꢁꢄꢑꢷꢄꢙꢰꢂꢄꢢꢘꢄꢢꢄꢀ꣨ꢄꢿꢦꢸꢙꢄꢙꢔꢐ꣟ꢆꢄꢔꢅꢼꢄꢁꢔꢒꢅꢄ  
soundsc(v,slim)  
ꢀꢗꢕꢜ꤀꣩ꢀꢗꢅꢚꢄꢀꢄꢔꢕꢖꢔ꣪ꢘꢄꢜ꣏ꢙꢄꢿꢽꢆꢔꢄꢀꢗꢕꢜ꤀꣩ꢜꢕꢆꢞꢢꢠꢄꢜꢂꢟꢞꢢꢠ꣪ꢄ  
Page 12 of 62  
ch−¬ng 2  
Ma trËn vµ c¸c phÐp to¸n ma trËn  
2.1V« h−íng, vector vµ ma trËn.ꢄ  
ꢫꢂꢄꢁꢃꢝꢆꢄꢞmatrixꢠꢄꢗꢭꢄꢌꢄꢬ꣈ꢆꢖꢄꢀꢩꢄꢜꢭꢄꢙꢼꢙꢄꢓꢔꢮꢆꢄꢁꣃꢄꢙꢰꢂꢄꢆ꣆ꢄꢿꢦꢸꢙꢄꢀ꣋ꢓꢄꢟ꣊ꢓꢄꢁꢔꢒꢅꢄꢔꢭꢆꢖꢄꢢꢭꢄ  
ꢄ
ꢙꢲꢁꢛꢄꢫꢂꢄꢁꢃꢝꢆꢄꢙꢔ꣮ꢄꢙ꣆ꢄꢜꢲꢁꢄꢔꢭꢆꢖꢄꢔꢅ꣏ꢙꢄꢜꢲꢁꢄꢙꢲꢁꢄꢿꢦꢸꢙꢄꢖ꣘ꢕꢄꢗꢭꢄꢢꢣꢙꢄꢁꢤꢄꢞvectorꢠꢛꢄꢫꢂꢄꢁꢃꢝꢆꢄꢙꢔ꣮ꢄꢙ꣆ꢄ  
ꢜꢲꢁꢄꢔꢭꢆꢖꢄꢢꢭꢄꢜꢲꢁꢄꢙꢲꢁꢄꢿꢦꢸꢙꢄꢖ꣘ꢕꢄꢗꢭꢄꢿ꣚ꢕꢄꢗꢦꢸꢆꢖꢄꢢꢥꢄꢔꢦꢧꢆꢖꢄꢞscalarꢠꢛꢄ  
ꢄ
ꢄ
꣗ꢻꢄꢔꢕꢺꢐꢄꢁꢅꢼꢆꢄꢔ꣘ꢙꢄꢜꢂꢄꢁꢃꢝꢆꢄꢢꢭꢄꢢꢒꢙꢁꢅꢃꢡꢄ  
Vector cét vc, vector hµng vr  
Ma trËn (  
mxn)  
v
a
a12 ꢀ a1n  
a22 ꢀ a  
1   
11  
v
a
2   
2n   
21  
Amn  
=
ꢄ
vc =  
ꢄ
ꢁ
ꢁ
ꢁ
ꢁ
vq  
am1 am2 ꢀ amn  
(
)
vr = v1 v2 ꢀ vp  
ꢄ
Víi aij (i=1..n; j=1..m), vk (k=1..q; k=1.p) lµ c¸c trÞ v« h−íng.  
KÝch th−íc cña ma trËn thÓ hiÖn b»ng kÝ hiÖu (m  
vµ n cét.  
kÝch th−íc cña vector vr lµ (1xp), cña vc lµ (qx1), cña v« h−íng lµ (1x1).  
xn), nghÜa lµ cã m hµng  
§é lín hay chiÒu dµi cña vector vr lµ p, cña vc lµ q.  
ꢄ
ꢄ
꣎ꢨꢄꢙꢔꢅꢄꢁꢕꢺꢆꢄꢁ꣌ꢄꢿ꣖ꢏꢄꢢꢯꢄꢀꢂꢐꢄꢖ꣘ꢕꢄꢜꢂꢄꢁꢃꢝꢆꢄꢿ꣚ꢕꢄꢵꢕꢺꢆꢄꢙꢔꢅꢄꢢꢒꢙꢁꢅꢃꢄꢔꢂꢏꢄꢢꢥꢄꢔꢦꢧꢆꢖꢛꢄ  
2.1.1NhËp gi¸ trÞ cho ma trËn.  
ꢄ
ꢄ
ꢄ
ꢄ
ꢄ
ꢄ
ꢴ꣆ꢄ꣼ꢄꢙꢼꢙꢔꢄꢢꢭꢅꢄꢵ꣐ꢄꢗꢕꢺꢐꢄꢙꢔꢅꢄꢙꢼꢙꢄꢬꢕ꣊ꢆꢄꢜꢂꢄꢁꢃꢝꢆꢛꢄ  
ꢄ
ꢄ
ꢄ
ꢄ
꣑꣍ꢕꢺꢁꢄꢑꣂꢄꢁꢃ꣔ꢙꢄꢁꢕ꣊ꢓꢄꢙꢼꢙꢄꢓꢔꢮꢆꢄꢁꣃꢄꢙꢰꢂꢄꢜꢂꢄꢁꢃꢝꢆꢛꢄ  
꣑꣎꣘ꢙꢄꢵ꣐ꢄꢗꢕꢺꢐꢄꢁ꣌ꢄꢜꢲꢁꢄꢎꢕꢗꢒꢄꢵ꣐ꢄꢗꢕꢺꢐꢛꢄ  
꣑ꢳꣃꢄꢵꢶꢆꢖꢄꢁꢅꢼꢆꢄꢁꣃꢄꢞꢡꢠꢛꢄ  
꣑꣦ꢭꢅꢄꢵ꣐ꢄꢗꢕꢺꢐꢄꢁꢃ꣔ꢙꢄꢁꢕ꣊ꢓꢄꢁ꣌ꢄꢬꢭꢆꢄꢓꢔꢻꢜꢛꢄ  
꤂ꢐꢕꢄꢁ꣋ꢙꢄꢿꢽꢆꢔꢄꢆꢖꢔꣳꢂꢄꢜꢂꢄꢁꢃꢝꢆꢄꢁꢃꢅꢆꢖꢄꢫꢂꢁꢗꢂꢬꢡꢄꢄ  
Tªn_matrËn=[a11,a12,a13; a21 a22,a23; a31 a32 a33];  
ꢄ
ꢄ
ꢄ
ꢄ
꣑꣭ꣂꢆ꣸ꢜꢂꢁꢃꢝꢆꢄꢿ꣏ꢁꢄꢁꢔꢒꢅꢄꣁꢐꢕꢄꢙꢼꢙꢔꢄꢁꣂꢆꢄꢬꢕ꣊ꢆꢛꢄ  
꣑ꣅꢲꢕꢄꢵꢐꢆꢖꢄꢁꢃꢅꢆꢖꢄꢆꢖꢅ꣏ꢙꢄꢢꢐꢥꢆꢖꢄ[ ] ꢿꢨꢄꢿꢽꢆꢔꢄꢆꢖꢔꣳꢂꢄꢙꢼꢙꢄꢓꢔꢮꢆꢄꢁꣃꢛꢄ  
꣑꣠꣧ꢐꢄꢙꢔ꣧ꢜꢄꢓꢔ꣟ꢏꢄꢞ꤃ꢠꢄꢵꣲꢆꢖꢄꢿꢨꢄꢓꢔ꣖ꢆꢄꢙꢼꢙꢔꢄꢙꢼꢙꢄꢔꢭꢆꢖꢛꢄꢴꢼꢙꢄꢖꢕꢼꢄꢁꢃꢽꢄꢁꢃꢅꢆꢖꢄꢔꢭꢆꢖꢄ  
ꢿꢦꢸꢙꢄꢓꢔ꣖ꢆꢄꢙꢼꢙꢔꢄꢬꣀꢕꢄꢵ꣧ꢐꢄꢓꢔ꣟ꢏꢄꢞꢘꢠꢘꢄꢔꢅ꣏ꢙꢄꢵ꣧ꢐꢄꢙꢼꢙꢔꢄꢞꢀꢓꢂꢙꢒꢄꢬꢂꢃꢠꢛꢄ  
ꢄ
ꢫꢂꢄꢁꢃꢝꢆꢄꢃ꣝ꢆꢖꢄꢿꢦꢸꢙꢄꢑꢻꢄꢔꢕꢺꢐꢄꢗꢭꢄ[ ].ꢄ  
a. LiÖt kª trùc tiÕp.  
ꢄ ꣍ꢭꢄꢙꢼꢙꢔꢄꢿꢽꢆꢔꢄꢆꢖꢔꣳꢂꢄꢿꢤꢆꢄꢖꢕ꣈ꢆꢄꢆꢔ꣧ꢁꢛꢄꢴꢼꢙꢄꢓꢔꢮꢆꢄꢁꣃꢄꢿꢦꢸꢙꢄꢗꢕꢺꢁꢄꢑꣂꢄꢁꢃꢅꢆꢖꢄꢵ꣧ꢐꢄꢆꢖꢅ꣏ꢙꢄ  
ꢢꢐꢥꢆꢖꢛꢄ  
>>c=5.8978;  
%v« h−íng  
%vector hµng  
%vector cét  
>>b=[1.2 -2 3.01 4+i*3.234];  
>>e=[-2;3;i];  
>>A=[1 0 0; 0 1 0; 0 0 1];  
%ma trËn 3x3  
ꢄ
ꢴ꣆ꢄꢁꢔꢨꢄꢟꢐꢩꢆꢖꢄꢵꣶꢆꢖꢄꢿꢨꢄꢓꢔ꣖ꢆꢄꢬꢕꢺꢁꢄꢁ꣌ꢆꢖꢄꢔꢭꢆꢖꢄꢙꢰꢂꢄꢜꢂꢄꢁꢃꢝꢆꢛꢄ  
>>e=[-2  
3
Page 13 of 62  
i];  
>>A=[1 0 0  
0 1 0  
0 0 1];  
ꢄ
ꢴ꣆ꢄꢁꢔꢨꢄꢿꢽꢆꢔꢄꢆꢖꢔꣳꢂꢄꢜꢂꢄꢁꢃꢝꢆꢄꢁ꣌ꢄꢙꢼꢙꢄꢓꢔꢮꢆꢄꢁꣃꢄꢑꢔꢼꢙꢛꢄ  
>>a=[1 2.4 -4.7];  
>>b=[1.3 -34.67 a];  
>>A=[a;a];  
Ma trËn b cã gi¸ trÞ lµ [1.3 -34.67 1 2.4 -4.7];  
Ma trËn A cã gi¸ trÞ lµ[1 2.4 -4.7;1 2.4 -4.7];  
꣭ꢃꢐꢏꢄꢙꢝꢓꢄꢙꢼꢙꢄꢓꢔꢮꢆꢄꢁꣃꢄꢙꢰꢂꢄꢜꢂꢄꢁꢃꢝꢆꢄꢬ꣰ꢆꢖꢄꢙꢼꢙꢔꢄꢀꣃꢄꢵꢶꢆꢖꢄꢙꢔ꣮ꢄꢀꢩꢄꢙꢰꢂꢄꢆ꣆ꢡꢄ  
>>s=a(3); %s cã gi¸ trÞ lµ -4.7  
>>a(1)=3.14; %phÇn tö thø nhÊt cña a thay ®æi tõ 1 thµnh 3.14  
ꢴ꣇ꢆꢖꢄꢙ꣆ꢄꢁꢔꢨꢄꢜꣀꢄꢃꢲꢆꢖꢄꢜꢂꢄꢁꢃꢝꢆꢄꢬ꣰ꢆꢖꢄꢙꢼꢙꢔꢄꢁꢔꣂꢜꢄꢙꢔꢅꢄꢆ꣆ꢄꢓꢔ꣟ꢆꢄꢁꣃꢄꢜꢧꢕꢛꢄꢴꢔꢅꢄꢙ꣖ꢐꢄꢗꢺꢆꢔꢄ  
ꢄ
ꢄ
ꢀꢂꢐꢡꢄ >>a(4)=-1.98;  
ꢜꢂꢄꢁꢃꢝꢆꢄꢂꢄꢗꢹꢙꢄꢆꢭꢏꢄꢀ꣨ꢄꢙ꣆ꢄ꣼ꢄꢓꢔꢮꢆꢄꢁꣃꢄa=[1 2.4 -4.7 -1.98]꤃ꢄ  
ꢄ
ꢙꣶꢆꢄꢢꢧꢕꢄꢙ꣖ꢐꢄꢗꢺꢆꢔꢡꢄ  
ꢄ >>a(7)=2.07꤃ꢄ  
ꢁꢔꢷꢄꢜꢂꢄꢁꢃꢝꢆꢄꢂꢄꢀ꣨ꢄꢙ꣆ꢄꣾꢄꢓꢔꢮꢆꢄꢁꣃꢘꢄꢙꢼꢙꢄꢓꢔꢮꢆꢄꢁꣃꢄꢂꢞꢇꢠꢘꢄꢂꢞꣽꢠꢄꢀ꣨ꢄꢁ꣔ꢄꢿꢲꢆꢖꢄꢆꢔꢝꢆꢄꢖꢕꢼꢄꢁꢃꢽꢄꢗꢭꢄꢋꢛꢄ  
ꢄ
b. §äc d÷ liÖu tõ 1 file d÷ liÖu.  
ꢄ
ꢄ
ꢳꣃꢄꢵꢶꢆꢖꢄꢔꢭꢜꢄloadꢄꢿꢨꢄꢁ꣈ꢕꢄꢵ꣐ꢄꢗꢕꢺꢐꢄꢁ꣌ꢄꢙꢼꢙꢄꢬꢕ꣊ꢆꢄꢿꣵꢄꢿꢦꢸꢙꢄꢁ꣚ꢅꢄꢁꢃꢦꢧꢙꢄꢿ꣆ꢛꢄ  
c. Sö dông to¸n tö (:). xem '2.4.4 C¸c thao t¸c ma trËn'  
ꢄ
d. Vµo sè liÖu trùc tiÕp tõ bµn phÝm.  
ꢄ
ꢄ
ꢳꣃꢄꢵꢶꢆꢖꢄꢙꢹꢄꢓꢔꢼꢓꢄꢆꢔꢝꢓꢄꢵ꣐ꢄꢗꢕꢺꢐꢄꢁ꣌ꢄꢬꢭꢆꢄꢓꢔꢻꢜꢄꢆꢔꢦꢄꢀꢂꢐꢡꢄ  
>>A=input('NhËp d÷ liÖu cho A: ');  
꣗ꢔꢕꢄꢁꢔ꣔ꢙꢄꢔꢕꢺꢆꢄꢄꢗꢺꢆꢔꢄꢆꢭꢏꢄꢜꢼꢏꢄꢀ꣨ꢄꢔꢕꢨꢆꢄꢁꢔꢽꢄꢟ꣖ꢐꢄꢑꢻꢄꢁ꣔ꢄꣷꣅꢔꢝꢓꢄꢵ꣐ꢄꢗꢕꢺꢐꢄꢙꢰꢂꢄ꣣ꢡꢄꣷꢄꢢꢭꢄꢿꢸꢕꢄ  
ꢆꢖꢦ꣞ꢕꢄꢀꣃꢄꢵꢶꢆꢖꢄꢆꢔꢝꢓꢄꢀꢩꢄꢗꢕꢺꢐꢄꢢꢭꢅꢛꢄꣅꢲꢕꢄꢵꢐꢆꢖꢄꢆꢔꢝꢓꢄꢗꢕꢺꢐꢄꢁꢦꢤꢆꢖꢄꢁ꣔ꢄꢙ꣖ꢐꢄꢗꢺꢆꢔꢄꢁꢔꢥꢆꢖꢄꢁꢔꢦ꣞ꢆꢖꢄ  
ꢙꢔꢅꢄꢜꢂꢄꢁꢃꢝꢆꢘꢄꢢꢭꢄꢙꢒꢗꢗꢛꢄ  
ꢄ
2.1.2 HiÓn thÞ ma trËn.ꢄ  
ꢄ
ꢫꢂꢁꢗꢂꢬꢄꢙꢔꢅꢄꢓꢔꢣꢓꢄꢔꢕꢨꢆꢄꢁꢔꢽꢄꢜꢂꢄꢁꢃꢝꢆꢄꢀꢩꢄꢁꢔꢒꢅꢄꢆꢔꢕꢯꢐꢄꢵ꣚ꢆꢖꢄꢑꢔꢼꢙꢄꢆꢔꢂꢐꢄꢢꢭꢄꢙꢐꢆꢖꢄꢙ꣧ꢓꢄ  
ꢆꢔꢕꢯꢐꢄꢔꢭꢜꢄꢿꢨꢄꢟꣃꢄꢗ꣓ꢄꢢꢕꢺꢙꢄꢔꢕꢨꢆꢄꢁꢔꢽꢄꢜꢂꢄꢁꢃꢝꢆꢄꢀꢩꢛꢄ  
ꢴꢼꢙꢄꢵ꣚ꢆꢖꢄꢁꢔ꣕ꢙꢄꢞꢎꢅꢃꢜꢂꢁꢠꢄꢬꢕꢨꢐꢄꢵꢕ꣛ꢆꢄꢀꢩꢡꢄ  
ꢄ
format short  
format long  
Sè dÊu phÈy cè ®Þnh, víi 5 ch÷ sè cã nghÜa sau dÊu ph¶y  
Sè dÊu phÈy cè ®Þnh, víi 15 ch÷ sè cã nghÜa sau dÊu ph¶y  
format short e Sè dÊu phÈy ®éng, víi 5 ch÷ sè cã nghÜa sau dÊu ph¶y  
format long e Sè dÊu phÈy ®éng, víi 15 ch÷ sè cã nghÜa sau dÊu ph¶y  
format short g Lùa chän tèt nhÊt phÈy cè ®Þnh hay ®éng víi 5 ch÷ sè cã  
nghÜa sau dÊu ph¶y  
format long g  
Lùa chän tèt nhÊt phÈy cè ®Þnh hay ®éng víi 15 ch÷ sè cã  
nghÜa sau dÊu ph¶y  
BiÓu thÞ sè thùc vÒ sè h÷u tØ gÇn nhÊt.  
format rat  
Page 14 of 62  
disp  
HiÓn thÞ néi dung cña biÕn.  
fprintf  
sprintf  
Hµm cho phÐp hiÓn thÞ theo c¸c khu«n d¹ng chØ ®Þnh.  
Hµm tr¶ vÒ x©u kÝ tù in theo c¸c khu«n d¹ng chØ ®Þnh.  
ꢄ
2.2ꢄC¸c ma trËn ®Æc biÖt.  
ꢄ
ꢬꢕꢺꢁꢛꢄ  
ꢫꢂꢁꢗꢂꢬꢄꢙꢐꢆꢖꢄꢙ꣧ꢓꢄꢆꢔꢕꢯꢐꢄꢔꢭꢜꢄꢗꢦꢐꢄꢙꢼꢙꢄꢔ꣰ꢆꢖꢄꢙꢔꢅꢄꢓꢔꢣꢓꢄꢀꢕꢆꢔꢄꢜꢂꢄꢁꢃꢝꢆꢄꢙ꣆ꢄꢵ꣚ꢆꢖꢄꢿ꣏ꢙꢄ  
Gi¶i thÝch  
Ma trËn mxn  
rezos(m,n)  
ones(m,n)  
eye(m,n)  
Ma trËn vu«ng nxn  
rezos(n)  
ma trËn kh«ng  
ma trËn ®¬n vÞ  
ones(n)  
ma tr©n ®¬n vÞ ®−êng chÐo më réng  
ma trËn ngÉu nhiªn ph©n bè ®Òu  
eye(n)  
rand(m,n)  
randn(m,n)  
rand(n)  
∈
(0,1)  
ma trËn ngÉu nhiªn ph©n bè chuÈn  
thuéc (-1,1) .  
randn(n)  
t¹o vect¬ n phÇn tö c¸ch ®Òu tõ a ®Õn b, m¨c ®Þnh n=100;  
vect¬ n phÇn tö hµm mò thËp ph©n tõ 10a ®Õn 10b, mÆc ®Þnh n=50;  
sinh ma trËn ma ph−¬ng cÊp n  
linspace(a,b,n)  
logspace(a,b,n)  
magic(n)  
pascal(n)  
sinh ma trËn Pascal  
compan(p)  
sinh ma trËn cã ®a thøc ®Æc tr−ng p  
[out1,out2,...] = gallery(matname, param1, param2, ...)% Xem help gallery.  
hadamard, hankel,hilb, invhilb, kron, roserr, toeplitz,vander, wilkinson.  
ꢄ
2.3 C¸c phÐp to¸n v« h−íng  
ꢄ
2.3.1BiÓu thøc sè häc cho v« h−íng.  
PhÐp to¸n  
Céng  
BiÓu thøc sè häc  
Matlab  
a+b  
a-b  
a+b  
a-b  
a.b  
a/b  
b:a  
ab  
Trõ  
Nh©n  
a*b  
Chia  
a/b  
Chia tr¸i  
Luü thõa  
a\b  
a^b  
2.3.2 Thø tù −u tiªn c¸c to¸n tö.  
ꢄ ꣗ꢔꢕꢄꢜꢲꢁꢄꢀꢩꢄꢁꢅꢼꢆꢄꢁꣃꢄꢑ꣊ꢁꢄꢔꢸꢓꢄꢁꢃꢅꢆꢖꢄꢬꢕꢨꢐꢄꢁꢔ꣕ꢙꢘꢄꢿꢕꢯꢐꢄꣁꢐꢂꢆꢄꢁꢃ꣘ꢆꢖꢄꢆꢔ꣧ꢁꢄꢗꢭꢄꢓꢔ꣈ꢕꢄꢬꢕ꣊ꢁꢄꢁꢔ꣕ꢄ  
ꢁ꣔ꢄꢦꢐꢄꢁꢕꣂꢆꢄꢙꢼꢙꢄꢁꢅꢼꢆꢄꢁꣃꢄꢁꢃꢅꢆꢖꢄꢬꢕꢨꢐꢄꢁꢔ꣕ꢙꢛꢄ  
Thø tù −u tiªn  
To¸n tö  
1
NgoÆc ®¬n ().  
2
3
4
5
Gäi hµm.  
Luü thõa.  
Nh©n, chia tõ tr¸i qua ph¶i.  
Céng, trõ tõ tr¸i qua ph¶i.  
ꢄ
2.3.3 C¸c phÐp to¸n vector.  
Page 15 of 62  
PhÐp to¸n  
C«ng thøc  
a+b  
a-b  
Matlab  
a+b  
a-b  
Céng  
Trõ  
Nh©n m¶ng  
a.b  
a/b  
a.*b  
a./b  
Chia ph¶i m¶ng  
Chia tr¸i m¶ng  
Luü thõa m¶ng  
a\b  
ab  
a.\b  
a.^b  
ꢄ
ꢴꢼꢙꢄꢓꢔꢣꢓꢄꢁꢅꢼꢆꢄꢁꢃꣂꢆꢄꢼꢓꢄꢵꢶꢆꢖꢄꢙꢔꢅꢄꢙꢼꢙꢄꢜꢂꢄꢁꢃꢝꢆꢄꢙꣲꢆꢖꢄꢑꢻꢙꢔꢄꢁꢔꢦꢧꢙꢘꢄꢔꢅ꣏ꢙꢄꢜꢂꢄꢁꢃꢝꢆꢄꢢꢧꢕꢄ  
ꢢꢥꢄꢔꢦꢧꢆꢖꢛꢄ  
꣦ꢻꢄꢵꢶꢡ  
A=[1 2 3;7 -6 5;3 0 -1];  
B=[1.1 2.4 5.6;-1 -3.6 8;-3 2 -1];  
a=[1 2 7 -2];  
b=[1 0 2 2];  
c=2;  
A.*B  
ans =  
a.*b  
ans =  
1.1000 4.8000 16.8000  
-7.0000 21.6000 40.0000  
1
0
14  
-4  
a./b  
Warning: Divide by zero.  
ans =  
-9.0000  
0
1.0000  
ꢄ
A.^B  
1.0000 Inf  
b.\a  
3.5000  
0.2857  
49  
-1.0000  
-1.0000  
ꢄ
ans =  
1.0e+005 *  
ans =  
1.0000  
a.^b  
0
1
0.0000 0.0001  
0.0000 0.0000 + 0.0000i  
0.0000  
0.0047  
3.9063  
-0.0000  
ans =  
1
0
4
ꢄ
A.^c  
ans =  
a+b  
ans =  
2
b.^c  
1
49  
4
36  
0
9
25  
1
2
9
0
9
ans =  
1
A+B  
0
4
4
ans =  
2.1000  
8.6000  
6.0000  
13.0000  
c.^a ꢄꢄ  
ans =  
4.4000  
2.0000 4.0000 128.0000  
0.2500  
a+c  
-9.6000  
2.0000  
ans =  
0
-
3
4
9
0
2.0000  
B+c  
ans =  
c./a  
ans =  
2.0000 1.0000 0.2857 -  
1.0000ꢄ  
3.1000  
7.6000  
1.0000  
10.0000  
-1.0000  
1.0000  
4.4000  
-1.6000  
4.0000  
ꢄ
2.4 C¸c phÐp to¸n ma trËnꢄ  
Page 16 of 62  
2.4.1 C¸c phÐp to¸n ma trËn.ꢄ  
ꢄ
PhÐp to¸n  
ChuyÓn vÞ  
C«ng thøc  
Bnm=ATmnvíiB={bÞj=aji}nm  
Bnm=ATmnvíi B={bÞj=a*ji}nm  
Matlab  
A.'  
ChuyÓn vÞ vµ lÊy liªn hîp phøc  
Céng, trõ  
A'  
{aij bij}mn=Amn Bmn  
n
A B  
A*B  
Nh©n  
{
c = a b }mp=Amn*Bnp  
∑
ij  
ik kj  
k=  
1
Chia ph¶i  
Chia tr¸i  
Luü thõa ma trËn  
Cnp=Amn/Bpm  
Cqn=Apq\Bpn  
A/B  
A\B  
A^p  
Ap , (p-nguyªn)  
nn  
A-1  
Ma trËn nghÞch ®¶o  
inv(A), A^(-1)  
pinv(A)  
nn  
Tnm=A-1  
Ma trËn tùa nghÞc ®¶o  
TÝch v« h−íng cña hai ma trËn  
mn  
sum(A.*B)  
tÝch v« h−íng cña  
c¸c cét  
m
Amn Bmn = c = a b  
∑
i
ij ij  
j=1  
1n  
ꢄ
꣗ꢻꢄꢔꢕꢺꢐꢄꢢ꣊ꢄꢓꢔ꣈ꢕꢄꢗꢭꢄꢑ꣊ꢁꢄꣁꢐ꣈ꢄꢙꢰꢂꢄꢓꢔꢣꢓꢄꢁꢅꢼꢆꢘꢄꢙꢔ꣮ꢄꢀꢩꢄꢜꢟꢆꢄꢗꢭꢄꢙꢔꢅꢄꢜꢂꢄꢁꢃꢝꢆꢄꢬ꣧ꢁꢄꢑꢷꢘꢄꢙꢔ꣮ꢄꢀꢩꢄꢄ  
ꢟꢆꢄꢗꢭꢄꢙꢔ꣮ꢄꢙꢔꢅꢄꢜꢂꢄꢁꢃꢝꢆꢄꢢꢐꢥꢆꢖ꤃ꢄꢁꢃꢅꢆꢖꢄꢙꢼꢙꢄꢙꢥꢆꢖꢄꢁꢔ꣕ꢙꢄꢙ꣆ꢄꢙꣲꢆꢖꢄꢙꢔ꣮ꢄꢀꢩꢄꢗꢭꢄꢬ꣋ꢁꢄꢬꢐꢲꢙꢄꢓꢔ꣈ꢕꢄ  
ꢆ
ꢁꢔꢅ꣈ꢄꢜꣵꢆꢄꢜꢧꢕꢄꢼꢓꢄꢵꢶꢆꢖꢄꢿꢦꢸꢙꢄꢓꢔꢣꢓꢄꢁꢅꢼꢆꢛꢄ  
꣦ꢻꢄꢵꢶꢡꢄ  
A=[1 2 3; 4 -5 6];  
D=[i 2 -6;5 4 2-i];  
E=[1;i;-3];  
B=[1 -1;-1 2];  
C=[1 -1;2 -1;-1 3];  
A+D  
ans =  
D.'  
ans =  
1.0000 + 1.0000i 4.0000 -3.0000  
9.0000  
0 + 1.0000i 5.0000  
2.0000  
-1.0000 8.0000 - 1.0000i  
4.0000  
2.0000 -  
-6.0000  
1.0000i  
D'  
A*E  
ans =  
B*A  
ans =  
ans =  
-3 7  
7 -12  
-3  
9
-8.0000 +  
2.0000i  
-14.0000 - 5.0000i  
0 - 1.0000i 5.0000  
2.0000  
-6.0000  
4.0000  
2.0000 - 1.0000i  
inv(B)  
ans =  
B^3  
pinv(A)  
ans =  
ans =  
5 -8  
-8 13  
0.0310 0.0471  
2
1
1
0.2291 -0.1006  
0.1702 0.0514  
1
B*inv(B)  
ans =  
A*pinv(A)  
ans =  
1.0000 -0.0000  
pinv(E)  
ans =  
0.0909 0 - 0.0909i -0.2727  
1
0
1
0.0000 1.0000  
0
Page 17 of 62  
A\D  
ans =  
A/D  
ans =  
0
-0.5556 + 0.2222i 0.0000  
0
0
-0.3023 - 0.0284i 0.3874 +  
0.0562i  
-1.1033 - 0.0920i 0.1669 + 0.0984i  
-1.5556  
+ 0.1111i  
0.3704 + 0.1852i 0.6667  
-0.9630 - 0.0741i  
ꢄ
2.4.2 C¸c thao t¸c ma trËn.ꢄ  
ꢄ
ꢄ
ꢫꢂꢁꢗꢂꢬꢄꢙꢐꢆꢖꢄꢙ꣧ꢓꢄꢆꢔꢕꢯꢐꢄꢔꢭꢜꢄꢙꢔꢅꢄꢓꢔꢣꢓꢄꢁꢔꢂꢅꢄꢁꢼꢙꢄꢁꢃꣂꢆꢄꢜꢂꢄꢁꢃꢝꢆꢡꢄ  
Hµm  
Rot90(A,n)  
Thao t¸c  
C¸c phÇn tö cña ma trËn A ®−îc quay mét gãc 90o ng−îc chiÒu  
kim ®ång hå. NÕu cã thªm tham sè n sÏ quay n lÇn.  
§¶o c¸c phÇn tö cña ma trËn A tõ tr¸i sang ph¶i.  
§¶o c¸c phÇn tö cña ma trËn A tõ trªn xuèng d−íi.  
fliplr(A)  
flipud(A)  
Bpq=Reshape(Amn)  
sort(A)  
§Þnh d¹ng l¹i ma trËn, yªu cÇu pxq=mxn.  
Tr¶ vÒ ma trËn víi c¸c cét ®−îc s¾p xÕp tõ nhá ®Õn lín. Sè phøc ®−îc  
xÐt b»ng modul cña nã.  
ChÝch c¸c phÇn tö cña ma trËn.  
LÊy c¸c phÇn tö cña ma trËn trªn ®−êng chÐo chÝnh, tr¶ vÒ vector cét  
diag(A)  
LÊy c¸c phÇn tö cña ma trËn trªn ®−êng chÐo tuú chän k:  
k=0 chän ®−êng chÐo chÝnh.  
diag(A,k)  
k>0 chän ®−êng chÐo thø k ë phÝa trªn ®−êng chÐo chÝnh.  
k<0 chän ®−êng chÐo thø k ë phÝa d−íi ®−êng chÐo chÝnh.  
Lµ ma trËn cïng cì víi A. LÊy c¸c phÇn tö trªn ®−êng chÐo chÝnh thø vµ  
phÝa trªn ®−êng chÐo chÝnh. c¸c vÞ trÝ kh¸c b»ng kh«ng.  
LÊy c¸c phÇn tö ë phÝa trªn ®−êng chÐo thø k vµ phÝa trªn.  
triu(A)  
triu(A,k)  
tril(A)  
Lµ ma trËn cïng cì víi A. LÊy c¸c phÇn tö trªn ®−êng chÐo chÝnh vµ n»m  
d−íi ®−êng chÐo chÝnh. c¸c vÞ trÝ kh¸c b»ng kh«ng.  
LÊy c¸c phÇn tö ë phÝa d−íi ®−êng chÐo chÝnh thø k vµ phÝa d−íi.  
tril(A,k)  
Thao t¸c c¸c phÇn tö cña ma trËn.  
Thao t¸c phÇn tö aij cña ma trËn A.  
[m,n]=size(A).  
A(i,j)  
i=1,m; j=1,n  
:
To¸n tö  
T¹o vector hµng x=a:d:b, t¹o cÊp sè p phÇn tö (p=[(b-a)/d], p lµ  
phÇn nguyªn) x(i)=a+i*d víi i=1,2,..,p.  
b>a nÕu d>0.  
x=a:d:b  
y=a:b  
b<a nÕu d<0.  
nÕu d=1 viÕt y=a:b;  
A(:,k)  
A(l,:)  
Thao t¸c cét thø k cña ma trËn.  
Thao t¸c hµng thø l cña ma trËn.  
Nèi c¸c cét cña ma trËn A thµnh vector cét.  
To¸n tö  
A(:)  
end  
A(end)  
A(end,k)  
A(l,end)  
Thao t¸c phÇn tö cuèi cïng cña ma trËn.  
Thao t¸c phÇn tö cuèi cïng cña cét thø k.  
Thao t¸c phÇn tö cuèi cïng cña hµng thø l.  
Page 18 of 62  
A(p:q,k)  
A(l,p:q)  
A(p:q,l:k)  
x(v)  
Thao t¸c phÇn tö tõ vÞ trÝ p ®Õn vÞ trÝ q cña cét thø k.  
Thao t¸c phÇn tö tõ vÞ trÝ p ®Õn vÞ trÝ q cña hµng thø l.  
Thao t¸c ma trËn con .  
Thao t¸c c¸c phÇn tö ®−îc cho trong vector chØ sè v.  
Thao t¸c c¸c cét chØ ®Þnh trong vector chØ sè v.  
Thao t¸c c¸c hµng chØ ®Þnh trong vector chØ sè v.  
A(:,v)  
A(v,:)  
ꢄ
ꢄ
ꢄ
ꢴꢼꢙꢄꢔꢭꢜꢄꢗ꣘ꢙꢄꢜꢂꢄꢁꢃꢝꢆꢡꢄ  
Tr¶ vÒ chØ sè c¸c phÇn tö cña A tho¶ m·n biÓu thøc  
logic exprA cña A.  
Tr¶ vÒ vÞ trÝ c¸c phÇn tö cña A kh¸c kh«ng.  
find(exprA)  
find(A)  
y=A(exprA)  
Tr¶ vÒ c¸c phÇn tö cña A tho¶ m·n biÓu thøc.  
y=A(A>0) Tr¶ vÒ c¸c phÇn tö cña A lín h¬n kh«ng.  
y=A(fimag(A)==0) Tr¶ vÒ c¸c phÇn tö cña A lµ sè thùc.  
y=A(a<=A&A<b) Tr¶ vÒ c¸c phÇn tö cña A thuéc ®o¹n [a,b).  
꣦ꢻꢄꢵꢶꢄꢌꢡꢄ  
A=[1 3 5 7; 2 4 6 8;9 8 7 6];  
B=rot90(A);  
C=rot90(A,2);  
E=fliplr(A);  
D=flipud(A);  
F=reshape(A,6,2);  
a=A(:);  
b=A(:,2);  
c=A(3,:);  
d=A(1:3,2);  
e=A(end);  
f=A(1:2,2:3);  
v=[1 3 5 7];  
g=A(v);  
A =  
8
6
6
7
4
2
2
9
3
4
8
5
6
7
1
3
5
7
8
9
2
4
6
8
D =  
9
8
7
6
9
2
1
8
4
7
6
8
B =  
6
7
5
3
1
8
6
4
6
7
3
5
7
F =  
8
1
5
2
9
2
9
3
4
8
6
7
C =  
7
8
6
6
8
7
7
6
5
8
4
3
9
2
1
7
8
6
b =  
E =  
a =  
3
4
7
5
3
1
1
Page 19 of 62  
8
9
8
6
v =  
c =  
e =  
f =  
1
3
9
5
4
7
8
7
6
g =  
d =  
1
5
3
3
4
5
4
6
꣦ꢻꢄꢵꢶꢄꢊꢡꢄ  
A=[1+i 1-3*i 12; 1 2.3 -4;2 -1-2.1*i -1; 1 2 4];  
rA=A(find(imag(A)==0));  
pA=A(find(A>0));  
nA=A(find(A<0));  
aAb=A(find(1<A&A<4));  
sA=sort(A);  
rA =  
1.0000  
-1.0000 - 2.1000i  
-4.0000  
-1.0000  
2.0000  
1.0000  
2.3000  
2.0000  
12.0000  
-4.0000  
-1.0000  
4.0000  
aAb =  
2.0000  
2.3000  
2.0000  
sA =  
pA =  
1.0000  
1.0000  
2.0000  
2.3000  
-1.0000  
4.0000  
1.0000 + 1.0000i  
1.0000  
2.0000  
1.0000 + 1.0000i -1.0000 - 2.1000i -4.0000  
2.0000 1.0000 - 3.0000i 12.0000  
1.0000  
ꢄ
ꢄ
ꢄ
ꢄ
ꢄ
ꢄ
1.0000 - 3.0000i  
2.3000  
2.0000  
12.0000  
4.0000  
nA =  
꣦ꢻꢄꢵꢶꢄ꣹ꢡꢄ  
A=[1+i 1-3*i 12; 1 2.3 -4;2 -1-2.1*i -1; 1 2 4];  
tuA=triu(A)  
tuA1=triu(A,2)  
tuA1=triu(A,-1)  
tlA=tril(A)  
tlA1=tril(A,2)  
tlA1=tril(A,-1)  
tuA =  
1.0000 + 1.0000i 1.0000 - 3.0000i 12.0000  
0
0
0
0
0
0
0
2.3000  
-4.0000  
-1.0000  
0
tuA1 =  
0
0
1.0000 + 1.0000i 1.0000 - 3.0000i 12.0000  
0
tuA1 =  
0
0
1.0000  
2.3000  
-1.0000 - 2.1000i -1.0000  
4.0000  
-4.0000  
0
0
0
0
12  
0
0
0
Page 20 of 62  
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